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MNRAS 514, 4629–4638 (2022) https://doi.org/10.1093/mnras/stac1112

Testing the third-body hypothesis in the cataclysmic variables LU

Camelopardalis, QZ Serpentis, V1007 Herculis and BK Lyncis

Carlos E. Chavez,

1 ‹Nikolaos Georgakarakos,

2 , 3 Andres Aviles,

4 Hector Aceves,

5 Gagik Tovmassian ,

5

Serge y Zhariko v,

5 J. E. Perez–Leon

4 and Francisco Tamayo

4

1

Universidad Auton

´

oma de Nuevo Le

´

on, Facultad de Ingenier

´

ıa Mec

´

anica y El

´

ectrica, San Nicol

´

as de los Garza 66451, NL, Mexico

2

New Yor k University Abu Dhabi, PO Box 129188, Saadiyat Island, Abu Dhabi, UAE

3

Center for Astro, Particle and Planetary Physics (CAP3), New Yo rk University Abu Dhabi, PO Box 129188, Saadiyat Island, Abu Dhabi, UAE

4

Universidad Auton

´

oma de Nuevo Le

´

on, Facultad de Ciencias F

´

ısico–Matem

´

aticas, San Nicol

´

as de los Garza 66451, NL, Mexico

5

Universidad Nacional Aut

´

onoma de M

´

exico, Instituto de Astronom

´

ıa, Ensenada 22860, BC, Mexico

Accepted 2022 April 11. Received 2022 April 5; in original form 2022 January 5

A B S T R A C T

Some cataclysmic variables (CVs) exhibit a very long photometric period (VLPP). We calculate the properties of a hypothetical

third body, initially assumed to be on a circular–planar orbit, by matching the modelled VLPP to the observed one of four CVs

studied here: LU Camelopardalis, QZ Serpentis, V1007 Herculis and BK Lyncis. The eccentric and low inclination orbits for

a third body are considered using analytical results. The chosen parameters of the binary components are based on the orbital

period of each CV. We also calculate the smallest corresponding semimajor axis permitted before the third body’s orbit becomes

unstable. A ﬁrst-order analytical post-Newtonian correction is applied, and the rate of precession of the pericentre is found,

but it cannot explain any of the observed VLPP. For the ﬁrst time, we also estimate the effect of secular perturbations by this

hypothetical third body on the mass transfer rate of such CVs. We made sure that the observed and calculated amplitude of

variability was also comparable. The mass of the third body satisfying all constraints ranges from 0.63 to 97 Jupiter masses.

Our results show further evidence supporting the hypothesis of a third body in three of these CVs, but only marginally in V1007

Herculis.

Key words: planets and satellites: dynamical evolution and stability – stars: individual: LU Camelopardalis –stars: individual:

QZ Serpentis –stars: individual: V1007 Herculis – stars: individual: BK Lyncis –nov ae, cataclysmic v ariables.

1 INTRODUCTION

A cataclysmic variable (CV) is a semidetached binary star system that

is particularly stable (Frank, King & Raine 2002 ). A CV consists of

a white dwarf (WD) primary star and a lower-mass main-sequence

secondary star, mainly an M star, although the spectral class can

range from a K to an L type star. The condition that deﬁnes the

distance between the two components is that the main-sequence star

ﬁlls its corresponding Roche lobe and loses matter through the L

1

Lagrangian point. The matter accretes on to the WD via an accretion

disc, unless the WD has a strong enough magnetic ﬁeld to prevent

the formation of the disc. If the system is disturbed for any reason, it

tends to restore to equilibrium.

Two mechanisms maintain the balance of a CV: the evolutionary

expansion of the secondary, and the decrease of the semimajor axis

of the binary due to the loss of angular momentum. The decrease

in angular momentum has two possible sources, depending on the

orbital period of the binary system. The ﬁrst source is magnetic

braking, and it is the dominant effect for systems that have orbital

periods abo v e 3 h (Whyte & Eggleton 1985 ; Livio & Pringle

1994 ). The second source is the emission of gravitational waves, for

E-mail: Carlos.ChavezPch@uanl.edu.mx

systems that have orbital periods below 2 h (Faulkner 1976 ; Chau &

Lauterborn 1977 ). In the 2–3 h period range, neither mechanism is

efﬁcient for the angular momentum removal. Hence, the number of

known systems in that range, called the period gap, is signiﬁcantly

smaller.

The material that the secondary loses through the L

1

point cannot

fall straight to the primary; instead, it forms an accretion disc (Frank

et al. 2002 ; Ritter 2008 ). This accretion disc is so luminous that

it outshines both stars. The disc brightness is proportional to the

mass transfer rate. Therefore, if the mass transfer rate changes, the

luminosity of the system changes. In particular, a change in the

location of the L

1

point will change the mass transfer rate and, as a

consequence, the luminosity of the whole system will change.

