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White paper: http://www.researchgate.net/profile/Patrice_Poyet August 24th, 2017, with DOI: 10.13140/RG.2.2.35677.92647

Calculating visual binaries’ orbits: should it ever be fully automated ?

Discussing 13 stars with 8 first-time orbits

Patrice Poyet

patricepoyet at yahoo.com

Observatoire du Jas de Tardivy à Caussols

Mosser* (43.748825 – 6.939806) & Cardoen (43.747978 – 6.936663) telescopes

Postal : 5, avenue Corniche d'Azur, 06100, Nice, France

Abstract

Many methods have been devised to compute orbits and they fall into two main categories:

geometric approaches and analytic ones. Geometric methods deliver orbital elements less

certain but are useful when the length of the arc observed so far is just more or less significant

of the entire trajectory to come. More and more often though, analytical methods are being used

to kind of automatically compute orbits for double stars having only travelled a small part of

their entire orbit. This is undoubtedly due to the widespread availability of computers and of

specialized computer programmes enabling the straightforward calculation of orbits out of a set

of observations without requiring from the operator a grasp of the binaries that he models finally

just in an abstract and distant way. One should remember that Couteau (1978) considered the

computation of orbits as a craftsman job for various reasons and Worley (1990) questioned the

usefulness of some orbits that appear more like computing efforts than good astronomical sense.

This paper will not be a quarrel of the ancients and the moderns, but will remind traditional

ways of computing orbits so that it will remain, for those who wish, a rewarding craftsman job

of playing with a wire and two pins, keeping a close contact with the specificity of the binaries

studied and thereof of the observations made, to first find with their best astronomical sense, the

foci of the apparent ellipse. The calculation of the orbits and ephemeris of 13 stars will serve as

example, and for eight of them first-time original orbits solutions are proposed, computed by

means of the geometrical method confirming its appropriateness and reliability.

I. Introduction

Many methods have been devised to compute orbits since Savary (1827), Herschel (1833, 1850),

Klinkerfues (1855) and they fall into two main categories: geometric approaches (van de Kamp,

1947), (Baize, 1954) and analytic ones (Van den Bos, 1962), (Arend, 1941, 1961, 1970),

(Dommanget, 1959), (Binnendijke, 1960) to quote a few and new methods keep emerging every

decade or so, as if the tens of methods available were neither satisfactory nor sufficient. This is the

proof that no method in itself offers “the” solution, each having strengths and weaknesses, but as

reminded by Couteau (1978), geometric methods deliver orbital elements less certain but are useful

when the length of the arc observed so far is just more or less significant of the entire trajectory to

come. Of the more than 100.000 binaries in the WDS, less than 2% have an orbit and just a quarter

of them have an orbit grade better than 3 and only a few of the latter have definitive orbital

parameters. It means that most of the work remaining to be done deals with stars having incomplete

trajectories over a limited portion of their trajectories and geometric methods remain in that case a

good if not the best choice. Nevertheless, the current trend, with the widespread availability of

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White paper: http://www.researchgate.net/profile/Patrice_Poyet August 24th, 2017, with DOI: 10.13140/RG.2.2.35677.92647

powerful PCs, is that more and more often analytical methods are being used to kind of

automatically compute orbits for double stars having only travelled a small part of their entire orbit

(Docobo and Ling, 2003), (Ling 2010), using advanced techniques that sometimes do not even

requires to get a value of the areal constant as for Docobo’s analytical method (1985), where

measures appear to be somehow weighted according to the size of the instrument and observing

means. Curiously enough, this would lead to affect the lowest value to Couteau’s measurements in

Docobo and Ling computations (2003) whereas Couteau is the discoverer himself in the first place

of all the stars studied, with the smallest instrument, i.e. a 50 cm refractor and a filar micrometer.

Recently some authors have devised new methods, for example Branham (2005) proposes an

analytic method based on semi-definite programming that incorporates the areal constant and that

should be insensitive to outliers when appropriate data reduction criterion are being used.

Nevertheless, one should remember that Couteau (1978) considered the computation of orbits as a

craftsman job for various reasons (some reminded in Appendix A) and Worley (1990) questioned

the usefulness of some orbits that appear more like computing efforts than good astronomical

practice or sense.

The geometrical method will be used in this paper to compute the orbits of 13 double stars, 9 of

them entirely new, paying a lot of attention to obtaining a somehow stable areal constant across

sectors, so that the projected law of areas be obeyed (Van de Kamp, 1961), before going any further

into the determination of the orbit elements and ephemeris. Sometimes, the initial ellipse one draws

through the cloud of observations is very far from the final ellipse that satisfies better the law of the

areas (e.g. A 552 in this paper). As a hint, Danjon (1952) reminds that the more the curve (t)

departs from a straight line, the more elliptical the appropriate solution will be, and he elaborates on

the case of A88, a very difficult object, for which the scatter could look nearly circular, whereas the

study of (t) suggests an eccentric ellipse as a final solution. This paper will not be a quarrel of the

ancients and the moderns, but will remind traditional ways of computing orbits so that it will

remain, for those who wish, a rewarding craftsman job of playing with a wire and two pins to first

find with their common sense the foci of the apparent ellipse. It also reminds us the great pedagogic

value of geometric methods for those wishing to start computing orbits, as they compel the operator

to be respectful of the characteristics of the observations and of the phenomenon observed and

represented. Then for each star, we will derive individual stellar masses whenever possible based

either on the trigonometric parallax, the dynamic parallax or both and will introduce “mass

convergence tables” where the absolute magnitudes, individual masses and parallax are determined

by small incremental refinements after a short ten step iteration.

Object ALPHA2K DELTA2K BD ADS HIP mag1 mag2 SpaOrbit trigo b ±e-trigo

RST 3340 00:05:05 -18,36-19,8447

TYC5841 579

1 10,4 10,4G0 NO

HU 616 06:54:01 33,4133,1427 554433145 9,1 9,5F0 NO 2,57 0,61

HU 841 07:40:00 66,0366,518 622837353 9 9G NO 7,54 1,44

A 552 08:44:04 -4,12-03,2454 696442863 7,5 8,5A2 NO 5,69 0,93

A 2554 08:53:09 1,4902,2088 707443676 8 10,2F0 2 14,79 1,02

BU 796 12:17:05 6,3607,2529 850059916 8 8,8F2 NO 7,21 0,45

RST 4502 12:34:09 -5,09-04,3307 61322 8,6 8,6A5 NO 3,76 0,61

A 1097 AB 14:02:00 57,1357,1478 908968554 7,8 8,1F5 3 6,82 0,69

HU 337 19:18:09 17,3617,3924 1230194911 8,6 9A3 NO 4,15 1,18

HO 137 20:40:05 29,4929,4131 14149102033 6,5 11A0 NO 13,47 0,46

BU 838 21:20:09 3,0802,4343 14880105399 7,6 9,6F8 1 14,09 0,99

STF 2822 AB 21:44:01 28,4528,4169 15270107310 4,7 6,1F5 3 44,97 0,43

HU 497 BC 23:17:05 16,5216,4896 16648115002 9,5 10F2III 3 6,74 1,45

Table 1: 13 stars of which 8 are first-time orbits are computed with latest observations

a all are main sequence unless otherwise indicated

b milli-arcsec or mas (SIMBAD)

c average error of 10% of mas up to 100 pc

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White paper: http://www.researchgate.net/profile/Patrice_Poyet August 24th, 2017, with DOI: 10.13140/RG.2.2.35677.92647

The main characteristics of the 13 stars studied in this paper, are summarized in Table1, with their

2000 celestial coordinates, catalogue references, i.e. (BD, ADS and HIP), their apparent magnitudes

and spectral types, existence of a published orbit in the USNO Sixth Catalog of Orbits of Visual

Binary Stars (Hartkopf and Mason, 2017), trigonometric parallax if any available (ESA, 1997), and

error on parallax (Perryman, 2009). Eight of them have no previously known orbits according to the

USNO Sixth Catalog of Orbits of Visual Binary Stars and we provide completely new original orbit

solutions and masses. As stated by Tokovinin (2016) the main reason for publishing here uncertain

first-time orbits remains the need to understand the orbital motion and to plan further observations

for their improvement, not leaving interesting stars neglected.

II. How to properly use the geometric method

Computing an orbit with the geometric method is very straightforward and the reader will refer to

the material published to get a thorough understanding of the underlying theory and practice (van de

Kamp, 1947), (Baize, 1954), (Couteau, 1978). Nevertheless, it can be worthwhile to remind the

successive stages to be performed, insisting on the careful handling of the various steps. The

position angles of the set of observations need to be corrected and brought back to the same

equinox as the celestial pole has drifted over the time span during which the measures have been

made, coordinates () are expressed in degrees, t0 being the epoch chosen for the equinox and t

the epoch of the observation.

= -0,0056 sin . sec (t – t0)

where the secant function sec = 1/cos . One should notice that by choosing t0 as close as possible

to the beginning of the series of measurements, corrections will be small or negligible. One should

also notice that the precession-correction is greater near the celestial pole and it shrinks rapidly once

you are more than 20° from the pole and is obviously null for = 0. For declinations less than about

80° the tiny correction can be most of the time ignored. Position angles become:

t = 0 -

Proper motion also slightly changes the position angles. The correction is written as follows:

= -0,00417 . . sin (t – t0)

where (i.e. the proper motion) is expressed in seconds of time per year for right ascension (RA)

as in here. Again, it is only for binaries having strong proper motions and located close to the pole

that the correction will yield significant values.

So, basically one might reasonably consider that for binaries having a declination lower than 70°,

raw data from the excel sheet might be used, otherwise they can easily be corrected using the

previous formulae. As the four binaries of this study have a < 60°, no correction have been

applied to the measurements.

The next step aims at generating a plot of all the observations, after having transformed the polar

coordinates (, ) into Cartesian or rectangular coordinates, so that they could be easily plotted on a

graph.

With x = cos and y = sin and

conversely = ( x2 + y2 )1/2 and = arctg y/x

The plot obtained can be printed out at different arbitrary scale so that it would offer a convenient

support for the most important part of the job, namely drawing the apparent ellipse with a thread

and two pins. This work is somehow subjective, as the drawing of the ellipse must make it pass

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through the most reliable observations (at least as close as possible) and globally limit the

deviations from most of the remaining points. While drawing the orbit through the set of

observations and checking the areal constant for various sectors, one should remember that is

more important and less error prone than as reminded by Jonckheere (1950), especially for close

binaries measured with a visual micrometer or sometimes just assessed on the Airy disk aspect. Of

course, many factors influence what one can expect of a measurement, 5 to even 10 degrees of

dispersion on on a tight binary with a significant difference of magnitude m between the pair is

less troublesome than a few degrees on an easy couple. Then, one must cope with the vast

heterogeneity of observers, instruments (i.e. mainly refractors and reflectors of all sizes and

configurations), micrometers (e.g. filar, double image, diffraction based, etc.) (Baize, 1949),

(Jonckheere, 1950), (Minois, 1984), webcams or video cams, CCDs (Buil, 1991), (Soulié and

Morlet, 1997), (Salaman and Morlet 2000), (Morlet and Salaman, 2005) or EMCCDs (Gili and

Agati, 2009), speckle (Bonneau, 1993) and now sophisticated cameras, e.g. (Gili et al., 2014) or a

combination of these as basic observations, the older being moreover often the most interesting

ones given the long duration of orbital motions. The tracing of the curves (t) and (t), especially

the latter, as suggested by Couteau (1978) and Dommanget (2000), and their adjustment in order to

contributing to establishing a satisfactory areal constant enables an assessment of the

representativeness of the observations and of their corresponding quality that does not enable an

automated procedure or programme, remaining blind.

The principal star A is not at the centre of the ellipse as we deal with the projection of the real

ellipse onto the sky along the observation line. Once a reasonable ellipse has been achieved, which

is not always obvious, far from it, the next step is to select 5 to 6 points corresponding to the best

observations (somehow equally distributed) and to link them to star A to use the planimeter to

measure in arbitrary units the surface of the sectors defined. The planimeter used is the model n°313

E from HAFF (https://www.haff.de/) and is very convenient.

Figure 1: Planimeter model n°313 E from HAFF

If Sn is the surface of a sector n and tn the time taken by the companion to run through it, the areal

constant will be obtained by: c = Sn / tn and this constant must be the same (at a few hundredth)

for the various sectors, this leading sometimes to significantly alter the original outline chosen for

the ellipse, adjusting the length of the wire, changing the position of the pins or both at the same

time until one gets a sense not of accomplishment but of minimal frustration as so rare are the orbits

that fit perfectly “head on” the observations. This constant c is the direct result of the second

Kepler’s law, i.e. that the radius vector sweeps out equal areas in equal times (Tatum, 2017). Once a

satisfactory apparent ellipse is drawn, the final position of the two pins must be spotted carefully

(among the various attempts leaving holes in the sheet) as they represent the foci of the ellipse and

all further work and measurements will depend on them. Then one should determine the scale,

usually in millimetres per arc sec.

The centre of the ellipse, C is of course located on the line joining the 2 foci and in the middle of

them and will serve as the origin of a new set of axis, () that will be used to measure the Thiele-

Innes constants directly on the drawing (Figure 2). Tracing a line through the centre C and star A

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(segment CA) will deliver a point on the ellipse, i.e. P the periastron and to find the conjugate

diameter to CP, one draws parallel strings to CP the middle of which lead to point R. As the

observed ellipse is a projection of the true ellipse, one should not expect P and R to necessarily fall

anywhere near the semi-major axis or semi-minor axis of the apparent ellipse. One just need to

work as neatly as possible, as one would do with a technical drawing. Now the eccentricity can be

readily computed by simply dividing two segments measured on the drawing:

e = CA/CP

Figure 2: The geometrical method, from Couteau (1978) corrected as North is down, East is right (the orbit is

as if observed through the instrument) and the new set of axis () is oriented similarly to (x,y).