CVs are notorious for variability on different time-scales and

magnitude scales. In this paper, we consider a speciﬁc case: a

relati vely lo w amplitude (0.07–0.97 mag) v ariability with periods

exceeding the orbital periods by hundreds to thousands of times. The

very long photometric period (VLPP) was ﬁrst singled out in FS

Aurigae (Chavez et al. 2012 , 2020 ), although the object shows many

other variabilities. More VLPPs have been identiﬁed in other CVs

(e.g. Thomas et al. 2010 ; Kalomeni 2012 ; Chavez et al. 2012 ; Yan g

et al. 2017 ; Chavez et al. 2020 ).

Different mechanisms have been proposed to explain VLPPs.

Thomas et al. ( 2010 ) found a long-term modulation with a period

© 2022 The Author(s)

Published by Oxford University Press on behalf of Royal Astronomical Society

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4630 C. E. Chavez et al.

MNRAS 514, 4629–4638 (2022)

of 4.43 d in the CV PX Andromedae, using eclipse analysis, and

proposed the disc precession period as the origin of the VLPP.

Another example is the DP Leonis system; for this, Beuermann et al.

( 2011 ) found a period of 2.8 yr, using eclipse time variations, and

concluded that a third body was the best explanation for the VLPP.

Honeycutt, Kafka & Robertson ( 2014 ) found a 25-d periodicity in

V794 Aquilae. Kalomeni ( 2012 ) disco v ered sev eral magnetic CVs

that have long-term variability, with a time-scale of hundreds of

days, and concluded that those VLPPs are likely to originate from

the modulation of mass transfer due to the magnetic cycles in the

companion star.

More recently, Chavez et al. ( 2012 , 2020 ), using dynamical

analysis, proposed that a third body can induce a VLPP by secular

perturbations on the inner binary. The third body can introduce

oscillations of the L

1 point of the close binary, and therefore the

mass transfer rate changes. This mechanism induces periods as long

as the VLPPs in the inner binary by means of secular perturbations

observed in the above-mentioned CVs. The VLPP was observed in

the long-term light curves of 10 CVs by Ya ng et al. ( 2017 ). As a

possible mechanism of the VLPP for ﬁve out of 10 systems, these

authors proposed a third body orbiting the close binary, with the

system being in Kozai–Lidov resonance (Kozai 1962 ; Lidov 1962 )

which requires an orbital inclination between the plane of the binary

and the orbit of the third object larger than 39 .

◦2. Thus, they were

able to estimate the possible orbital period of the third body.

Our main goal in this research is to investigate whether a third body

can explain the observed VLPP of four CVs, rather than obtaining a

precise value on the mass of the third body.

This paper is organized as follows. In Section 2 , we provide

information about the CVs considered in this work, and their initial

parameters. In Section 3 , we give the properties of the third body that

result from our analysis, in order to explain the observed VLPPs. In

Section 4 , we brieﬂy address the potential role of a post-Newtonian

correction on the VLPPs. In Section 5 , we address the effects of a

probable third body on the mass transfer rate and brightness of the

four CVs. In Section 6 , we present our results and a discussion, and

we provide ﬁnal comments on this work in Section 7 .

2 THE CATACLYSMIC VARIABLES STUDIED

AND THEIR INITIAL PARAMETERS

Yang et al. ( 2017 ) matched 344 out of 1580 known CVs,

and extracted their data from the Palomar Transient Factory

(PTF) data repository. These images were combined with

the Catalina Real-Time Transit Surv e y (CRTS) light curves.

They found 10 systems with unknown VLPPs: BK Lyncis

(BK Lyn; 2MASS J09201119 + 3356423), CT Bootis, LU

Camelopardalis (LU Cam; 2MASS J05581789 + 6753459),

QZ Serpentis (QZ Ser; SDSS J15565447 + 210719.0), V825

Herculis (2MASS J17183699 + 4115511), V1007 Herculis

(V1007 Her; 1RXS J172405.7 + 411402), Ursa Majoris 01

(UMa 01; 2MASS J09193569 + 5028261), Coronae Borealis 06

(2MASS J15321369 + 3701046), Herculis 12 (Her 12; SDSS

J155037.27 + 405440.0) and VW Coronae Borealis (USNO-B1.0

1231 −00276740). They analyse each system and, depending on

the value of its VLPP, propose a most likely origin, such as the

precession of the accretion disc, hierarchical three-body systems and

magnetic ﬁeld change of the companion star. They argue that if the

long-term period is less than several tens of days, the disc precession

explanation is preferred. Ho we ver, the hierarchical three-body

system or the variations in the magnetic ﬁeld are fa v oured for longer

periods. Six out of those 10 systems they propose to be a hierarchical

triple: BK Lyn, LU Cam, QZ Ser, V1007 Her, Her 12 and UMa 01.