Now let’s consider the Thiele-Innes constants, noted A, B, F, G, C, H (which determine the plane of

the true orbit and its size) and

F1 = F . cos and G1 = G . cos where = arcsin e

and e the eccentricity already obtained ;

one should first take note that if we take the centre of the apparent ellipse C as the origin of a new

set of axis () parallel to the old ones and oriented in a similar way, then for a point of true

anomaly u:

= A . cos u + F1 . sin u

= B . cos u + G1 . sin u

and the Thiele-Innes constants A and B are simply measured on the graph as the coordinates of P

with respect to the centre C axis () and the measure of the coordinates of the point R delivers:

r = F1 and r = G1

In the example of Figure 2, coordinates of P, P = A > 0 and P = B > 0, R is found following the

arrow along the motion of the companion, and in that case r = F1 < 0 and r = G1 > 0. It is

important to pay attention to the signs of the coordinates of A, B, F1, G1, as filling the spreadsheet

cells with values having the wrong signs obviously ruins the computation.

Knowing F1, G1 and cos enable to calculate F = F1 / cos and G = G1 / cos .

We now have c, e, A, B, F, G and can calculate xc = -Ae and yc = -Bc and CA=(xc2+yc2)1/2

5

The dynamic elements: P period in years, n the mean motion, T the epoch of the passage at

periastron, e eccentricity, a semi-major axis will be obtained by simple calculations made with

measurements on the drawing. The geometric elements of Campbell: i the inclination, the angle

of position of the line of nodes (i.e. the intersection of the object's orbital plane with the plane of

reference), also referred to as the position angle of the first node, i.e. the node whose position angle

is less than 180º, known as the longitude of ascending node, and the angle between the node and

periastron (i.e. the argument of periastron defines the orientation of the ellipse in the orbital plane,

as an angle measured from the ascending node to the periastron in the direction of motion of the

secondary), will be obtained by various calculations using the Thiele-Innes constants, i.e. A, B, F

and G..

Once the orbit is drawn and the coordinates of P and R have been measured we have made a

significant part of the work, obtaining:

P = S / c is done by dividing in arbitrary surface units, the surface of the ellipse by the areal

constant. The surface of the ellipse is the result of a direct measure with the planimeter and c, the

areal constant, has been determined by an average of several ci obtained measuring successive

sectors ci = Si / ti with Si = surface of sector i travelled during time ti and finally we have c = (i

Si / ti ) / n for n sectors. One can also create a double diagram [(,) ; (,t)] as already proposed

Herschel (1933) to check the areal constant as reminded by Dommanget (2000) ;

n = 360 / P, it is the annual mean motion in degrees also referred to as = 2 /P in radians;

T = average of several measures (T1,T2,T3) where T1= tB1 + (area B1AP) / c ; T2= tB2 + (area

B2AP) / c ; T3= tB3 + (area B3AP) / c areas measured with the planimeter on the drawing of the

apparent ellipse. T can also directly be read on the curve (t) ;

e = CA / CP directly measured on the orbit’s drawing ; also e = sin where the ellipse is the

projection of the principal circle (eccentric circle or, preferably, the auxiliary circle) from a plane

tilted of an angle ;

a = the semi-major axis is directly measured on the orbit’s drawing and given the scale converted in

arc sec or else given by a = [(A . G – B . F) / cos i ]1/2 ;

then the measurement on the orbit’s drawing of the coordinates of P and R according to axis (),

leads to the Thiele-Innes constants A, B, F1, G1, we can also obtain (given = arcsin e) F and G

therefore the first four constants from which we will compute i, , , let’s see how.

Let’s introduce the constants C1 = (B – F) / (A + G) and C2 = (B + F) / (A – G) then is given by:

= ( arctg C1 + arctg C2 ) / 2 ;

= arctg C1 - i.e. the argument of periastron (ω);

Let’s finally introduce the constant C3 = [(B + F) . sin (+)] / [(B - F) . sin (- )]

as tg2 (i/2) = C3 one can easily compute that:

i = arcos [(C3 – 1) / (C3 + 1)] or else i = 2 arctg (C3)1/2

By inputting the values P, scale (mm/arc sec), e, A, B, F1, G1 one gets in an excel spreadsheet

immediately , , i and a in arc sec. We provide the spreadsheet as complementary data to the

present paper. Inputting in a different excel sheet P, n, T, e, a, i, , , t0, t, one gets the ephemeris

and the drawing of the corresponding ellipse.

Let’s see how one can get an ephemeris using the aforementioned orbital elements. We have two

equations for that:

tg ( - ) = tg (v + ) . cos i (1, II) from which we will derive and where v is the true anomaly ;

= r . [ cos (v + ) / cos (- ) ] (2, II) where r = [a . (1 – e2)] / (1 + e . cos v) (3, II).

6

Using (1, II) to compute , and (2, II) to compute r for , one will need v, the true anomaly, the only

unknown:

v = 2 arctg( [(1+e)/(1-e)]1/2 . tg u/2) (4, II) where u is the eccentric anomaly.

For a long time, astronomers have used a table of the keplerian motion, that gives the true anomaly

v (or of (v – M), i.e. equation of the centre) as a function of M the mean anomaly (i.e. M = n(t-T))

for any possible eccentricity (Danjon, 1952). We are going to detail how one can in 3 small steps

perform an iteration in a spreadsheet that will provide an acceptable value of u in (4, II) that we

need in order to compute v, the true anomaly.

As: u – e . sin u = M (5, II) we have u = M + e . sin u and we will make a first approximation with:

u0 = M + e . sin M to get a first value of u0 and compute the error made 0 using M instead of u.

As: e . sin u0 = u0 – M therefore, the error 0 = e . sin u0 – (u0 – M) (6, II)

by differentiation of tg v/2 = [(1+e)/(1-e)]1/2 . tg u/2 we get

du/dt = / (1 – e . cos u) (7, II) with = 2 /P (in radians)

hence u1 = u0 + du/dt (8, II) ;

replacing in (8, II) du/dt with its value in (7, II) we get:

u1 = u0 + 0 / (1 – e . cos u0) (9, II) ; u1 from which we compute 1 = e . sin u1 – (u1 – M) (10, II)

reused to compute u2 as in (9, II), then 2 as in (10, II) and so on until n = 0. Usually three steps are

sufficient to make the process converge and get a stable value of u3 (with residuals 3 ± = 0) which

can be used in (4, II) to get v, which was the only unknown in (1, II) and (2, II), thus obtain an

ephemeris in (, ). This is how works the spreadsheet attached as a dataset to this paper. Note that

eq. (5, II) is referred to as Kepler’s equation and can be obtained by trivial derivation of (7, II) as

demonstrated in Appendix D, more easily than what is proposed for example by Tatum (2017, eq.

9.6.5).

Kepler’s equation is transcendental in u and must be solved numerically, convergence by iteration

as aforementioned is also proposed in (Meuss, 2009) and (Boulé et al., 2017). We’ve also added a

sheet “u & v fct of M” which computes any value of u (eccentric anomaly) for any given e and M

(input number in yellow cells), and one can check the results against the Table p.432 of Danjon

(1952), using a less convergent yet simpler iteration (u0=M+e sinM, u1=M+e sin u0, u2=M+e sin u1,

… un=M+e sin un-1) than what is used to compute the ephemeris and described above (example is

given in Appendix E for e=0,35 and M=0,25).

Let’s now remind of the various pitfalls one should avoid in order to smoothly move from the

orbit’s drawing to obtaining the dynamic elements, the elements of Campbell and finally the

ephemeris. The first thing to pay attention to is the orientation of the axis on which the scatter of

measures is plotted then that the plot preserves equal scales on both axis (beware of software

adjustments due to legends or else on the print). Facing the plot, the North should be down (X axis

positive readings) and the East at the right (Y axis positive readings) somehow as if you were

observing through the eyepiece of an instrument.

All measurements of coordinates should be made according to the new set of axis () passing

through C. Coordinates of P are (A,B) and of R (F1, G1) according to (). One should observe that

R is obtained from P in the direction of rotation of the companion star which can be direct (values

of increase over time) or retrograde (values of decrease over time). The smallest error in the

drawing, the slightest measurement error spoils the work and ruins the result. Drawing orbits,

measuring Thiele-Innes constants and deriving orbital parameters is like directly measuring binaries

with a filar micrometer, its a work to be done with care and application.

The great advantage of the geometrical method is that obtaining the orbital parameters requires to

7

maintain a relationship with the data and with the apparent ellipse that one tries to fit to the

observations. It's not just putting a batch of observations, filtered or weighted according to the

observers and / or their observing means or techniques, into a computer letting it operate as a black

box and delivering orbital parameters for sparsely defined trajectories over short segments of an

apparent orbit.

III. Obtaining dynamic parallaxes and masses

Computing orbits aims at measuring accurate stellar masses to test evolutionary models of star

populations and star formation theories. The short presentation under this section will aim at

introducing all the parameters that will be later used in the excel spreadsheet that comes together

with this paper. Given the trigonometric parallax, one can immediately compute the sum of the

masses for the binary by using the third Kepler’s law (the square of the orbital period of a planet is

directly proportional to the cube of the semi-major axis of its orbit), (Tatum, 2017):

A + B = a3 / p3 P2 (1, III)

where A and B are the masses of the components A and B given in Solar mass (taken as unity), a3

is the semi-major axis in arcsec as explained in the previous section, p3 is the parallax (i.e.

trigonometric or dynamic) and P2 the period as given by the orbital parameters. As one will notice

the relation gives access to the total mass of the system A + B and not individual masses.

An approximate relation between the mass and luminosity of main-sequence stars, was predicted by

Eddington (1924) and is obtained from a graph of absolute bolometric magnitude against the

logarithm of mass (in solar units), i.e. M /M, for a large number of binary stars. Most points lie on

an approximately straight line. Since a star's absolute bolometric magnitude is a function of the

logarithm of its luminosity (in solar units), i.e. L /L, this line is represented by the M-L relation:

log(L / L) = n log(M / M) and L / L = (M / M)n

where n averages about 3 for bright massive stars, about 4 for Sun-type stars, and about 2.5 for dim

red dwarfs of low mass. This Mass-Luminosity Relation (MLR) (Eddington, 1924, 1926) leads to

the calculation of dynamic parallaxes (Kuiper, 1938), (Russell, 1938), (Russel and Moore, 1940),

(Baize, 1943), (Baize et Romani, 1946), (Baize, 1947), (Harris et al., 1963), (Couteau, 1971),

(Dommanget, 1976) enabling the knowledge of each individual mass of the binary system, A and

B. Using this method and coupling it to simple iterative calculations in a spreadsheet enable to

derive in less than ten iterations stable absolute bolometric magnitudes for each star A and B,

individual masses for A and B, the dynamic parallax and the sum of masses to serve as a cross-

check.

It is proposed by Mullaney (2005) to initialise the process taking A = B = 1 therefore s =2 and

let converge the process, while the author had independently the same idea starting the iteration

with the calculation of the absolute bolometric magnitudes. We will see that both approaches lead to

quasi identical results to 10-3 on the individual masses. So let’s formulate the relationship that

express the log of the mass as of the bolometric absolute magnitude:

log10 = -k(M- M) (2, III)

where = the mass in solar units, k used to be 0,1117 (Couteau 1978) but has been revised taking

into account hot stars of spectroscopic systems as k=0,0987 (McAlister and Hartkopf, 1988) and

accepted as such by Couteau (1988, 1994) leading to decrease computed masses (of one to five

percents), M the bolometric absolute magnitude of the star and M is absolute magnitude of the

Sun = 4,77. The absolute magnitude is the magnitude of the star would it be observed from a 10

parsecs distance. One should notice that we can write:

8

A + B = A (1 + B / A) and using (2, III) we get

A + B = A (1 + 10–k.(MB-M) + k.(MA-M)) = A (1 + 10–k.(MB-MA)) = A (1 + 10–k.m)

we pose D = 1 + 10–k.m then A + B = A D

The absolute magnitude, the apparent magnitude and the parallax are linked as follows in the

Pogson’s formulae:

M = m + 5 + 5 log10 p (3, III) where m should be a corrected apparent bolometric magnitude mb is:

mb = mv + Cv where Cv is a function of the spectral type or of the temperature of the star as given in

a table as the one from Harris (1963) in Appendix B.

M can be replaced in (2, III) with its value from (3, III)

log10 = -k(m + 5 + 5 log10 p - M) = -k(mvA + CVA + 5 - M) – 5k log10 p (4, III)

besides from (1) and (2) we have A + B = A (1 + 10–k.m ) = a3 / p3 P2 (5, III)

to simplify we pose = p3 P2 then we can write:A D = / p3 (6, III)

from (6, III) we can write: log10A + log10 D = 3 log10 – 3 log10 p therefore

log10A = -3 log10 p + 3 log10 – log10 D (7, III) and by matching (4, III) and (7, III) we get:

– 5k log10 p - k(mvA + CVA + 5 + M) = -3 log10 p + 3 log10 - log10 D

and finally the expression of the parallax:

(1 – 5/3 k) log10 p = log10 + k/3 (mvA + CVA + 5 - M) – 1/3 log10 D

with = (mvA + CVA + 5 - M) we get:

(1 – 5/3 k) log10 p = log10 + k/3 – 1/3 log10 D (8, III)

that we will use in our spreadsheet to compute the parallax as p = 10 (log10 p)

from (4,III) we can extract the value of log10 p:

log10 p = -1/5 . (mvA + CVA + 5 - M) - log10A / 5k

and replace it in (7,III), then we can write:

log10A = 3/5 . (mvA + CVA + 5 - M) + 3/5k . log10A + 3 log10 - log10 D

log10A - 3/5k . log10A = 3/5 . + 3 log10 - log10 D thus multiplying by 5

5 log10A - 3/k . log10A = 3 + 15 log10 - 5 log10 D hence multiplying by –1 and factoring:

[(3 – 5k)/k] . log10A = 5 log10 D – 3 – 15 log10 (9, III)1

for star B we have a similar equation where is replaced by , i.e. = (mvB + CVB + 5 - M) and

notation D is replaced by D’ defined as:

D’ = 1 + 10+k.m (10, III)

Equations (9, III) and (10, III) directly give the individual masses of A and B without having the

need to resort to the parallax, whereas the dynamic parallax only gives their sum (1, III). Equations

(8, III) and (9, III) suppose that the spectrum of both components are known. Most of the times we

only know the global spectrum and make the assumption that they are identical. The error is

negligible as to very different spectrums correspond high m, and in that case log10 D tends to zero.