UMa 01 has a long orbital period of P

1 = 404.10 ±0.30 min

(6.735 h), which is long compared with other systems in the sample.

According to the orbital period distribution for CVs, the number of

systems with such a period or larger is very small and therefore most

of the statistical results cannot be applied to them (Knigge 2006 ;

Knigge, Baraffe & Patterson 2011 ). UMa 01 has been presumably

formed recently and there are no good enough estimates of its

parameters (e.g. mass, radius, temperature) of either component.

Additionally, Her 12 was identiﬁed as a CV by Adelman-McCarthy

et al. ( 2006 ), but we do not model it because its period is not well

constrained, possibly being in the range P

1 = 76–174 min (Yang

et al. 2017 ). We study each of the four systems remaining (i.e. LU

Cam, QZ Ser, V1007 Her and BK Lyn) to learn more about their

dynamical attributes.

The values reported by Knigge et al. ( 2011 ) are used for calculating

the mass of each member of the CV. We did so because Knigge et al.

give all the parameters that we later use in this research, such as mass,

radius and semimajor axis, for each component of the binary. In their

study, they used eclipsing CVs and theoretical restrictions to obtain

the semi-empirical donor sequence for CVs with orbital periods P

1

< 6 h. They estimate all key physical, photometric and spectral-type

parameters of the secondary and primary as a function of the orbital

period. We use the data from their tables 6 and 8 (Knigge et al. 2011 )

to obtain the parameters of the CVs in our selection. In practice, we

use the online version of those tables (which are far more complete)

to obtain the adequate values for the CVs studied here. If the systems

hav e an y peculiar features, we will point it out in the te xt and we will

state the reference used for such a value.

2.1 White dwarf mass

First, we brieﬂy describe the mass value used by Knigge et al. ( 2011 ).

In their research, they explain that they used the mean value of

M

1

= 0 . 75 ±0 . 05 M

. In 2011, the new data pointed to a mean

value for the WD in CVs of M

1

= 0 . 79 ±0 . 05 M

. They stated

that because they had already begun to assemble the grid of donors

sequence and evolution tracks ‘we chose to retain M

1

= 0 . 75 ±

0 . 05 M

as a representative of WD mass’. More recently, in a re vie w

by Zorotovic & Schreiber ( 2020 ), it was reported that the mean

WD value could be even higher, M

1

= 0 . 82 –0 . 83 ±0 . 05 M

. We

decided to use the Knigge et al. ( 2011 ) values for all parameters

of the WD to be consistent throughout the paper, and also because

the y pro vide estimates for M

1 and R

1 (the WD’s mass and radius)

corresponding to the orbital period of each CV (both values are

necessary for the calculations in the following sections).

To understand how this affects the calculations, we w ould lik e to

point out that in Chavez et al. ( 2012 ), the calculations were done

with M

1

= 0 . 7 M

, which we then updated to M

1

= 0 . 75 M

(a

change of 7 per cent) in Chavez et al. ( 2020 ). The minimum in the

middle panel of ﬁg. 8 of Chavez et al. ( 2012 ) –the semimajor axis

versus the mass of the third body –has a value of M

3

= 50 M

J

, while

when M

1

= 0 . 75 M

is used (see ﬁg. 3 of Chavez et al. 2020 ) the

minimum corresponds to M

3

= 30 M

J

. This is a 40 per cent decrease

of the mass of the third body at the minimum.

2.2 LU Camelopardalis

LU Cam is a dwarf nova CV and the ﬁrst spectrum of this system was

obtained by Jiang et al. ( 2000 ). Its orbital period was ﬁrst reported by

Sheets et al. ( 2007 ) to be P

1

= 0.1499686(7) d = 3.599246 h. They

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Testing the third-body hypothesis in four CVs 4631

MNRAS 514, 4629–4638 (2022)

Figure 1. Results for the LU Cam system. The numerical integrations

performed are represented by black points. To p panel: period of the long-term

modulation (secular period) as a function of the third-body mass. Each blue

curve joining black points correspond to different P

2

/ P

1

ratios. The black line

around 2.4 corresponds to the observed VLPP. Only numerical integrations

that can explain the observed VLPP are shown (middle panel). The blue curve

corresponds to the planar and circular planar analytical solution. The green

line represents the analytical planar systems with an eccentricity of 0.2 and the

red line represents the planar systems with an eccentricity of 0.5. The doted

line represents the inner stability limit calculated by Georgakarakos ( 2013 )

and the grey solid line denotes that of Holman & Wieger t ( 1999 ). Bottom

panel: similar quantities as in the middle panel, but for a circular orbit with

different inclinations. The third-body values consistent with observations

obtained here are M

3

= 97 M

J

and P

2

= 1.06 d.

point out that the averaged spectrum shows a strong blue continuum.

Yang et al. ( 2017 ) report a VLPP of 265.76 d and point out that

the hierarchical triple system explanation is their best candidate to

explain it.

Using data from Knigge et al. ( 2011 ), we obtain M

1

= 0 . 75 M

,

M

2

= 0 . 26 M

. We show all the parameters of the system in Ta ble 1 .

2.3 QZ Serpentis

QZ Ser is a system that has been classiﬁed as a dwarf nova. The

system has an orbital period of P

1

= 119.752(2) min = 1.99584 h,

according to Thorstensen et al. ( 2002a ). They found that the system

Figure 2. Results for the QZ Ser system. In this system, the third body is

found to have a mass M

3

= 0.63 M

J

and P

2

= 1.04 d.

is not a usual CV, as it is one of a few objects known with a short

orbital period and a secondary non-standard K-type star. This K-type

secondary has a much smaller mass than a usual K star because

of unstable thermal-scale mass transfer evolution. There are other

examples of this type of CV. For instance, Thorstensen et al. ( 2002b )

found a K4 in the dwarf nova 1RXS J232953.9 + 062814, while

Ashley et al. ( 2020 ) found a K5 around a CV with a period of

4.99 h.

Thorstensen et al. ( 2002a ) used evolutionary models to estimate

the parametersof QZ Ser, such as M

2

= 0 . 125 ±0 . 025 M

, which

yielded R

2

= 0 . 185 ±0 . 013 R

, where R

2

is the secondary’s radius.

They also used a typical WD mass value of M

1

= 0 . 7 M

, widely

used in 2002 (Jiang et al. 2000 ; Thorstensen et al. 2002a ).

Thorstensen et al. ( 2002a ) estimated from observations of the

ellipsoidal variations that the inclination (with respect to sky’s plane)

of the system must be i = 33 .

◦7 ±4

◦. They decided to use this estimate

to constrain the secondary’s mass, and then proceeded to check mass

ratios between the primary and the secondary between 0.1 and 0.4

for this system. Using the secondary’s velocity amplitude, they give a

mass function of f = 0 . 075(5) M

. The inclination can be calculated

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4632 C. E. Chavez et al.

MNRAS 514, 4629–4638 (2022)

Figure 3. Results for the V1007 Her system. Integrations yield for the third

body a mass of M

3

= 148 M

J

and P

2

= 1.08 d.

from the masses and the mass function using the following equation:

i = arcsin

( M

1

+ M

2

)

2

f

M

3

1 1 / 3

. (1)

Taking M

2

= 0 . 125 M

and M

1

= 0 . 7 M

, Thorstensen et al.

( 2002a ) obtained a value of i = 32

◦.

If we calculate the statistical values obtained by Knigge et al.

( 2011 ), for the parameters of this CV (using the orbital period to do

so) we ﬁnd that M

1

= 0 . 75 M

, R

1

= 0 . 0107 R

, M

2

= 0 . 15 M

and R

2

= 0 . 1923 R

. Therefore, these M

2

and R

2

estimates are both

well within the uncertainties of the estimates of Thorstensen et al.

( 2002a ). As we pointed out earlier, we decided to use the Knigge

et al. ( 2011 ) values to be consistent throughout the paper as we need

estimates for M

1

and R

1

; both values will be used in the following

sections.

Additionally, using these values in equation ( 1 ) we obtain a value

for the inclination i = 31 .

◦7, which is well within the observational

inclination uncertainty estimated by Thorstensen et al. ( 2002a ) and

very close to the value they provide.

Yang et al. ( 2017 ) found a VLPP of 277.72, which is the longest

among the four systems studied, and concluded that a hierarchical

triple system is the best scenario to explain this period. Ta ble 1 shows

the parameters used for this system in this work.

2.4 V1007 Herculis

This CV was disco v ered by Greiner et al. ( 1998 ), who found that it is

a polar system with an orbital period of P

1

= 404.10 ±0.30 min =

1.9988 h. Because it is a polar system, there is no disc around it,

and there are no periods associated with the disc. Using the orbital

period, Green et al. ( 1998 ) estimated the mass of the secondary to be

M

2

= 0 . 16 M

; to do so, they assumed a mass–radius relationship

for main-sequence stars using Patterson ( 1984 ).

Using the parameters of Knigge et al. ( 2011 ) for this CV, we ﬁnd

that M

1

= 0 . 75 M

and M

2

= 0 . 15 M

, also shown in Table 1 along

with the rest of the parameters. The VLPP observed by Ya ng et al.