One will notice the importance of the visual magnitude beyond good orbital parameters, therefore

good dynamic parallaxes and reliable masses depend on good photometric measurements.

Finally using (8, III) and replacing the parallax by its expression as a function of the absolute and

1 This equation is incorrect in Couteau (1978) eq (60,VI) p. 166

9

visual magnitude and expressing the period P, one gets:

(1 – 5/3 k) log10 p = log10 (a3 / P2 )1/3 + k/3 (mvA + CVA + 5 - M) – 1/3 log10 D therefore

(1 – 5/3 k) log10 p = log10 a – 2/3 log10 P + k/3 (mvA + CVA + 5 - M) – 1/3 log10 D then

log10 P = 3/2 log10 a – 1/2 log10 D – k/2 MO + 5k/2 + k/2 (mvA + CVA) – 3/2 p (1-5/3k) (11, III)

replacing p using (3, III) let’s develop the term – 3/2 p (1-5/3k):

-3/2 p (1-5/3k) = – (1-5/3k)(MA/5).3/2 + (1-5/3k) ((mvA + CVA)/5).3/2 + (1-5/3k).3/2

i.e. MA((k/2)-3/10) + 3/10 (mvA + CVA) – k/2 (mvA + CVA) +3/2 – 5k/2

now replacing the term – 3/2 p (1-5/3k) by its value in (11, III) we get:

log10 P = 3/2 log10 a – 1/2 log10 D – k/2 M + MA((k/2)-3/10) + 3/10 (mvA + CVA) + 3/2 (13, III)

As long as the MLR is followed by the star, equation (13, III) where MA is taken from Appendix C

can be used to get an indicative value of the Period P.

From (1) one can also easily get the dynamic parallax:

p = a3 / [((A + B) . P2)]1/3 (13, III)

Now, using simple iterative calculations in a spreadsheet enables to derive in less than ten iterations

stable absolute bolometric magnitudes for each star A and B using equation (3, III) (the first value

of the parallax in (3, III) is either trigonometric if available or derived from (8, III)) with Cv taken

from Appendix B and constants from Baize and Romani (1946), then individual masses for A and

B using (2, III) and then a new value of the parallax p from (14, III) which will replace the first

parallax used in order to compute new values of absolute bolometric magnitudes for each star A and

B, and so on, until stable values are obtained usually in less than 6 steps starting with the

trigonometric parallax and nearly immediately with the dynamic parallax.

One can according to what is proposed by Mullaney (2005) do the same as above but initialise the

process by choosing arbitrary initial masses equal to that of the Sun, i.e. 1, and m=2, compute a

first parallax using (13, III) that will be reused to compute absolute bolometric magnitudes for each

star A and B using equation (3, III), then individual masses for A and B using (2, III) and then a new

value of the parallax p from (13, III), and so on until one gets convergent values in less than 5 or 6

steps.

MAMBAB mMAMBAB m

3,695 4,886 1,3185 0,9706 0,00570 2,29 1,000 1,000 0,00596 2,00

2,920 4,111 1,6094 1,1847 0,00533 2,79 3,018 4,209 1,569 1,155 0,00537 2,72

2,776 3,967 1,6702 1,2295 0,00526 2,90 2,794 3,985 1,662 1,224 0,00527 2,89

2,749 3,940 1,6818 1,2381 0,00525 2,92 2,752 3,943 1,680 1,237 0,00525 2,92

2,744 3,935 1,6840 1,2397 0,00525 2,92 2,744 3,935 1,684 1,239 0,00525 2,92

2,743 3,934 1,6844 1,2400 0,00525 2,92 2,743 3,934 1,684 1,240 0,00525 2,92

2,743 3,934 1,6845 1,2400 0,00525 2,92 2,743 3,934 1,684 1,240 0,00525 2,92

2,743 3,934 1,6845 1,2400 0,00525 2,92 2,743 3,934 1,684 1,240 0,00525 2,92

2,743 3,934 1,6845 1,2400 0,00525 2,92 2,743 3,934 1,684 1,240 0,00525 2,92

2,743 3,934 1,6845 1,2400 0,00525 2,92 2,743 3,934 1,684 1,240 0,00525 2,92

Table2: Example of convergence of MA, MB, A, B, , m for HU 497 BC following the method

described by the author i.e. left (starting with trigonometric parallax), or Mullaney (2005) right.

Trying to know the influence of a small variation of the factor k on the masses is interesting as k

evolved from 0,1117 in (Couteau, 1978) to 0,0987 for (McAlister and Hartkopf, 1988) as was

already mentioned. The spreadsheet has been programmed so that k is a parameter, the value of

10

which can be easily changed. But let’s try to assess the impact of factor k on the computation of the

mass of A taking the partial derivative with respect to k of (9, III). First, log10 D will be replaced in

(9, III) by the two first terms of its development in series near the origin:

log10 D = log10 2 – k/m m (14, III) hence

log10A = (5k.log10 2)/(3 – 5k) – (5k2 . m)/ 2.(3-5k) – (3k/(3 – 5k)). – (15k/(3 – 5k)).log10

let’s now take the partial derivative with respect to k:

∂ log10A / ∂ k =

15.log10 2/(3 – 5k)2 – ((60-50k).k.m)/4(3 – 5k)2 – 9./(3 – 5k)2 – (45.log10 )/(3 – 5k)2

∂ log10A/∂ k = [15.log10 2 – [((6-5k)10k.m)/4] - 9. – 45.log10 )] / (3 – 5k)2 (15, III)

equation (15, III) can be re-arranged as:

∂ log10A/∂ k = (5/ (3 – 5k)2 ) . [ 3.log10 2 – (k(6-5k).m)/2 – 9./5 – 9.log10 ) ] (16, III)2

The lower the value of ∂ log10A/∂ k, the more the mass is independent of the value of k. It is

asserted by Couteau (1978) that the variation of the mass is independent of k if (16,III) tends to zero

and that this condition is close to be perfectly satisfied for binaries of spectral type G, and not far

for most other types for which the residuals are small. Couteau (1978) asserts that the success of the

method of dynamic parallaxes and masses is a result of that condition being satisfied and of a low

dependency on small departures from the ML relation. The small sample we have in this paper

shows that ∂ log10A/∂ k does not necessarily tend to zero and that the dynamic parallaxes, but even

more the masses, are somehow sensitive to k.

IV. Stellar evolution assessed with binaries’ orbits

The Mass-Luminosity Relation used in (2, III) is somehow rough as it does not take into account the

chemical composition of the stars, e.g. their metallicity, nor the slow but steady change of the kernel

which leads to a slight but continuous increase in the stars’ brightness with their age. In fact, this

relation applies to what is referred to as the Zero Age (ZA), the time when the stars start having

their nuclear reactions burning Hydrogen which corresponds to the Main Sequence (MS) and which

lasts for 90% of the lifespan of the stars. Therefore, equation (2, III) is well suited as long as the

system observed is not too old for having left the ZAMS. More generally, what we have presented

in the previous section, which leads to dynamic parallaxes and masses, works well as long as the

stars observed did not leave the ZAMS and does not suit very well the case of massive stars for

which the evolution will be accelerated and the lifespan reduced. Couteau (1985, 1988, 1994)

started promoting a fundamental relationship between the mass and the luminosity which applies to

double stars having a known orbit and which does not make any hypothesis as in (2, III) nor

requires the parallax. We start the reasoning with (5, III) and further define:

the ratio R = B / and A = (1 – R) / p3 hence p3 = [(1 – R) ] / A thus

3 log10 = log10 ( (1 – R) ) – log10 A => 3 log10 p = log10 (1 – R) + log10 – log10 A

using (3, III) we have log10 p = (MA - mA – 5)/5 and replacing p by its value:

3/5 MA – 3/5 mA – 3 = log10 (1 – R) + 3 log10 - log10 A, hence multiplying by 5/3

MA – mA – 5 = 5/3 log10 (1 – R) + 5 log10 - 5/3 log10 A thus

MA = mA + 5 + 5 log10 + 5/3 log10 (1 – R) - 5/3 log10 A (1, IV)3

2 This equation is mistyped in Couteau (1978) eq (62,VI) p. 167

3 This equation is incorrect in Couteau (1988), eq (10,XI) p. 221

11

Using the terminology adopted by Couteau (1994), the reduced absolute magnitude is noted M* and

knowing that

M*A = MA + 5/3 log10 A (2, IV)

one can infer replacing MA by its value (1, IV) that:

M*A = mA + 5 + 5 log10 + 5/3 log10 (1 – R) (3, IV) 4

The nice property of equation (3, IV) is that the reduced absolute magnitude depends only on the

apparent magnitude (measured in a photometric system), the orbital constant , and the mass ratio

R. There is no hypothesis or need of an empirical relationship as the MLR in (2, III), and (3, IV) is

based on fundamental data, except R which varies only slightly between 0,45 and 0,5 and has a

limited effect through its logarithm. Exactly in the same way as one can obtain bolometric

magnitude M, applying the bolometric correction Cvb to the visual magnitude M = V + Cvb, we have

for the reduced magnitudes M* = V* + Cvb , with Cvb from (Harris et al., 1963) or

V* = M* - Cvb (4a, IV) or

V* = V + 5/3 log10 (4b, IV)

with V the visual absolute magnitude according to Rakos (1984) or

V = V* - 5/3 log10 (4c, IV) that we can use to compute the parallax with:

log10 p = [(VA – mA ) / 5] –1 (5, IV)

The diagram (B-V), M* or (B-V), V* applied to double stars, where (B-V) are colour indices linked

to the spectral type, gives information on the luminosity of the stars, without knowing their parallax

which, notwithstanding its ignorance, fully plays its role throughout the orbital constant . We will

refer to it as CHR-diagram, for Couteau-Hertzprung-Russell as it shares its main properties with

the original HR-diagram, except that no need of the trigonometric parallax is required and therefore

no assumption is made with respect to the absolute magnitude of the stars. Couteau (1994) applied

this approach to thirty primary components of COU double stars and could already identify on this

small sample very clearly three spectral classes, i.e. giants (sub-giants of class IV and some

probable class III), main sequence V, and sub-dwarfs (class VI). The great value of this CHR-

diagram is to sort out double stars according to their spectral type and therefore limit the usage of

dynamic parallaxes and masses to those falling into the main sequence part.

Using (2, III) with k=0,0987 (McAlister and Hartkopf, 1988) and (2, IV) one can get the flow

equation from the reduced absolute magnitudes to the absolute magnitudes:

MA = M*A + (5k/3) MA – (5k/3) . 4,77 hence MA (1- (5k/3)) = M*A – 0,7846 then

0,8355 MA = M*A – 0,7846 thus

MA = 1,1968 M*A – 0,9390 (6, IV)

Replacing in (2, III) M*A by its value as defined in (2, IV) we will obtain a new relation of Zero

Age between the mass and the reduced absolute magnitude:

log10 A = -k M*A + 5k/3 log10 A + k M thus log10 A - 5k/3 log10 A = -k M*A + k MO hence

(1-5k/3) log10 A = -k M*A + k MO therefore:

log10 A = (-k M*A + k M) / (1-(5k/3)) (7, IV)

and again replacing k by its value (i.e. 0,0987, though k is a parameter in the spreadsheet to enable

sensitivity studies) we get:

log10 A = -(0,0987/0,8355) M*A + (0,0987/0,8355) M, hence with the absolute magnitude of the

sun M=4,77 we get:

4 There is a typesetting error with a – instead of a + in this equation in Couteau (1994).

12

log10 A = - 0,1181 M*A + 0,5634

The strength of (7, IV) is that M*A is known precisely from (3, IV) and in this new relation, the

parallax, be it known or not, plays its role through the orbital constant . The mass which is

calculated in that manner supposes that the star is still in its ZAMS stage. To confirm it, one just

needs to see whether the mass matches what one can expect from its spectral type, or (B-V) as

given in Appendix F. If it does not, it means that the star has evolved, is not “zero age” any longer

and has left the main sequence, and its brilliance is sur-luminous and can be evaluated. One just

needs to assign to the star a mass corresponding to its (B-V) in Appendix F, then using (2, IV) one

gets a visual absolute magnitude, the difference of which with the zero age value gives an indicator

of evolution.

A more representative CHR-diagram is provided in (Couteau, 1988), where 150 double stars having

trustworthy orbits have been used, provided that measures of their (B-V) was available. Abscissa is

(B-V) which has the advantage to be a measure a opposed to the spectral type which is a

classification (or a temperature which remains somehow arbitrary) and ordinate is V*. This is a

fundamental diagram has the position of the stars does not depend on any hypothesis or measure of

parallax (and few could have been plotted in a HR-diagram has the absolute magnitude requires the

parallax). This representation shows the grouping around the ZAMS, and few stars fall below as

they are sub-dwarfs, but a number of them lie above the ZAMS, some being very remote and

correspond to old stars getting more and more luminous as their photosphere grows. The knowledge

of V* at the same time as one computes the orbit enables to position of the star on the CHR-

diagram and to determine at once if the object is of zero age and if one can reliably apply to it the

computation of dynamic parallax and masses.