( 2017 ) is 170.59 d.

2.5 BK Lyn cis

BK Lyn is a nova-like CV, which was disco v ered by Green et al.

( 1998 ). The calculated orbital period is P

1

= 107.97 ±0.07 min =

1.7995 h, found by Ringwald et al. ( 1996 ). In addition, the secondary

was found to be an M5V star by Dhillon et al. ( 2000 ), using infrared

spectroscopy. The accretion rate was found to be in the range

˙

M

WD

≈

10

−8

–10

−9

M

yr

−1

, constraining the mass of the WD in a wide

range of values between 0.4 and 1.2 M

. Yan g et al. ( 2017 ) found

that the VLPP for this system is 42.05 d (the lowest among all CVs

studied here) and ruled out other possible e xplanations e xcept for a

hierarchical triple system.

Using Knigge et al. ( 2011 ), as pointed out in Section 2.1.4, we

obtain M

1

= 0 . 75 M

and M

2

= 0 . 13 M

, with all the parameters

of the system shown in Ta bl e 1

3 THREE-BODY CATACLYSMIC VARIABLE

As pointed out earlier, Ya ng et al. (

2017 ) proposed the hierarchical

triple system hypothesis for the four systems studied here after ruling

out other e xplanations. The y e xplored the Lido v–Kozai resonances as

a possible explanation for the VLPP observed, and found the possible

semimajor axis of the third body. The mutual inclination between the

inner binary orbital plane and the third-body orbital plane should be

greater than 39 .

◦2 for this mechanism to be ef fecti ve in disturbing the

inner binary ef fecti vely.

Here we explore a new possibility, namely that the secular

perturbation by a low eccentricity and low inclination third object

explains the VLPP and also the change of magnitude observed in

these four CVs.

3.1 Third body on a close near-circular planar orbit

While investigating the system FS Aurigae, Chavez et al. ( 2012 ) ruled

out that the VLPP could correspond directly to the period of a third

body, because the object would be too distant to have an important

effect on the inner binary. A series of numerical integrations were

performed and showed that indeed the effect is minimal and could

not explain the VLPP of the CV FS Aurigae.

It was concluded that a third body on a close near-circular planar

orbit could produce perturbations on the central binary eccentricity,

and these are modulated at three different scales: the period of the

binary P

1

, the period of the perturber P

2

and the much longer secular

period, the VLPP. Secular perturbations have been studied both

analytically and numerically by Georgakarakos ( 2002 , 2003 , 2004 ,

Testing the third-body hypothesis in four CVs 4633

MNRAS 514, 4629–4638 (2022)

Tab l e 1. Initial parameters and magnitudes for all systems are calculated using data from Knigge et al. ( 2011 ). We show the observed minimum magnitude

( M

B min

), maximum ( M

B max

) and o v erall change ( M

B

) due to the VLPP (Yang et al. 2017 ).

Name of the CV Binary period M

1 M

2 R

1 VLPP M

2

/ M

1 a M

B max M

B min M

B log (

˙

M

2

)

(h) (M

) (M

) (R

) (d) (au) (M

yr

−1

)

LU Cam 3.5992 0.75 0.26 0.011 265.76 0.34 0.0055 15.55 16.10 0.55 −9.02

QZ Ser 1.99584 0.75 0.15 0.011 277.72 0.20 0.0036 17.43 17.50 0.07 −10.09

V1007 Her 1.99883 0.75 0.15 0.011 170.59 0.20 0.0036 17.83 18.80 0.97 −10.09

BK Lyn 1.7995 0.75 0.13 0.011 42.05 0.17 0.0033 14.40 15.08 0.68 −10.14

2006 , 2009 ). A third body prevents the complete circularization of the

orbit due to tides by producing a long-term eccentricity modulation

(e.g. Mazeh & Shaham 1979 ; Soderhjelm 1982 ; Soderhjelm 1984 ;

Chavez et al. 2012 , 2020 ). From Georgakarakos ( 2009 ), it is possible

to estimate the amplitude of such eccentricity by using

e

1

∝ q

3

P

1

P

2

8 / 3

e

2

1 −e

2

2

−5 / 2

, (2)

where P

2 is the period of the third body around the inner binary,

e

2 is the eccentricity of the orbit and q

3 = M

3

/( M

1 + M

2 + M

3

).

Therefore, any changes over time on the eccentricity e

2

, such as the

modulations studied in Chavez et al. ( 2012 ), will have an effect on

the eccentricity e

1

of the CV, modulating and changing the position

of the L

1 point and hence changing the brightness of the system.

The details of the numerical modelling are given in the following

subsection.