V. Visual Binaries Studied

RST 3340

WDS 00055-1835, BD -19,8447, 10.4-10.4, Sp G3V, PM +004+005. This is a southern object as the

declination makes the binary extremely difficult for northern observers given the close separation. 8

observations have been used to compute the orbit, starting with the first observation and discovery

by Rossiter, R.A. in 1935,706, and last by Mason et al. (2011) on 2001,5701. Unfortunately we do

not know the trigonometric parallax. Given the observations, one could easily draw three competing

ellipses having one of the foci (West) slightly shifted to the South.

RST 3340 various apparent ellipses drawn (left) and as produced by the ephemeris in the spreadsheet (right).

Scale of apparent ellipse on the drawing: 1 arcsec = 198,05mm. Axes and pass through C.

13

One way to assess the adequacy of these ellipses is to compute for each of them the standard

deviation of the set of areal constants obtained for each sector, therefore measuring the dispersion.

One remembers that the dispersion should be minimum. The classical formula is:

= [n x2 – (x)2 / n(n-1) ]1/2 (1, V)

where n is the number of sectors measured and the xi are the areal constants for each sector i. By

doing so, one finds for our scale with arbitrary HAFF units that ellipse1=1,162 whereas ellipse3=1,650,

therefore ellipse1 fits better the data than the subsequent attempts. Equation (13, III) gives a

probable period at discovery of 153,14, which is a good preliminary estimate for the real computed

period of 129,54. Take note that motion of the companion is retrograde, i.e. goes descending. The

orbital elements are as follow:

Orbital elements

P=129,54 a"=0,283 A"=0,1717 C"= 0,1929

n=2,77906 i°=132,86 B"=-0,1161 H"= 0,0765

T=1959,25 25,68 F"=-0,2065 F1"= -0,2020

e=0,207 68,36 G"=-0,1781 G1"= -0,1742

The following abbreviated ephemeris is derived:

HU 616 - HIP 33145 - [Poy2017]

Year

2017,5 251,31 0,27

2018,5 249,27 0,27

2019,5 247,18 0,26

2020,5 245,04 0,26

2021,5 242,85 0,26

2022,5 240,61 0,26

2023,5 238,32 0,25

2024,5 235,98 0,25

2025,5 233,58 0,25

2026,5 231,13 0,24

Spectral type G3V and the position occupied by the star on the CHR diagram (0,56 – 5,26) confirms

its class V status though the star is situated slightly under the ZAMS line. A is in accordance with

spectral type G3 as given for the main sequence by Pecker and Schatzman (1959), i.e. 0,99. As

the trigonometric parallax is not available, one cannot compute (1, III). Masses computed with

the dynamic parallax without or with the convergence algorithm match very well. The mass

computed with (9, III), i.e. 1,01 or (7,IV), i.e. 0,9 also corresponds well to that found in the

Appendix F for Sp=G3, i.e. 0,97. Note that ∂log10A/∂k is minimum for this spectral type and

class.

Parallaxes, Masses & Stellar evolution

V Mag 10,08 log (8,III) -2,0586 dyn Iter/Conv. 0,00900 P prob (13,III) 153,14

(B-V) color idx 0,560 (dyn) 0,00874 A by Iter/Conv. 0,93 M* (3,IV) 5,18

mA10,40 log A (9, III) 0,00622 by Iter/Conv. 0,93 V* (4a, IV) 5,26

mB10,40 A dyn 1,01 m by Iter/Conv. 1,86 A from (7,IV) 0,90

Sp G3V log B (10, III) -0,03159 d parsec 111,17 VA (4c, IV) 5,34

CvA-0,08 B dyn 0,93 d light years 362,59 from (5,IV) 0,00971

trigo (mas) NO A dyn+B dyn 1,94 a UA 32,41 CHR diagram "≈ -" ZAMS

trigo (1,III) ##### dyn (1,III) 2,03 k (MLR) 0,0987 ∂log10A/∂k 0,08

14

Final dynamical values computed are the following: dyn of 9,222mas ±0,684, (A+B)dyn=1,944

±0,085, Adyn=0,947±0,061, Bdyn=0,930±0,034.

HU 616

WDS 06541+3341, BD 33,1427, ADS 5544, HIP 33145, apparent magnitudes 10.0-10.4, Sp F0, PM

-008-016. No orbit is listed by (Hartkopf and Mason, 2017). Only 5 observations have been used to

compute the orbit, but they are well distributed along the trajectory of the companion, starting with

the first observation and discovery by Hussey W. J. in 1902,76, and last by R. Gili on 1990,148.

HU 616 apparent ellipse, drawn (left) and as produced by the ephemeris in the spreadsheet (right). Scale of

apparent ellipse on the drawing: 1 arcsec = 405,55mm. Axes and pass through C.

Equation (13, III) gives a probable period at discovery of 128,01, which is a reasonable preliminary

estimate for the real computed period of 181,27. The orbital elements are as follow:

Orbital elements

P=181,27 a"=0,155 A"=-0,0925 C"= 0,1139

n=1,98599 i°=54,03 B"=0,0493 H"= 0,0521

T=2011,11 99,85 F"=-0,0132 F1"= -0,0123

e=0,356 65,41 G"=-0,1451 G1"= -0,1356

The following abbreviated ephemeris is derived:

HU 616 - HIP 33145 - [Poy2017]

Year

2017,5 191,40 0,06

2018,5 198,44 0,06

2019,5 205,20 0,06

2020,5 211,57 0,07

2021,5 217,49 0,07

2022,5 222,92 0,07

2023,5 227,87 0,07

2024,5 232,37 0,08

2025,5 236,46 0,08

2026,5 240,16 0,09

15

Spectral type F0V and the position occupied by the star on the CHR diagram (0,230 – 3,13)

confirms its class V status and falls perfectly on the ZAMS line. Mass of A as computed from (7,VI)

matches perfectly that estimated according spectral type F0 as given for the main sequence by

Pecker and Schatzman (1959), i.e. 1,58. Masses computed with (9,III) and (10, III) seem a bit

over-rated, whereas dynamic results, i.e. the dynamic parallax and masses computed with the

convergence algorithm or without (7,IV) are close at around A≈1,64 which is slightly above that

found in the Appendix F for Sp=F8, i.e. 1,49.

The former trigonometric parallax 3,5±1,3 seemed to slightly under-evaluate the distance and leads

to slightly under-estimated masses (1, III), though the parallax computed with (5, IV) is very

close to this trigonometric value. Overall all dynamic results are in close accordance and show little

deviations. Latest trigonometric parallax values from “Gaia” as published by (Brown et al., 2016)

and accessed through SIMBAD give a trigonometric parallax of 2,57±0,61, which gives a value of

=6,65 with (1,III) in the very wide range of [3,51-14,98] and seems this time to over-

estimate the sum of the masses, though our value based on the dynamic parallax (1, III) at 3,84 still

falls within the new range.

Parallaxes, Masses & Stellar evolution

V Mag 9,34 log (8,III) -2,5105 dyn Iter/Conv. 0,00328 P prob (13,III) 128,01

(B-V) color idx 0,230 (dyn) 0,00309 A by Iter/Conv. 1,68 M* (3,IV) 3,05

mA10,00 log A (9, III) 0,30229 by Iter/Conv. 1,53 V* (4a, IV) 3,13

mB10,40 A dyn 2,01 m by Iter/Conv. 3,21 A from (7,IV) 1,60

Sp F0V log B (10, III) 0,18484 d parsec 305,18 VA (4c, IV) 2,79

CvA-0,08 B dyn 1,53 d light years 995,37 from (5,IV) 0,00362

trigo (mas) 2,57 A dyn+B dyn 3,54 a UA 50,14 CHR diagram "=" ZAMS

trigo (1,III) 6,65 dyn (1,III) 3,84 k (MLR) 0,0987 ∂log10A/∂k 3,63

Final dynamical values computed are the following: dyn of 3,354mas ±0,378, (A+B)dyn=3,527

±0,315, Adyn=1,759±0,217, Bdyn=1,531±0,119.

HU 841

WDS 07400+6603, BD 66,518, ADS 6228, HIP 37353, apparent magnitudes 9.4-9.4, Sp G, PM

-005-022. No orbit is listed by (Hartkopf and Mason, 2017). 15 observations have been used to

compute the orbit, starting with the first observation and discovery by Hussey W. J. in 1904,94, and

last by Heintz, W.D. on 1986,17. No more recent observation is available.

16

HU 841 apparent ellipse, drawn (left) and as produced by the ephemeris in the spreadsheet (right). Scale of

apparent ellipse on the drawing: 1 arcsec = 257,81mm. Axes and pass through C.

Equation (13, III) gives a probable period at discovery of 97,17, which is a reasonable preliminary

estimate for the real computed period of 155,58. The orbital elements are as follow:

Orbital elements

P=155,58 a"=0,253 A"=0,1435 C"= -0,1780

n=2,31392 i°=50,28 B"=-0,1086 H"= -0,0788

T=2010,24 87,60 F"=0,0751 F1"= 0,0465

e=0,785 246,11 G"=0,2285 G1"= 0,1416

The following abbreviated ephemeris is derived:

HU 841 - HIP 37353 - [Poy2017]

Year

2017,5 254,10 0,10

2018,5 259,21 0,11

2019,5 263,41 0,13

2020,5 266,95 0,14

2021,5 269,99 0,15

2022,5 272,64 0,16

2023,5 274,97 0,18

2024,5 277,04 0,19

2025,5 278,91 0,20

2026,5 280,60 0,20

Spectral type F8V and the position occupied by the star on the CHR diagram (0,345 – 3,61)

confirms its class V status though it lies slightly above the ZAMS line. Mass of A is above that

estimated according to spectral type F8 as given for the main sequence by Pecker and Schatzman,

(1959), i.e. 1,19. All dynamic results, the dynamic parallax and masses computed without or with

the convergence algorithm match very well. The mass computed with (9, III) or (7,IV) is above that

found in the Appendix F for Sp=F8, i.e. 1,13. The trigonometric parallax 7,54±1,44 seems to

under-evaluate the distance and leads to under-estimated masses (1, III), but with a wide range

[0,93-2,95] given the ±1,44 uncertainty on the value of the trigonometric parallax. masses

determined by iteration/convergence at 2,80 falls within that range.

17

Parallaxes, Masses & Stellar evolution

V Mag 8,95 log (8,III) -2,2248 dyn Iter/Conv. 0,00621 P prob (13,III) 97,17

(B-V) color idx 0,490 (dyn) 0,00596 A by Iter/Conv. 1,40 M* (3,IV) 3,69

mA9,40 log A (9, III) 0,19979 by Iter/Conv. 1,40 V* (4a, IV) 3,77

mB9,40 A dyn 1,58 m by Iter/Conv. 2,80 A from (7,IV) 1,34

Sp F8V log B (10, III) 0,14663 d parsec 161,11 VA (4c, IV) 3,55

CvA-0,08 B dyn 1,40 d light years 525,47 from (5,IV) 0,00678

trigo (mas) 7,54 A dyn+B dyn 2,99 a UA 42,49 CHR diagram "+" ZAMS

trigo (1,III) 1,56 dyn (1,III) 3,17 k (MLR) 0,0987 ∂log10A/∂k 2,42

Final dynamical values computed are the following: dyn of 6,367mas ±0,577, (A+B)dyn=2,986

±0,183, Adyn=1,443±0,126, Bdyn=1,402±0,071.

A 552

WDS 08441-0412, BD -03,2454, ADS 6964, HIP 42863, apparent magnitudes 7.55-9.37, Sp A2,

PM -019+002. 10 observations from the discovery date by Aitken in 1903,04 to last interferometric

measures by Hartkopf, W.I. in 1996,8666. I found it very surprising that no orbit had yet been

computed as the companion seems to have made a complete rotation since the discovery. No orbit is

listed by (Hartkopf and Mason, 2017). This star is the typical example for which successive orbits,

more elliptic each time, had to be envisaged until the law of the areas could be better satisfied. The

final ellipse proposed is quite far from the initial attempts. It is a preliminary orbit.

A 552 apparent ellipse, drawn (left) and as produced by the ephemeris in the spreadsheet (right). Scale of

apparent ellipse on the drawing: 1 arcsec = 362,50. Axes and pass through C.

Equation (13, III) gives a probable period at discovery of 91,32, which is a good preliminary

estimate for the real computed period of 106,58. The orbital elements are as follow:

Orbital elements

P=106,58 a"=0,197 A"=-0,1186 C"= 0,0071

n=3,37774 i°=60,13 B"=-0,1572 H"= -0,1708

T=1957,40 54,15 F"=0,0747 F1"= 0,0676

e=0,427 177,63 G"=-0,0640 G1"= -0,0579

18

The following abbreviated ephemeris is derived:

A 552 - HIP 42863 - [Poy2017]

Year

2017,5 57,73 0,28

2018,5 58,51 0,27

2019,5 59,29 0,27

2020,5 60,09 0,27

2021,5 60,91 0,27

2022,5 61,74 0,26

2023,5 62,59 0,26

2024,5 63,46 0,26

2025,5 64,36 0,25

2026,5 65,29 0,25

Spectral type A2V and the position occupied by the star on the CHR diagram (0,170 – 2,18)

confirms its class V status as it lies very close to the ZAMS line. Mass of A is in good accordance

with spectral type A2 as given for the main sequence by Pecker and Schatzman (1959), i.e. 2,50.

All dynamic results, the dynamic parallax and masses computed without or with the convergence

algorithm match well. The mass computed with (7,IV) matches perfectly that found in the Appendix

F for Sp=F8, i.e. 2,19. The trigonometric parallax 5,69±0,93 gives very comparable results, i.e.

-trigo=3,66 to those obtained by iteration/convergence, i.e. iter/conv=3,88. Obviously,

∂log10A/∂k increases for this spectral type.

Parallaxes, Masses & Stellar evolution

V Mag 7,30 log (8,III) -2,2634 dyn Iter/Conv. 0,00558 P prob (13,III) 91,32

(B-V) color idx 0,170 (dyn) 0,00545 A by Iter/Conv. 2,34 M* (3,IV) 1,93

mA7,55 log A (9, III) 0,39846 by Iter/Conv. 1,55 V* (4a, IV) 2,18

mB9,37 A dyn 2,50 m by Iter/Conv. 3,88 A from (7,IV) 2,17

Sp A2V log B (10, III) 0,18929 d parsec 179,29 VA (4c, IV) 1,62

CvA-0,25 B dyn 1,55 d light years 584,78 from (5,IV) 0,00651

trigo (mas) 5,69 A dyn+B dyn 4,05 a UA 36,15 CHR diagram "≈" ZAMS

trigo (1,III) 3,66 dyn (1,III) 4,16 k (MLR) 0,0987 ∂log10A/∂k 4,63

Final dynamical values computed are the following: dyn of 5,982mas ±0,749, (A+B)dyn=4,031

±0,138, Adyn=2,336±0,168, Bdyn=1,546±0,074.

A 2554

WDS 08539+0149, BD 02,2088, ADS 7074, HIP 43676, apparent magnitudes 7.44-9.64, Sp F0, PM

+010-040. There are two orbits, one by Zirm, H. (Zir2007) with P=83,6y, the other by Tokovinin, A.

(Tok2015c) with P=43,86y. To be honest, the data plot used by Zirm wds08539+0149a.png is easily

recognisable even though we made slightly different choices with respect to the foci, whereas the

one by Tokovinin, i.e. wds08539+0149o.png cannot be matched with the one we produce with the

16 observations we have. The solution we propose with P=82,6y is way closer to the one from Zirm

even though the orientation of the orbit differs significantly. 16 observations have been used to

compute the orbit, starting with the first observation and discovery by R. G. Aitken in 1913,11, and

last on 1996,8665 by W.I. Hartkopf.

19

A 2554 apparent ellipse, drawn (left) and as produced by the ephemeris in the spreadsheet (right). Scale of

apparent ellipse on the drawing: 1 arcsec = 212,50mm. Axes and pass through C.

Equation (13, III) gives a probable period at discovery of 93,26, which is a good guess for the real

computed period of 82,60. Take note that motion of the companion is retrograde, i.e. goes

descending. The orbital elements are as follow:

Orbital elements

P=82,60 a"=0,359 A"=0,0706 C"= -0,2740

n=4,35835 i°=126,24 B"=0,2212 H"= 0,0939

T=1980,5 12,41 F"=0,3466 F1"= 0,2824

e=0,580 288,92 G"=0,0058 G1"= 0,0047

The following abbreviated ephemeris is derived:

A 2554 - HIP 43676 - [Poy2017]

Year

2017 262,65 0,35

2018 260,35 0,35

2019 258,10 0,35

2020 255,90 0,36

2021 253,76 0,36

2022 251,66 0,37

2023 249,61 0,37

2024 247,62 0,38

2025 245,68 0,38

2026 243,78 0,38

Spectral type A9V and the position occupied by the star on the CHR diagram (0,345 – 3,61)

confirms its class V status. Mass of A is in accordance with spectral type A9 as given for the main

sequence by Pecker and Schatzman (1959), i.e. 1,62. All results either computed with the

trigonometric parallax or with the dynamic parallax without or with the convergence algorithm

match very well. The mass computed with (9, III) or (7,IV) also corresponds well to that found in

the Appendix F for Sp=A9, i.e. 1,57. The trigonometric parallax 14,79±1,02 gives a range of

[1,72-2,60] given the ±1,02 uncertainty within which all dynamic values fall easily.

20

Parallaxes, Masses & Stellar evolution

V Mag 7,23 log (8,III) -1,8542 dyn Iter/Conv. 0,01426 P prob (13,III) 93,26

(B-V) color idx 0,345 (dyn) 0,01399 A by Iter/Conv. 1,46 M* (3,IV) 3,51

mA7,44 log A (9, III) 0,18862 by Iter/Conv. 0,88 V* (4a, IV) 3,61

mB9,64 A dyn 1,54 m by Iter/Conv. 2,34 A from (7,IV) 1,41

Sp A9V log B (10, III) -0,05333 d parsec 70,13 VA (4c, IV) 3,36

CvA-0,10 B dyn 0,88 d light years 228,75 from (5,IV) 0,01527

trigo (mas) 14,8 A dyn+B dyn 2,43 a UA 25,67 CHR diagram "=" ZAMS

trigo (1,III) 2,10 dyn (1,III) 2,48 k (MLR) 0,0987 ∂log10A/∂k 2,04

Final dynamical values computed are the following: dyn of 14,628mas ±0,903, (A+B)dyn=2,417

±0,070, Adyn=1,471±0,068, Bdyn=0,884±0,031.

BU 796

WDS 12175+0636, BD 07,2529, ADS 8500, HIP 59916, apparent magnitudes 8.38-9.46, Sp F2, PM

+027-040. No orbit is listed by (Hartkopf and Mason, 2017). 34 observations have been used to

compute the orbit, starting with the first observation and discovery by W. Burnham in 1881,34, and

last on 1989,28 by W.D. Heintz. This system is somehow frustrating as it has been measured for

more than a century by the best observers using the largest instruments of their time and the

measures look like a very heterogeneously scattered plot, showing an odd disposition, but does not

resemble a drift due to proper motion. So we clearly have an orbital system, but the tentative orbit

proposed, which should probably be even more elliptical, has just the objective to draw the attention

on this star, and is a very preliminary attempt. The situation might be more complex and a third

companion, or more, could disturb the trajectories (or else ?), and the discussion is open and

criticisms welcome.

BU 796 apparent ellipse drawn (left) and as produced by the ephemeris in the spreadsheet (right). Scale of

apparent ellipse on the drawing: 1 arcsec = 208,33mm. Axes and pass through C.

The orbital elements are as follow:

Orbital elements

P=110,26 a"=0,414 A"=0,0192 C"= 0,1651

n=3,26501 i°=69,97 B"=0,3792 H"= 0,3522

T=1966,70 77,98 F"=-0,1622 F1"= -0,1368

e=0,538 25,12 G"=-0,1452 G1"= -0,1224

21

The following abbreviated ephemeris is derived:

BU 796 - HIP 59916 - [Poy2017]

Year

2017,0 264,87 0,60

2018,0 265,32 0,60

2019,0 265,78 0,60

2020,0 266,24 0,59

2021,0 266,71 0,59

2022,0 267,18 0,58

2023,0 267,67 0,58

2024,0 268,16 0,57

2025,0 268,66 0,57

2026,0 269,17 0,56

All results are summarized in the following table. Spectral type F2 and class V is confirmed by the

position occupied on the CHR diagram (0,400 - 4,35), though the star lies slightly below the

ZAMS. Mass of A computed with (9, III), 1,33 , is in reasonable accordance with spectral type

A5 as given for the main sequence by Pecker and Schatzman (1959), i.e. 1,46. The trigonometric

parallax as given by SIMBAD 7,21±0,45mas gives extremely dubious results, i.e. =15,58 (1,

III), very far from dynamic calculations and common sense as two massive B7 stars (which is not

the case) will hardly go further than 7,68 (Appendix F). Results offered by (8, III), (9, III), (10,

III), (1, III) with dynamic parallax, parallax and masses obtained by iteration / convergence and

finally (7, IV) are very consistent. Finally, Adyn is just slightly under that found in the Appendix F,

i.e. 1,38, for Sp=F2.

Parallaxes, Masses & Stellar evolution

V Mag 7,92 log (8,III) -1,8692 dyn Iter/Conv. 0,01409 P prob (13,III) 172,40

(B-V) color idx 0,400 (dyn) 0,01351 A by Iter/Conv. 1,17 M* (3,IV) 4,30

mA8,38 log A (9, III) 0,12299 by Iter/Conv. 0,92 V* (4a, IV) 4,35

mB9,46 A dyn 1,33 m by Iter/Conv. 2,09 A from (7,IV) 1,14

Sp F2V log B (10, III) -0,03794 d parsec 70,98 VA (4c, IV) 4,26

CvA-0,05 B dyn 0,92 d light years 231,49 from (5,IV) 0,01500

trigo (mas) 7,21 A dyn+B dyn 2,24 a UA 30,64 CHR diagram "-" ZAMS

trigo (1,III) 15,58 dyn (1,III) 2,37 k (MLR) 0,0987 ∂log10A/∂k 1,38

Final dynamical values computed are the following: dyn of 14,257mas ±1,051, (A+B)dyn=2,232

±0,139, Adyn=1,211±0,102, Bdyn=0,916±0,052.

RST 4502

WDS 12349-0509, BD -04,3307, apparent magnitudes 8.49-9.34, Sp A5, PM -030-005. No orbit is

listed by (Hartkopf and Mason, 2017). 18 observations have been used to compute the orbit, starting

with the first observation and discovery by R.A. Rossiter in 1938,475, and last on 1997,1317 by

W.I. Hartkopf.

22

RST 4502 apparent ellipse drawn (left) and as produced by the ephemeris in the spreadsheet (right). Scale of

apparent ellipse on the drawing: 1 arcsec = 397,62mm. Axes and pass through C.

The orbital elements are as follow:

Orbital elements

P=147,54 a"=0,172 A"=0,0126 C"= 0,0718

n=2,44002 i°=50,70 B"=-0,1559 H"= 0,1122

T=2029,46 72,56 F"=0,1154 F1"= 0,1119

e=0,244 32,60 G"=0,0609 G1"= 0,0591

The following abbreviated ephemeris is derived:

RST 4502 - BD -04,3307 - [Poy2017]

Year

2017,5 264,33 0,13

2018,5 267,13 0,12

2019,5 270,03 0,12

2020,5 273,05 0,12

2021,5 276,21 0,12

2022,5 279,53 0,11

2023,5 283,03 0,11

2024,5 286,73 0,11

2025,5 290,64 0,10

2026,5 294,77 0,10

All results are summarized in the following table. Spectral type A5 and class V is confirmed by the

position occupied on the CHR diagram (0,260 – 2,22), though the star lies slightly above the

ZAMS. Mass of A computed with (7, IV), 2,08 , is in good accordance with spectral type A5 as

given for the main sequence by Pecker and Schatzman (1959), i.e. 1,94 . The trigonometric

parallax 3,76±0,61mas gives results, i.e. =4,41 (I, III), close to what is obtained by dynamic

calculations, i.e. =4,33. Results offered by (8, III), (9, III), (10, III), (1, III) with dynamic

parallax, parallax and masses obtained by iteration / convergence and finally (7, IV) are very

consistent. Finally, the mass computed by dynamic means is just slightly above that found in the

Appendix F for Sp=A5, i.e. 1,90.

23

Parallaxes, Masses & Stellar evolution

V Mag 8,02 log (8,III) -2,4305 dyn Iter/Conv. 0,00387 P prob (13,III) 86,76

(B-V) color idx 0,260 (dyn) 0,00371 A by Iter/Conv. 2,21 M* (3,IV) 2,07

mA8,49 log A (9, III) 0,39998 by Iter/Conv. 1,82 V* (4a, IV) 2,22

mB9,34 A dyn 2,51 m by Iter/Conv. 4,03 A from (7,IV) 2,08

Sp A5 log B (10, III) 0,26056 d parsec 258,22 VA (4c, IV) 1,69

CvA-0,15 B dyn 1,82 d light years 842,21 from (5,IV) 0,00437

trigo (mas) 3,76 A dyn+B dyn 4,33 a UA 46,38 CHR diagram "+" ZAMS

trigo (1,III) 4,41 dyn (1,III) 4,58 k (MLR) 0,0987 ∂log10A/∂k 4,76

Final dynamical values computed are the following: dyn of 4,039mas ±0,464, (A+B)dyn=4,316

±0,275, Adyn=2,268±0,220, Bdyn=1,822±0,113.

A 1097 AB

WDS 14020+5713, BD 57,1478, ADS 9089, HIP 68554, apparent magnitudes 7,76-8.24, Sp F5, PM

-008-011. There have been three orbits published for this system, i.e. Cou1960c (P=126y), Jnv1960

(P=333,3y), Sca2000a (P=224,7y). With a period of 144,81y the solution we propose differs

significantly from the one from Scardia (2000) and is closer to that of Couteau (1960), though

observations have since elongated much the ellipse towards the West (last observation used by

Couteau was in 1959,44). 58 observations have been used to compute the orbit, starting with the

first observation and discovery by Aitken R. G., in 1905,77, and last on 2000,49 by Docobo, J.A.

The final observation listed in the WDS (2010) fits well in the plot.

A 1097 AB apparent ellipse drawn (left) and as produced by the ephemeris in the spreadsheet (right). Scale

of apparent ellipse on the drawing: 1 arcsec = 202,76mm. Axes and pass through C.

The orbital elements are as follow:

Orbital elements

P=144,81 a"=0,394 A"=0,2318 C"= -0,3019

n=2,48602 i°=53,92 B"=0,1036 H"= 0,1024

T=1894,5 84,15 F"=-0,0361 F1"= -0,0345

e=0,294 288,73 G"=0,3793 G1"= 0,3625

24

The following abbreviated ephemeris is derived:

A 1097 AB - HIP 68554 - [Poy2017]

Year

2017 276,41 0,34

2018 278,31 0,33

2019 280,33 0,32

2020 282,47 0,31

2021 284,76 0,30

2022 287,22 0,29

2023 289,87 0,28

2024 292,76 0,27

2025 295,91 0,26

2026 299,37 0,24

All results are summarized in the following table. Spectral type F5 and class V is confirmed by the

position occupied on the CHR diagram (0,461 – 3,96). Mass of A is in good accordance with

spectral type F5 as given for the main sequence by Pecker and Schatzman (1959), i.e. 1,35 . The

trigonometric parallax 6,82±0,69mas seems under-estimated and gives results far from the well

grouped and consistent results offered by (8, III), (9, III), (10, III), (1, III) with the dynamic

parallax, or parallax and masses obtained by iteration / convergence and finally (7, IV). The mass

computed with by iteration / convergence or (7, IV) corresponds well to that found in the Appendix

F for Sp=F5, i.e. 1,25. ∂log10A/∂k is rather small for this spectral type and class (the star falls

nearly on the ZAMS line in CHR diagram), hence the MLR is well obeyed which comforts the

dynamic results.