3.2 Numerical modelling for the circular case

We performed dynamical simulations of the CVs with a hypothetical

third body. The high-order Runge–Kutta–Nystrom RKN 12(10) 17M

integrator of Brankin et al. ( 1989 ) was used for the equations of

motion of the complete three-body problem in the barycentre inertial

reference frame. The total energy was conserved to 10

−5 or better

for all numerical experiments.

As in Chavez et al. ( 2012 ), tidal deformation of the stars in the close

binary is not important for CVs in general and the two objects can be

considered as point masses. Hence, all three bodies are considered as

point masses in our integrations. The binary is initially on a circular

orbit, and the third mass mo v es initially on its own circular orbit

around the inner binary in the same plane. The mass M

3

and its orbital

period P

2

are chosen across an ensemble of numerical experiments.

We proceed as follows. We ﬁx the value of the period of the

third body P

2

, we change its mass M

3

, we perform the numerical

integrations, and then the eccentricity e

1 is calculated as a func-

tion of time. We obtain the secular period on each integration

from e

1 using a Lomb–Scargle periodogram (Lomb 1976 ; Scar-

gle 1982 ). All this shows the effect that the mass has on the secular

period.

In Figs 1 –4 , we show, as a function of mass, the VLPPs and

semimajor axis obtained from our numerical experiments for each

of the four CVs studied. Each curve represents a given P

2 period

that remains constant as we change the mass. We joined the points

by using an interpolated curve (spline method) on each case. A

black point that appears, for example, in Fig. 1 (middle panel)

represents a system that can explain the observed VLPP; that

is, an y giv en point represents a combination of semimajor axis

and mass that can produce, by secular perturbations, the observed

VLPP.

Figure 4. Results for the BK Lyn system. Here, the third body has a mass of

M

3

= 88 M

J

and period of P

2

= 0.44 d.

3.3 Analytical modelling of the third body on an eccentric and

inclined orbit

F ollowing Chav ez et al. ( 2020 ), we also inv estigate the effect that

eccentricity and inclination of the third body may have on the

resulting VLPP and the expected parameters of mass and semimajor

axis of the third body.

We decided to use pre viously deri ved analytical results to see

the effect of eccentricity and inclination. The orbital evolution of

hierarchical triple systems has been studied in a succession of

4634 C. E. Chavez et al.

MNRAS 514, 4629–4638 (2022)

Tab l e 2. GR periods for all systems obtained using the ﬁrst-order post-

Newtonian correction.

Name of the CV VLPP GR period GR period

(d) (d) (yr)

LU Cam 265.76 27851.13 76.25

QZ Ser 277.72 11445.08 31.33

V1007 Her 170.59 11245.42 30.79

BK Ly n 42.05 9626.77 26.36

papers (Georgakarakos

2002 , 2003 , 2004 , 2006 , 2009 , 2013 ; Geor-

gakarakos, Dobbs-Dixon & Way 2016 ). Some of these studies were

focused on the secular evolution of such systems. These analytical

results can give us estimates about the inner binary’s frequency and

period of motion. Hence, we can determine which mass values and

orbital conﬁgurations of a potential third-body companion can give

rise to the secular periods observed in each CV.

We use the results of Georgakarakos ( 2009 ) for a coplanar

perturber on a low eccentricity orbit, and for coplanar systems with

eccentric perturbers we make use of Georgakarakos ( 2003 ). Finally,

for systems with low eccentricity and low mutual inclinations (with

i

m

< 39 .

◦23) the results of Georgakarakos ( 2004 ) are used.

The analytical expressions for the frequencies and periods can be

found in the Appendix, while details of deri v ations can be found in

the papers mentioned abo v e.

In Figs 1 –4 , we show the analytical estimates as curves in

different colours depending on the third-body’s initial eccentricity

or inclination.

4 EFFECT OF POST-NEWTONIAN

CORRECTION

Here, we also consider other dynamical effects that may produce the

long-term signal we observe in the light curve of the stellar binaries.

We study the effect of a ﬁrst-order post-Newtonian general relativity

(GR) correction to the orbit of the stellar binary.

For all stellar pairs under investigation, the small semimajor axis

of the orbit makes it an interesting case to include a post-Newtonian

correction to describe the system’s motion more accurately. Inclusion

of a post-Newtonian correction to our orbit produces an additional

precession of the pericentre at the following rate (e.g. Naoz et al.

2013 ; Georgakarakos & Eggl 2015 ):

˙ =

3 G

3 / 2

( M

1

+ M

2

)

3 / 2

c

2

a

5 / 2

1

(1 −e

2

1

)

. (3)

Here, G is the gravitational constant, c is the speed of light in vacuum,

a

1

is the semimajor axis of the inner binary and e

1

is the eccentricity

of the inner binary. Based on the precession rate given in the abo v e

equation, the post-Newtonian pericentre circulation period for all

systems is shown in Tab le 2 . The periods calculated are too long to

e xplain an y of the VLPPs.