Parallaxes, Masses & Stellar evolution

V Mag 7,80 log (8,III) -2,0069 dyn Iter/Conv. 0,01049 P prob (13,III) 142,29

(B-V) color idx 0,461 (dyn) 0,00984 A by Iter/Conv. 1,31 M* (3,IV) 3,92

mA8,51 log A (9, III) 0,20177 by Iter/Conv. 1,22 V* (4a, IV) 3,96

mB8,83 A dyn 1,59 m by Iter/Conv. 2,53 A from (7,IV) 1,26

Sp F5 log B (10, III) 0,08641 d parsec 95,30 VA (4c, IV) 3,79

CvA-0,04 B dyn 1,22 d light years 310,82 from (5,IV) 0,01139

trigo (mas) 6,82 A dyn+B dyn 2,81 a UA 40,08 CHR diagram "≈" ZAMS

trigo (1,III) 9,23 dyn (1,III) 3,07 k (MLR) 0,0987 ∂log10A/∂k 2,41

Final dynamical values computed are the following: dyn of 10,617mas ±1,093, (A+B)dyn=2,805

±0,268, Adyn=1,389±0,177, Bdyn=1,220±0,099.

HU 337

WDS 19188+1736, BD 17,3924, ADS 12301, HIP 94911, apparent magnitudes 9.27-10.14, Sp A3,

PM -005+007. No orbit is listed by (Hartkopf and Mason, 2017). 16 observations have been used to

compute the orbit, starting with the first observation and discovery by Hussey W. J. in 1901,51, and

last on 1989,69 by Le Beau, J. Three observations from Van Biesbroeck are dubious with respect to

the separation namely VBS 1951,816, 1954,05, 1954,8 and are given at 0,”09. VBS 1951,816 is

reported by the author himself has having an uncertain position angle. Notice should be taken that

measures made at Mac Donald (T200) by Van Biesbroeck were systematically to short in separation

as reported by Couteau (1988) in comparative Table (3, VIII), p.127.

25

Close to the minimum separation, we have two measures from Van den Bos with the T200 at Mac

Donald 1958,65 (0,”12) and 1961,69 (0,”10) even though Couteau (1988) states that Van den Bos

was reluctant to measure down to that limit. Couteau (1988) also reminds that Van den Bos was

probably the greatest double stars observer of all times. We also have one measure by Couteau

1970,48 (336°,0,”11) with the 74cm refractor in Nice, that he evaluates as uncertain, but which

appears to be magic work, right on spot. Having partially recounted the history of the measures on

HU 337, the trajectory seems long and well shaped enough to compute a preliminary orbit, but the

star moves to its new separation maximum around 0,”20 during the period (2017-2027) and

CCD/EMCCD or speckle observations would enable to get better elements.

HU 337 apparent ellipse drawn (left) and as produced by the ephemeris in the spreadsheet (right). Scale of

apparent ellipse on the drawing: 1 arcsec = 303,70mm. Axes and pass through C.

The orbital elements are as follow:

Orbital elements

P=228,90 a"=0,234 A"=0,0395 C"= 0,1584

n=1,57276 i°=126,32 B"=-0,1679 H"= -0,1026

T=1989,1 60,78 F"=-0,1618 F1"= -0,1581

e=0,213 122,92 G"=-0,1348 G1"= -0,1317

The following abbreviated ephemeris is derived:

HU 337 - HIP 94911 - [Poy2017]

Year

2017,5 235,89 0,20

2018,5 234,69 0,20

2019,5 233,50 0,20

2020,5 232,31 0,20

2021,5 231,13 0,20

2022,5 229,94 0,20

2023,5 228,75 0,20

2024,5 227,57 0,20

2025,5 226,37 0,20

2026,5 225,18 0,20

All results are summarized in the following table. Spectral type A3 and class V is confirmed by the

position occupied on the CHR diagram (0,310 – 3,09), though the star lies slightly above the main

sequence and probably shows a beginning of evolution from the ZAMS. In any case, mass of A is in

26

reasonable accordance with spectral type A3 as given for the main sequence by (Pecker and

Schatzman, 1959), i.e. 1,70. The mass computed with (9, III), i.e. 2,05, matches very well the

value found in Appendix F for Sp=A3. Latest value of trigonometric parallax (SIMBAD), i.e.

4,15±1,18mas (±14,22% error), leads to a very wide a range of trigo[1,62-9,32] at a 68,2%

confidence interval, therefore dynamic results are well within that range.

Parallaxes, Masses & Stellar evolution

V Mag 8,79 log (8,III) -2,3938 dyn Iter/Conv. 0,00422 P prob (13,III) 304,65

(B-V) color idx 0,310 (dyn) 0,00404 A by Iter/Conv. 1,80 M* (3,IV) 2,89

mA9,27 log A (9, III) 0,31082 by Iter/Conv. 1,47 V* (4a, IV) 3,09

mB10,14 A dyn 2,05 m by Iter/Conv. 3,27 A from (7,IV) 1,67

Sp A3 log B (10, III) 0,16825 d parsec 237,10 VA (4c, IV) 2,72

CvA-0,20 B dyn 1,47 d light years 773,31 from (5,IV) 0,00489

trigo (mas) 4,15 A dyn+B dyn 3,52 a UA 58,00 CHR diagram "≈" ZAMS

trigo (1,III) 3,43 dyn (1,III) 3,72 k (MLR) 0,0987 ∂log10A/∂k 3,68

Final dynamical values computed are the following: dyn of 4,465mas ±0,604, (A+B)dyn=3,504

±0,228, Adyn=1,836±0,192, Bdyn=1,473±0,097.

HO 137

WDS 20406+2948, BD 29,4131, ADS 14149, HIP 102033, apparent magnitudes 6.13-9.26, Sp A2V,

PM +020+046. No orbit is listed by (Hartkopf and Mason, 2017). 26 observations have been used to

compute the orbit, starting with the discovery and first observation by G.W. Hough in 1885,83, and

last on 2007,7918, : 351°,8, :0”,74, i.e. unpublished measure with a filar micrometer by the

author with a T40,6 Cassegrain reflector (F/D=26,87) with a flat supporting the hyperbolic (Poyet,

2014).

HO 137 apparent ellipse drawn (left) and as produced by the ephemeris in the spreadsheet (right). Scale of

apparent ellipse on the drawing: 1 arcsec = 92,34mm. Axes and pass through C.

27

The orbital elements are as follow:

Orbital elements

P=228,90 a"=0,234 A"=0,0395 C"= 0,1584

n=1,57276 i°=126,32 B"=-0,1679 H"= -0,1026

T=1989,10 60,78 F"=-0,1618 F1"= -0,1581

e=0,213 122,92 G"=-0,1348 G1"= -0,1317

This orbit is pretty odd and should be considered as very preliminary, time will tell. The following

abbreviated ephemeris is derived:

HO 137 - HIP 102033 - [Poy2017]

Year

2017,0 3,54 0,69

2018,0 5,01 0,68

2019,0 6,52 0,67

2020,0 8,07 0,66

2021,0 9,67 0,65

2022,0 11,31 0,64

2023,0 13,00 0,62

2024,0 14,75 0,61

2025,0 16,56 0,59

2026,0 18,45 0,57

All results are summarized in the following table. Spectral type A2 and class V is confirmed by the

position occupied on the CHR diagram (0,310 – 3,09), though the star lies slightly below the main

sequence. In any case, mass of A is in reasonable accordance The mass computed with (9, III), i.e.

1,69 is not too far from the value found in Appendix F for Sp=A2, i.e. 2,19, but is lower than

that given for the main sequence by (Pecker and Schatzman, 1959), i.e. 2,50 (Sp=A2). All

dynamic values are in good accordance and we can wonder from these a significant error affecting

the trigonometric parallax, given as 12,2±0,65mas which leads to totally unrealistic results, i.e.

(A+B)trigo of 14,35. Data from SIMBAD (van Leeuwen, 2007) 13,47±0,46mas only slightly

improve the picture lowering (A+B)trigo at 10,66 in the range [9,64-11,83].

Parallaxes, Masses & Stellar evolution

V Mag 6,02 log (8,III) -1,6618 dyn Iter/Conv. 0,02200 P prob (13,III) 544,53

(B-V) color idx 0,160 (dyn) 0,02179 A by Iter/Conv. 1,64 M* (3,IV) 3,21

mA6,13 log A (9, III) 0,22791 by Iter/Conv. 0,81 V* (4a, IV) 3,46

mB9,26 A dyn 1,69 m by Iter/Conv. 2,45 A from (7,IV) 1,53

Sp A2V log B (10, III) -0,09401 d parsec 45,45 VA (4c, IV) 3,15

CvA-0,25 B dyn 0,81 d light years 148,23 from (5,IV) 0,02540

trigo (mas) 13,47 A dyn+B dyn 2,50 a UA 55,19 CHR diagram "-" ZAMS

trigo (1,III) 10,66 dyn (1,III) 2,52 k (MLR) 0,0987 ∂log10A/∂k 2,39

Final dynamical values computed are the following: dyn of 23,593mas ±2,556, (A+B)dyn=2,487

±0,038, Adyn=1,619±0,083, Bdyn=0,805±0,031.

28

BU 838

WDS 21209+0307, BD 02,4343, ADS 14880, HIP 105399, apparent magnitudes 7.92-10.02, Sp F8,

PM +133+004. There exists an orbit by Zirm, H. (Zir2015a) with a much longer period, i.e. 1280y,

than what is proposed here of 803y. 47 observations have been used by the author to compute the

orbit, starting with the discovery and first observation by W. Burnham in 1881,66 and last used on

2007,627, : 153°,1, :1”,775, i.e. unpublished measure by the author with a T40,6 Cassegrain

reflector (Poyet, 2014), with a flat supporting the hyperbolic (F/D=26,87), a CCD and the Fourier

Transform (FT) of the logarithm of the power spectrum (autocorrelation) as explained by Buil

(1991), then “Reduc” (Losse, 2005) and “Surface” (Morlet and Salaman, 2005) algorithms, then

°= arctg[(x1-x2)/(y1-y2)]*(180/) and =[(x1-x2)2+( y1-y2)2]1/2 /2.

Object 1

BU 838 as imaged by the author on 2007,627 : 153°,07±0,044, :1”,75±0,021

BU 838 apparent ellipse drawn (left) and as produced by the ephemeris in the spreadsheet (right). Scale of

apparent ellipse on the drawing: 1 arcsec = 32,00mm. Axes and pass through C.

The orbital elements are as follow:

Orbital elements

P=802,90 a"=2,095 A"=1,1719 C"= -1,6426

n=0,44837 i°=56,58 B"=-0,5625 H"= 0,5989

T=2414,12 30,85 F"=1,4868 F1"= 1,3438

e=0,428 290,03 G"=1,3485 G1"= 1,2188

29

The following abbreviated ephemeris, with a three year interval as it is a long term period couple, is

derived:

BU 838 - HIP 105399 - [Poy2017]

Year

2016,0 155,29 1,87

2019,0 156,13 1,88

2022,0 156,95 1,89

2025,0 157,77 1,90

2028,0 158,58 1,91

2031,0 159,37 1,93

2034,0 160,16 1,94

2037,0 160,94 1,95

2040,0 161,71 1,96

2043,0 162,47 1,97

Spectral type F8 and class V are confirmed by the position occupied on the CHR diagram (0,510 –

4,56), as the star falls perfectly on the ZAMS line. Latest parallax available from SIMBAD (van

Leeuwen, 2007) is 14,09±0,99 which gives (A+B)trigo=5,10, and that seems way out of the

expected boundaries for such a F8V binary as taking values from Pecker and Schatzman (1959), we

get A=B=1,19, i.e. expected 2,38, the trigonometric parallax giving a value of more than

double. Expected values from Appendix F gives A=B=1,13, i.e. expected 2,26. All

dynamical values computed are very close and coherent, be it using (9, III), (10, III), by iteration/

convergence or using (7, IV).

Parallaxes, Masses & Stellar evolution

V Mag 7,86 log (8,III) -1,7006 dyn Iter/Conv. 0,02003 P prob (13,III) 940,37

(B-V) color idx 0,510 (dyn) 0,01993 A by Iter/Conv. 1,09 M* (3,IV) 4,50

mA7,92 log A (9, III) 0,04619 by Iter/Conv. 0,68 V* (4a, IV) 4,56

mB10,02 A dyn 1,11 m by Iter/Conv. 1,77 A from (7,IV) 1,08

Sp F8V log B (10, III) -0,16817 d parsec 49,92 VA (4c, IV) 4,50

CvA-0,06 B dyn 0,68 d light years 162,80 from (5,IV) 0,02073

trigo (mas) 14,1 A dyn+B dyn 1,79 a UA 105,13 CHR diagram "=" ZAMS

trigo (1,III) 5,10 dyn (1,III) 1,80 k (MLR) 0,0987 ∂log10A/∂k 0,32

Dynamical values computed are the following: dyn of 20,329mas ±0,571, (A+B)dyn=1,789

±0,015, Adyn=1,094±0,018, Bdyn=0,679±0,008. Note that the probable period as determined from

(13, III) of 904,37 is closer to the period of 802,9 y of this orbit than that of (Zir2015a) and that

∂log10A/∂k is small for this spectral type/class and position in the CHR diagram, which comforts

the dynamical results.