5 EFFECT OF THE THIRD BODY ON THE

MASS TRANSFER RATE AND BRIGHTNESS

5.1 Non-magnetic cases

It is possible to estimate how the modulation of the inner binary, due

to the secular perturbation of the third body, affects the mass transfer

and the brightness of the system. First we focus our attention on

the non-magnetic cases: LU Cam, QZ Ser and BK Lyn. We follow

Chavez et al. ( 2020 ), and a brief re vie w is provided here.

To calculate the mass loss of the secondary, it is necessary to make

use of the deﬁnition of R

L

(2). It is difﬁcult to calculate the volume of

the Roche lobe directly, so it is better to deﬁne an equi v alent radius

of the Roche lobe as the radius, R

L

(2), of a sphere with the same

volume as the Roche lobe. Sepinsky, Willems & Kalogera ( 2007 )

generalized the deﬁnition of R

L

(2) including eccentric binaries, as

R

L

(2) = r

12

( t)

0 . 49 q

2 / 3

0 . 6 q

2 / 3

+ ln (1 + q

1 / 3

)

, (4)

where r

12

is the distance between the two stars at any given time. We

can obtain r

12

from our numerical integrations for each system.

Now we want to know the change in magnitude that produces

this particular combination of parameters, and then we can compare

with the observed magnitude change in the light curve. Therefore,

we can ﬁnd the system in each case that better explains observations

according to our calculations.

We proceed as follows to estimate the change in magnitude due to

the previous choice of parameters. We can calculate the maximum

R

L

(2)

max

, shown as a blue horizontal line in Fig. 5 and the minimum

R

L

(2)

min

, shown as a red horizontal line in Fig. 5 for each system

directly from our numerical results. From here, we can estimate the

mass transfer rate

˙

M (2) and hence the value of the luminosity of each

CV.

Assuming that the secondary is a polytrope of index 3/2 and that the

density around L

1

is decaying exponentially, it is possible to estimate

the mass transfer rate using equation (2.12) of Wa rn er ( 1995 ):

˙

M (2) = −C

M(2)

P

1 R

R(2)

3

. (5)

Here, C is a dimensionless constant ≈10 −20, R (2) is the secondary

stellar radius, R is the amount by which the secondary o v erﬁlls its

Roche lobe, R = R (2) −R

L

(2), and P

1

is the inner binary period.

The R (2) distance needs to be calculated carefully as the equation for

˙

M (2) is very sensitive to the amount of o v erﬁll. We decided to adjust

R (2) to obtain the

˙

M (2) value that we report here in Tab le 1 ; in

Fig. 5 , the value of R (2) is represented by a purple horizontal line.

Because R

L

(2) is a function of time, we instead use its mean value,

R

L

(2)

mean

, shown as a green line in Fig. 5 . Hence, we adjust the value

R (2) for each integration (in Fig. 5 , the system is LU Cam), until

the difference given by R = R (2) −R

L

(2)

mean

is correct, such that

log

˙

M (2) is as in Table 1 .

We can calculate the maximum and minimum of the mass transfer

rate by using the values of R

L

(2)

max

and R

L

(2)

min

to obtain

˙

M (2)

max

and

˙

M (2)

min

.

There are two main sources of the luminosity of CVs: the hotspot

and the disc. The luminosity resulting from the so-called hotspot is

produced when a stream of stellar mass crosses the L

1 point and

collides with the disc. Its expression (Warner 1995 ) is given by

L ( SP ) ≈GM(1)

˙

M (2)

r

d

, (6)

where L ( SP ) is the luminosity due to the hotspot, and the radius of the

disc is typically r

d

≈0.40 ×a

1

with a

1

being the semimajor axis of

the inner binary (see Table 1 ). Applying this equation to our extreme

values on R

L

(2), we obtain the L ( SP )

max

and L ( SP )

min

values.

Alternatively, the luminosity due to the accretion disc using

equation (2.22a) of Warne r ( 1995 ) is

L ( d) ≈1

2

GM(1)

˙

M (2)

R

1

. (7)

Testing the third-body hypothesis in four CVs 4635

MNRAS 514, 4629–4638 (2022)

Figure 5. Method used to calculate the change of magnitude due to the third body. The time evolution of R

L

(2) for the CV LU Cam is shown as an example.

The blue horizontal line is the maximum value for the R

L

(2) that the system reaches, R

L

(2)

max

, the red line corresponds to the minimum value, R

L

(2)

min

, the

green line is the mean value, R

L

(2)

mean

, and the purple line is the R (2) value. See text for more details.