STF 2822 AB

WDS 21441+2845, BD 28,4169, ADS 15270, HIP 107310, apparent magnitudes 4.75-6.18, Sp

initially given as F6V+G2V but last available in SIMBAD is F3V+F6V that we will use, PM +260-

243. 326 observations have been used to compute the orbit, from the first starting in 1780,840 made

by W. Herschel, to the last made by the author in 2007,624 with a T62cm at F/D=24,53 (Poyet et

al., 2014) with a CCD and same reduction techniques as shortly mentioned above for BU 828. Only

one observation by C.P. Olivier, has been corrected, as the measure was inverted by 180°, i.e.

1947,840, : 79°,40, : 1”,2. There are 3 known orbits by W.D. Heintz and J.A. Docobo, i.e.

30

Hei1966 (P=507,5 y), Doc1987b (P=713,19 y) and Hei1995 (789 y). The orbit proposed here has a

shorter period than all published orbits, i.e. of 428,0 y. Usually the lack of observations hinders the

drawing of the ellipse, this case is the contrary as too many observations make control of each

difficult and even hide somehow the final ellipse chosen.

STF 2822 AB as imaged by the author with the T62cm (composite of 19 images), unpublished measure as

autocorrelogram of FT(log(power spectrum)) gives 2007,624, : 312°,583±0,151, :1”,806±0,0024

STF 2822 AB apparent ellipse drawn (left) and as produced by the ephemeris in the spreadsheet (right).

Scale of apparent ellipse on the drawing: 1 arcsec = 22,59mm. Axes and pass through C.

The orbital elements are as follow:

Orbital elements

P=428,00 a"=4,635 A"=0,8189 C"= 2,5470

n=0,84112 i°=75,60 B"=-3,7849 H"= -3,6970

T=1974,00 111,93 F"=1,8625 F1"= 1,4830

e=0,605 145,44 G"=-2,0849 G1"= -1,6600

The following abbreviated ephemeris is derived:

STF 2822 AB - HIP 107310 - [Poy2017]

Year

2017 148,18 1,40

2018 150,16 1,37

2019 152,24 1,34

2020 154,42 1,31

2021 156,70 1,28

2022 159,09 1,25

2023 161,58 1,22

2024 164,17 1,20

2025 166,86 1,18

2026 169,65 1,16

Spectral type F3+F6 and class V are confirmed by the position occupied on the CHR diagram

(0,470 – 3,99), as the star falls nearly perfectly on the ZAMS line (just a little bit above). Latest

parallax available from SIMBAD (van Leeuwen, 2007) is 44,97±0,43 and let us believe that high

accuracy has been obtained for this close star. Strangely enough, (A+B)trigo=5,98 and that seems

way out of the expected boundaries for such a F3V+F6V binary as taking values from Pecker and

31

Schatzman (1959), we get A=1,42 and B=1,28 i.e. expected 2,70, the trigonometric

parallax giving a value of more than double. Expected values from Appendix F give A=1,32 and

B=1,22 i.e. expected 2,54. All dynamical values computed are very close and coherent

(9, III), (10, III), by iteration/ convergence or using (7, IV).

Parallaxes, Masses & Stellar evolution

V Mag 4,50 log (8,III) -1,2122 dyn Iter/Conv. 0,06276 P prob (13,III) 506,97

(B-V) color idx 0,470 (dyn) 0,06135 A by Iter/Conv. 1,28 M* (3,IV) 3,95

mA4,75 log A (9, III) 0,13579 by Iter/Conv. 0,92 V* (4a, IV) 3,99

mB6,18 A dyn 1,37 m by Iter/Conv. 2,20 A from (7,IV) 1,25

Sp F3 log B (10, III) -0,03537 d parsec 15,93 VA (4c, IV) 3,83

CvA-0,05 B dyn 0,92 d light years 51,97 from (5,IV) 0,06554

trigo (mas) 44,97 A dyn+B dyn 2,29 a UA 75,55 CHR diagram "≈" ZAMS

trigo (1,III) 5,98 dyn (1,III) 2,35 k (MLR) 0,0987 ∂log10A/∂k 1,49

Dynamical values computed are the following: dyn of 63,445mas ±2,958, (A+B)dyn=2,281

±0,078, Adyn=1,298±0,061, Bdyn=0,922±0,030. Note that the probable period as determined from

(13, III) is closer to the period of 428 y of this orbit than to any other periods of previous orbits

before.

HU 497 BC

WDS 23175+1652, BD 16,4896, ADS 16648, HIP 115002, apparent magnitudes 9.5-10.0, Sp F2III,

PM +008-022. The author computed a first orbit for HU 497 BC in 2007 using the geometrical

method, further to an exchange with Gili (2007) who had provided his latest measures. It happens

that 3 orbits have been published since by Ling (2010) (i.e. IAUDS 161 Lin2007a, Lin2009b,

Lin2010c) using the method and programs by Docobo (1985) the latest orbits being less elliptical

than the first one (e2007a=0,357, e2009b=0,313, e2010c=0,313). As the last one appears still too elliptical

as compared to the one the author computed in 2007 (e=0,178), it seems as a good test to assess the

relevance of geometrical methods as compared to others approaches. The closer BC pair was

distinguished by W. J. Hussey at the Lick Observatory in 1901,78 and has since described an arc of

about 165°, latest observation used was by Gili, R. in 2007,822.

HU 497 BC apparent ellipse (left) and as produced by the ephemeris in the spreadsheet (right). Scale of

apparent ellipse on the drawing: 1 arcsec = 185,71mm. Axes and pass through C and are in that example

close to N and E.

32

It is now approaching its second separation maximum according to Tokovinin (1997, 1999). The

orbital elements are as follow:

Orbital elements

P=310,40 a"=0,344 A"=-0,2100 C"= -0,2481

n=1,15981 i°=131,86 B"=0,1131 H"= -0,0645

T=1974,60 40,41 F"=0,2161 F1"= 0,2127

e=0,178 255,43 G"=0,2599 G1"= 0,2558

The following abbreviated ephemeris is derived:

HU 497 BC - HIP 115002 – [Poy2017]

Year

2017,5 260,27 0,24

2018,5 258,79 0,25

2019,5 257,35 0,25

2020,5 255,94 0,26

2021,5 254,58 0,26

2022,5 253,25 0,26

2023,5 251,95 0,27

2024,5 250,69 0,27

2025,5 249,46 0,27

2026,5 248,26 0,28

As suggested by spectral type F2III, at least A is a giant. From what was developed in section IV

one knows that the MLR might not be followed well to compute satisfactorily dynamic parallax and

masses. The giant status is well confirmed by the position of the star in the (B-V,V*) CHR diagram

(0,494 – 3,27) and we get a striking difference between the =0,78 computed with (1, III) as based

on trigonometric parallax and =2,77 as determined by the dynamic parallax (convergence

algorithm) or =3,11 based on (1, III) and the dynamic parallax. Moreover, given the significant

uncertainty affecting the trigonometric parallax, i.e. 8,14±2,28mas, the possible range for the total

mass of the system, at a 68,2% confidence interval (p±error-in-), is as vague as [0,37-2,10]. In

any case, as this star does not obey well the MLR (i.e. it is a giant), one should be cautious reading

the dynamic results listed in the following table.

Parallaxes, Masses & Stellar evolution

V Mag 8,77 log (8,III) -2,2886 dyn Iter/Conv. 0,00535 P prob (13,III) 230,17

(B-V) color idx 0,494 (dyn) 0,00514 A by Iter/Conv. 1,57 M* (3,IV) 3,22

mA9,19 log A (9, III) 0,24608 by Iter/Conv. 1,20 V* (4a, IV) 3,27

mB10,38 A dyn 1,76 m by Iter/Conv. 2,77 A from (7,IV) 1,53

Sp F2III log B (10, III) 0,07868 d parsec 187,08 VA (4c, IV) 2,96

CvA-0,05 B dyn 1,20 d light years 610,17 from (5,IV) 0,00567

trigo (mas) 6,74 A dyn+B dyn 2,96 a UA 66,89 CHR diagram "++" ZAMS

trigo (1,III) 1,38 dyn (1,III) 3,11 k (MLR) 0,0987 ∂log10A/∂k 2,86

Dynamical values computed are the following: dyn of 5,408mas ±0,373, (A+B)dyn=2,946

±0,169, Adyn=1,620±0,125, Bdyn=1,199±0,063.

33

VI. Conclusion

The geometrical method has been used in this paper to compute the orbits of 13 double stars, 9 of

them being first-time orbits of medium to long term periods’ systems. Two observations can be

made: one dealing with the parallaxes, the other with the influence of the factor k of the Mass-

Luminosity Relation. The next Table present a summary of the various dynamic parallaxes

computed in this paper and their corresponding trigonometric parallaxes as found in SIMBAD:

Object dyn trigo ±e-trigo (dyn-trigo)in % dyn

RST 3340 9,222

HU 616 3,354 2,57 0,61 0,784 23,38%

HU 841 6,367 7,54 1,44 1,173 18,42%

A 552 5,982 5,69 0,93 0,292 4,88%

A 2554 14,628 14,79 1,02 0,162 1,11%

BU 796 14,257 7,21 0,45 7,047 49,43%

RST 4502 4,039 3,76 0,61 0,279 6,91%

A 1097 AB 10,617 6,82 0,69 3,797 35,76%

HU 337 4,465 4,15 1,18 0,315 7,05%

HO 137 23,593 13,47 0,46 10,123 42,91%

BU 838 20,329 14,09 0,99 6,239 30,69%

STF 2822 AB 63,445 44,97 0,43 18,475 29,12%

HU 497 BC 5,408 6,74 1,45 1,332 24,63%

Mean of in %: 22,86%

± 15,66%

dyn minus trigo expressed in % of dyn and corresponding ±

Their differences, i.e. dyn minus trigo, have been expressed in % of dyn . One will notice that four

stars have a good match whereas the remaining show very large differences that lead to compute

values of with (1, III) and trigo, which are well outside a reasonable range of acceptable masses

for the systems. A standard deviation of ±15,66% of these hides the fact that we have two groups:

one for which the figures are acceptable, the other delivering wild results, completely outside the

expected range of reasonable values for the corresponding spectral types (all are main sequence

stars, except HU 497 BC). This is a troublesome conclusion as neither the announced ±e-trigo nor

any other information can be used to judge a priori of the appropriateness of these trigo, for the

computation of reliable s.

Object Sp B-V ∂log10A/∂k

RST 3340 G3V 0,560 0,08

HU 616 F0V 0,230 3,63

HU 841 F8V 0,490 2,42

A 552 A2V 0,170 4,63

A 2554 A9V 0,345 2,04

BU 796 F2V 0,400 1,38

RST 4502 A5V 0,260 4,76

A 1097 AB F5V 0,461 2,41

HU 337 A3V 0,310 3,68

HO 137 A0 0,160 2,39

BU 838 F8 0,510 0,32

STF 2822 ABF5 0,470 1,49

HU 497 BC F2III 0,494 2,86

Correlation -0,73

Negative correlation of ((B-V), ∂log10A/∂k)

34

The second observation deals with the factor k of the Mass-Luminosity Relation (MLR). Obtaining

reliable masses by the dynamic parallax method supposes that the MLR be followed by the systems

studied, which is considered to be the case for ZAMS binaries, and in that case ∂log10A/∂k should

tend to zero or at least be small. Positioning the binaries studied in the CHR-diagram has confirmed

that only one giant belong to this set, i.e. HU 497 BC.

Even though the residuals of the partial derivative ∂log10A/∂k were small, they did not really tend

towards zero and obviously presented some correlation with the value of (B-V) as the Table above

shows, the negative correlation of ((B-V), ∂log10A/∂k) amounts to –0,73 which is very significant.

The higher the value of (B-V) the lower the influence of k on the computation of the masses. The

sample is small and goes from G3V to A0V, but it would be interesting to observe how the

relationship evolves on a larger sample and on stars in the G4V to M4V range and to perform a

sensitivity study to varying values of k.

VII. Discussion

Unless a complete revolution has been made and faultless data are available, an orbit always

remains a choice trying to satisfy competing and even sometimes conflicting objectives. Fitting best

the ellipse to the observations does not always lead to a satisfactory solution, i.e. one that best

satisfies the law of the areas as we have stressed and others also did long before us, e.g. (Danjon,

1952), (Couteau, 1978). This clearly means that in such a case, where a close system can lead to

significant deviations, particularly in , a solution that satisfies better the 2nd Kepler’s law at the

increase of the residuals (observed minus calculated, i.e. O-C) is preferable.

Minimizing the residuals is a straightforward means for the software developer to quantify in some

way a solution delivered by a computer programme, but as we have reminded it should only be used

after that a consistent proposal be obtained with respect to the law of the areas. Most of the time,

though, the areal constant c is not as constant as one would hope and a significant dispersion is

shown across the various sectors, as they are progressively tested, increasing their size and variety.

Once a given c is adopted, eventually after that competing orbits be tested, using a technique such

as described for RST 3340 and minimizing the standard deviation across sectors, ci, for these

several potential orbits, a period is immediately obtained P=S/c and should be checked with

common sense, against the observations whenever possible. When a value of T has been obtained,

i.e. passage at periastron, from an average of several measures, one should be ready to eventually

adjust it slightly (e.g. say max up to 1,5% of P) if this later improves the ephemeris and enable to

reduce the residuals, especially in , as they are the most significant. Even though the calculations

deliver perfect solutions in mathematical terms, once the orbital elements are chosen, one should

remind that we start from a set of observations, often made to the limits of the instruments used by

skilled observers doing their best fighting variable seeing, disturbing the measures. Of course,

modern detectors and all sorts of CCDs and speckle techniques deliver more objective data, e.g.

(Tokovinin et al., 2014), (Tokovinin, 2016, 2017), but it will be long before we can compute orbits

with intermediate periods [70-200+y] that do not rely at all on old but precious measures made

sometimes more than a century ago, most of the time with a filar micrometer by a skilled observer,

or even more often for close binaries, being simply estimated at the limit of the diffraction image

observed. At the time, each astronomer who populated the files used today to compute orbits had

spent more than a year learning the aspect of binaries at various magnifications when they were

closer than the diffraction limit, i.e. 0,85*14/D (D diameter in cm of the objective). It is amazing to

notice that many exceptional observers have estimated correctly the aspect of the binaries below

50% of the resolving power of their instrument. This can be asserted from the comparison of their

measures to the currently known (good grade) orbits, e.g. Schiaparelli estimating with the 50cm

35

refractor 1728 in 1898,461 at (190°/0,”15) when the orbit gives (191°/0,”13), Baize delivering

consistent measures down to 0,”18 with a 38cm, e.g. Leo estimated in 1956,31 at (224°,1/0,”19)

when the orbit gives (223°,2/0,”18). Still, from time to time various error factors happen, including

but not limited to inversion of measures when m is small (i.e. contradictory quadrants), existence

of unknown 3rd bodies (Genet et al., 2015) and computing an orbit requires to go back to how the

measures were made, to who did them, and whenever possible to go back to the notes

accompanying the measures or the estimations when the values delivered are well below the

resolution power.

Even though large telescopes and speckle techniques provide with higher quality measures, or even

old refractors with latest cameras, e.g. EM-CCD (Gili et al., 2014), (Mason, 2017), open new

horizons to fainter systems with more objective and impersonal results, our bulk of data will long

remain dependent on the pioneers who made double star observing a exciting discipline. In that

sense, this paper suggests that computing orbits could somehow remain a craft work for

intermediate to long term period’s orbits, as it will be difficult for computer programmes to get a

feel of all what plays a role in opting for a final orbital solution. Bringing the art and science of

calculating orbits together, whatever the progress made, seems probably to remain as actual than

ever.

Acknowledgements

This article is a modest tribute to Dr. Paul Couteau (1923-2014), who gave me access to the large refractor in

Nice (74cm) in 1979 when I was so young and afforded me support and hospitality at the Nice Observatory.

The author would also like to express his kindest thanks to Jean-Pierre Vida and Joseph Abad who made the

construction of the facilities used at the “Jas de Tardivy Observatory” - Caussols (close to the OCA- Calern

observatory), i.e. the Mosser (43.748825 – 6.939806) and Cardoen (43.747978 – 6.936663) telescopes

possible. Thanks also go to René Gili who over the nearly 40 years I have been knowing him has always

been welcoming and encouraging by his absolute determination and calm steadfastness. I also thank my wife

Béatrice for her patience to bear with me and holding the pins to help me draw the orbits. This research has

made use of the Washington Double Star Catalog and of the Sixth Catalog of Orbits of Visual Binary Stars

maintained at the U.S. Naval Observatory and the author is grateful for their availability. An extensive use of

SIDONIe (Le Contel et al., 2017) has also been made for 10 years and the author is very grateful for this

extended web version of the original Couteau’s files. This work also used the SIMBAD service operated by

Centre des Données Stellaires (Strasbourg, France) and made an extensive use of the bibliographic

Astrophysics Data System maintained by SAO/NASA

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39

Appendix A : on calculating sound orbits

in “Computing a double star orbital elements” (Couteau, 1978), p.119: The difficulty comes from

measurements errors. Only a well trained observer, knowing the history of the observations, is able

to find reasonable orbital elements from disparate measurements, too rare at some time and too

numerous at others… But one needs years of experience and practice before being able to judge one

or another observation of a given binary. Modern numerical processing methods are not suited to

such a job. Computing an orbit will ever remain a craftsman job.

in “Calcul des éléments d’une orbite d’étoile double” (Couteau, 1978, p.119): La difficulté vient des

erreurs de mesures. Seul un observateur bien entraîné, connaissant l’histoire des observations, est à

même de trouver des éléments à peu près corrects à partir de mesures disparates, trop rares à

certaines époques, surabondantes à d’autres… Mais il faut des années de pratique avant de savoir

juger telle ou telle observation de tel couple. Les méthodes modernes de traitement numérique ne

sont guère exploitables. Un calcul d’orbite d’étoile double restera toujours un travail d’artisan »

in « Differential correction of the elements » (Couteau, 1978), p.145 : The sort and importance of

the deviations depend on the difficulty of the binary… This is one of the reasons why computing

orbits should only be performed by regular observers, knowing well the difficulties of the

measurements.

in « Correction différentielles des éléments » (Couteau, 1978), p.145 : L’allure et l’importance des

écarts dépendent de la difficulté de la binaire… C’est l’une des raisons pour lesquelles les calculs

d’orbites ne devraient être entrepris que par des observateurs réguliers, connaissant bien les

difficultés des mesures.

in “Is This Orbit Really Necessary?” Worley (1990) laments the tendency to publish orbits of

double stars that are neither reliable nor useful and imply that if more time were spent observing

and less time computing, we would all be better off.

Appendix B : Correction Cv to mv as a function of Sp type

Table Cv Harris III (1963) et Baize (1943) for M types

Sp Cv Sp Cv Sp Cv Sp Cv

B5 -1,39A1 -0,32F5 -0,04K3 -0,25

B6 -1,21A2 -0,25G0 -0,06K5 -0,71

B7 -1,04A3 -0,20G5 -0,10K7 -1,02

B8 -0,85A5 -0,15G8 -0,15M0 -1,22

B9 -0,66A7 -0,12K0 -0,19M2 -1,43

A0 -0,40F0 -0,08K2 -0,25M4 -1,67

40

Appendix C : and M as a function of Spectral Type Sp

Sp M Sp M

B7 -0,60 3,98 G1 4,50 1,70

B8 -0,30 3,68 G2 4,70 1,20

B9 0,00 3,41 G3 4,80 0,99

A0 0,30 3,16 G4 5,00 0,94

A1 0,90 2,71 G5 5,10 0,92

A2 1,30 2,50 G6 5,30 0,87

A3 1,70 2,15 G7 5,50 0,83

A4 2,00 2,04 G8 5,60 0,81

A5 2,20 1,94 G9 5,80 0,77

A6 2,40 1,84 K0 6,00 0,73

A7 2,60 1,75 K1 6,20 0,69

A8 2,80 1,66 K2 6,50 0,66

A9 2,90 1,62 K3 6,90 0,58

F0 3,10 1,58 K4 7,20 0,54

F1 3,20 1,50 K5 7,50 0,46

F2 3,30 1,46 K6 7,90 0,45

F3 3,40 1,42 K7 8,20 0,41

F4 3,50 1,39 K8 8,50 0,38

F5 3,60 1,35 K9 8,90 0,35

F6 3,80 1,28 M0 9,20 0,32

F7 3,90 1,25 M1 9,70 0,28

F8 4,10 1,19 M2 10,10 0,25

F9 4,20 1,16 M3 10,60 0,22

G0 4,40 1,10 M4 11,30 0,19

Mass according to Baize and Romani (1946) and absolute bolometric magnitude M as a function

of Spectral Type Sp, data compiled for the main sequence from Pecker and Schatzman (1959).

Appendix D : Kepler’s Law

Starting from the polar equation to an ellipse, and thus equation (7, II)

du/dt = / (1 – e . cos u)

dt = du . (1 – e . cos u)

dt = du – e . cos u du

thus by trivial integration :

∫ dt = ∫ du – ∫ e . cos u du

∫ dt = ∫ u – e . ∫ cos u du

∫ dt = u – e . sin u

(t – T) = u – e . sin u

M = u – e . sin u ; equation known as Kepler’s Law.

41

Appendix E : Keplerian’s Motion

Compute any value of u (eccentric anomaly) for any given e and M

(input cells in yellow)

e 0,35

20M 0,34906585

u0 0,4687729 u0=M+e sinM

u1 0,50719302 u1=M+e sin u0

u2 0,51906979 u2=M+e sin u1

u3 0,522691 0,735810 42,158823

u4 0,523791 0,737288 42,243504

u5 0,524124 0,737736 42,269172

u6 0,524225 30,035882 0,737872 42,276951

u7 0,524256 30,037636 0,737913 42,279307

u8 0,524265 30,038167 0,737926 42,280022

u9 0,524268 30,038328 0,737929 42,280238

u10 0,524269 30,038377 0,737931 42,280304

u11 0,524269 30,038391 0,737931 42,280323

u12 0,524269 30,038396 0,737931 42,280329

u13 0,524269 30,038397 0,737931 42,280331

u14 0,524269 30,038398 0,737931 42,280332

u15 0,524269 30,038398 0,737931 42,280332

u16 0,524269 30,038398 0,737931 42,280332

u17 0,524269 30,038398 0,737931 42,280332

u18 0,524269 30,038398 0,737931 42,280332

u19 0,524269 30,038398 0,737931 42,280332

u20 0,524269 30,038398 0,737931 42,280332

u21 0,524269 30,038398 0,737931 42,280332

u22 0,524269 30,038398 0,737931 42,280332

u23 0,524269 30,038398 0,737931 42,280332

u24 0,524269 30,038398 0,737931 42,280332

u25 0,524269 30,038398 0,737931 42,280332

u26 0,524269 30,038398 0,737931 42,280332

u27 0,524269 30,038398 0,737931 42,280332

u28 0,524269 30,038398 0,737931 42,280332

u29 0,524269 30,038398 0,737931 42,280332

un 0,524269 30,038398 0,737931 42,280332

u rad u deg v rad v deg

v = 2 arctg ( sqrt( (1+e)/(1-e) ) * tg u/2 )

42

Appendix F : (B-V), V, Cb, Mb, , V* as a function of Sp V

Sp Temp (B-V) V Cb Mb1log V*3

B7 14500 -0,12 -0,11 -1,04 -1,15 0,584 3,84 0,86

B8 13400 -0,09 0,22 -0,85 -0,63 0,533 3,41 1,11

B9 12400 -0,06 0,55 -0,66 -0,11 0,482 3,03 1,35

A0 10800 0,00 1,10 -0,40 0,70 0,402 2,52 1,77

A1 10200 0,03 1,26 -0,32 0,94 0,378 2,39 1,89

A2 9730 0,06 1,57 -0,25 1,32 0,341 2,19 2,14

A3 9260 0,09 1,75 -0,20 1,55 0,318 2,08 2,28

A4 8940 0,12 1,93 -0,18 1,75 0,298 1,99 2,42

A5 8620 0,15 2,10 -0,15 1,95 0,278 1,90 2,56

A6 8405 0,18 2,22 -0,14 2,09 0,265 1,84 2,66

A7 8190 0,20 2,34 -0,12 2,22 0,252 1,79 2,76

A8 7795 0,24 2,57 -0,11 2,46 0,223 1,67 2,94

A9 7481 0,28 2,82 -0,09 2,73 0,197 1,57 3,15

F0 7240 0,33 3,10 -0,08 3,02 0,173 1,49 3,39

F1 7085 0,36 3,26 -0,07 3,19 0,157 1,43 3,52

F2 6930 0,38 3,41 -0,06 3,35 0,140 1,38 3,64

F3 6732 0,40 3,51 -0,05 3,46 0,122 1,32 3,71

F4 6603 0,42 3,65 -0,05 3,60 0,108 1,28 3,83

F5 6540 0,45 3,82 -0,04 3,78 0,098 1,25 3,98

F6 6540 0,47 3,94 -0,04 3,90 0,086 1,22 4,08

F7 6320 0,50 4,11 -0,04 4,07 0,069 1,17 4,23

F8 6200 0,53 4,28 -0,05 4,23 0,053 1,13 4,37

F9 6060 0,57 4,47 -0,06 4,42 0,035 1,08 4,53

G0 5920 0,60 4,66 -0,06 4,60 0,017 1,04 4,69

G1 5850 0,62 4,77 -0,07 4,70 0,007 1,02 4,78

G2 5780 0,64 4,87 -0,07 4,80 -0,003 0,99 4,87

G3 5666 0,65 4,89 -0,08 4,81 -0,014 0,97 4,87

G4 5610 0,66 4,97 -0,09 4,88 -0,020 0,96 4,93

G5 5610 0,68 5,09 -0,10 4,99 -0,022 0,95 5,05

G6 5514 0,69 5,12 -0,12 5,00 -0,032 0,93 5,06

G7 5475 0,70 5,20 -0,13 5,07 -0,038 0,92 5,13

G8 5490 0,72 5,33 -0,15 5,18 -0,040 0,91 5,26

G9 5365 0,77 5,64 -0,17 5,47 -0,068 0,86 5,52

K0 5240 0,81 5,94 -0,19 5,75 -0,097 0,80 5,78

K1 5010 0,87 6,17 -0,22 5,95 -0,116 0,77 5,98

K2 4780 0,92 6,40 -0,25 6,15 -0,136 0,73 6,17

K3 4590 0,98 6,70 -0,35 6,35 -0,156 0,70 6,44

K4 4280 1,08 7,13 -0,53 6,60 -0,180 0,66 6,83

K5 3970 1,18 7,56 -0,71 6,85 -0,205 0,62 7,22

K6 3769 1,24 7,94 -0,75 7,20 -0,228 0,59 7,58

K7 3631 1,29 8,23 -0,77 7,46 -0,244 0,57 7,86

K8 3503 1,33 8,52 -0,80 7,72 -0,259 0,55 8,14

K9 3346 1,39 8,91 -0,84 8,07 -0,279 0,53 8,50

M0 3236 1,44 9,20 -0,86 8,33 -0,294 0,51 8,78

M1 3070 1,51 9,67 -0,91 8,76 -0,317 0,48 9,23

M2 2948 1,57 10,05 -0,94 9,11 -0,334 0,46 9,60

M3 2809 1,64 10,53 -0,99 9,54 -0,355 0,44 10,05

M4 2635 1,75 11,18 -1,05 10,13 -0,383 0,41 10,68

1 Absolute bolometric magnitude Mb=V+Cb

2 Unit in Solar Mass

3 Reduced absolute magnitude V*=V+(5/3) . log10

43