Using this equation we can obtain the extreme values of L ( d )

max

and L ( d )

min

for each system. The total luminosity for each extreme

is found by adding the estimated luminosity of the hotspot plus

the luminosity of the disc, obtaining L ( d)

T

max

and L ( d)

T

min

for each

system.

Then, it is possible to calculate the bolometric magnitude using

M

bol

= −2.5 log ( L / L

0

), with L

0

= 3.0128 ×10

28

W used as a standard

luminosity for comparison. From the extreme values, we obtained

M

B max

and M

B min

, leading to a magnitude difference M

B

.

5.2 Magnetic case

V1007 Her is the only magnetic system in our selection , which,

according to Wu & Kiss ( 2008 ), is a polar system. The accretion

luminosity of an accreting WD is given by

L

acc

= −GM (1)

˙

M (2)

R

1

. (8)

For polars in a high state, L

acc is much higher than the intrinsic

luminosity of the two stars. Thus, we have L

bol ≈L

acc

. Polars are

Roche lobe ﬁlling systems, with the mass transfer rate given by

equation ( 5 ), again using Sepinsky et al. ( 2007 ) to calculate R

L

(2)

directly from the integration. Therefore, from equations ( 5 ) and ( 8 ),

it is possible to estimate the change in brightness for V1007 Her from

L

acc max

and L

acc min

.

6 RESULTS AND DISCUSSION

We studied an ensemble of initial conditions for a hypothetical third

body in each system, and the way it affects both the VLPP and the

change of brightness. All the results of the numerical integrations are

shown as black points in Figs 1 –4 , which correspond to LU Cam,

QZ Ser, V1007 Her and BK Lyn, respectively.

The upper panel of each ﬁgure shows the resulting secular periods

of the binary eccentricity as a function of the mass of the perturber.

Each curve corresponds to different P

2

/ P

1

ratios. The thick horizontal

line corresponds to the VLPP value of each system.

For a given P

2

/ P

1

ratio (i.e. a given curve), as we change the mass

of the system, some of our integrations produce secular perturbations

that never reach the VLPP line. We argue that only systems that cross

the VLPP line can explain the long-term change in the light curve.

The middle panel is a plot of the perturber’s semimajor axis

against its mass. The black points denote the results of the numerical

integrations, while the solid curves are analytical solutions from

Georgakarakos ( 2009 ) ( e

2 = 0, blue curve) and Georgakarakos

Tab l e 3. Summary of values used to estimate the integration that best ﬁts the

VLPP and the change of magnitude for each system.

Variable LU Cam QZ Ser V1007 Her BK Lyn

P

2

/ P

1 7.1 12.5 13.0 5.9

M

3

(M

J

) 97 0.63 148 88

a

2

(au) 0.021 0.019 0.021 0.011

M

B 0.55 0.07 0.73 0.68

(

2003 ) (eccentric cases, green and red curves). The straight line

denotes the orbital stability limit as given in Holman & Wiegert

( 1999 ), while the dotted line is the stability limit based on the results

of Georgakarakos ( 2013 ). In contrast to Holman & Wi egert ( 1999 ),

Georgakarakos ( 2013 ) does not assume a massless particle for any

of the three bodies. Hence, two branches of the dotted line are due

to the dependence of the stability limit on the mass of the perturber.

The lower panels in Figs 1 –4 are similar to what we present

in the middle panels, but the inclination is varied here. For the

coplanar case (blue curve), we use Georgakarakos ( 2009 ), while

for the three-dimensional cases (green and red curves), we make use

of Georgakarakos ( 2004 ).

Tab le 3 lists the ratio between the period of the third body

compared with the period of the inner binary (i.e. P

2

/ P

1

), the mass

of the third body (in Jupiter masses M

J

), and the semimajor axis of

the third body ( a

2

in au) and the change of magnitude of each system

( M

B

). We can compare the magnitude change for each system to

the observed change of magnitude, as given in Ta bl e 1 .

Now we discuss some details of the results for each CV. We

searched for all the numerical integrations whose secular period

matched the observed period of the system, and then made all the

required calculations in order to estimate the change in magnitude

that arises from the perturbations of the third body. A search was

done until a system was found that matched the observed change of

magnitude of the system. This led to a system that can simultaneously

explain the VLPP and the change in magnitude.

6.1 LU Camelopardis

This CV has an observed VLPP of 265.76 d, with M

2

/ M

1 = 0.34,

which is the largest ratio among the CVs studied here. Fig. 1 shows

our numerical results for this system.

The stability limits given by Holman & Wiegert ( 1999 ) and

Georgakarakos ( 2013 ) are also shown. Holman & Wiegert rule out

any a < 0.013 au (grey horizontal line), while Georgakarakos rules