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A Phenomenological Approach to Searching for Earth Sized Exoplanets

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  • Independent Researcher

Abstract

The technology used to search for and validate exoplanets is improving to the point that more and more Earth-sized planets are being discovered. Using the existing data for large (Jupiter sized) exoplanets, a phenomenological approach to estimating the number of Earth-sized exoplanets is outlined in this paper. Central to this approach is the two parameter turbulence model for exoplanetary formation. The existing body of data for exoplanets may permit an estimate for the number of Earth sized planets orbiting K or G type stars within the Milky Way.
A Phenomenological Approach to Searching for Earth
Sized Exoplanets
by Patrick Bruskiewich
Director, Vancouver Institute for Advanced Studies
Abstract
The technology used to search for and validate exoplanets is improving to the point
that more and more Earth-sized planets are being discovered. Using the existing
data for large (Jupiter sized) exoplanets, a phenomenological approach to
estimating the number of Earth-sized exoplanets is outlined in this paper. Central
to this approach is the two parameter turbulence model for exoplanetary formation.
The existing body of data for exoplanets may permit an estimate for the number of
Earth sized planets orbiting K or G type stars within the Milky Way.
An Outsider’s Look at the Solar System
If a distant astronomer were to view our solar system with instruments that could
only detect Jupiter sized planets (0.05 < Massplanet < 1 MassJupiter) which would span
the mass of Uranus, Neptune, Saturn and Jupiter, how might phenomenology
permit an estimate of the number of Earth-sized planets in the solar system.
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Uranus is about 0.046 that of the Mass of Jupiter. Uranus is 15 times more massive
than the Earth.
The two parameter turbulence model for planetary formation is a model that allows
for a variety of planetary size based on the Reynold’s number for different planet
types.1 If we assume this to be the phenomenological effect that links together
planetary types, then we can estimate the existence of Earth-sized planets using a
power law. It is understood that the type of nebular materials will ultimately
determine the number, distribution and types of exoplanets. 2
A phenomenological approach for estimating the number of Earth-sized planets in
the solar system is to use a less than or equal measure (Jupiter Mass) and a count.
Mass Less than or Equal
(Jupiter Mass)
Number Planets
1 4 Jupiter, Saturn, Neptune and Uranus
0.31 3 Saturn, Neptune and Uranus
0.06 2 Neptune and Uranus
Table 1: Planets less than or Equal (Jupiter Mass)
Let us plot this data and fit a power law to model the distribution of planet sizes.
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0 0.2 0.4 0.6 0.8 1 1.2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
f(x) = 4.00816825515041 x^0.24880562653527
R² = 0.99992214776275
Jupiter Sized Planet Count - Solar System
less than (mass Jupiter)
Number
Fig. 1: A Power law model the distribution of Solar System planet sizes
The power law for planets of varied size in the solar system is (x is Jupiter Mass)
y = 4.00 x 0.2488
This power law can be used to estimate the number of Earth-sized planets in the
solar system. An Earth-sized planet would have a x value of 1/318 Jupiter masses,
that is x = 0.0031, from which we arrive at
y = 4.00 (0.0031)0.2488 = 4.00 (0.238) = 0.95 ~ 1.0
Let us now consider a Venus-sized planet (0.82 Earth Masses), that is x = 0.82 /
318 = 0.0026 from which we arrive at
3
y = 4.00 (0.0027)0.2488 = 4.00 (0.227) = 0.91 ~ 1.0
Let us now consider a Mars sized planet (0.11 Earth Masses), that is x = 0.11 / 318
= 0.00035 from which we arrive at
y = 4.00 (0.00035)0.2488 = 4.00 (0.14) = 0.55 ~ 1.0
It appears then that this simple phenomenological power law provides for a
predictor of an Earth-sized, a Venus-sized and a Mars-sized planets in the solar
system, based on the two parameter turbulence model and the detection of Jupiter
sized planets.
Applying this approach to Exoplanetary systems within 35 pc of the Sun
Exoplanets are being found around a variety of star types. Many exoplanetary
systems have been discovered around small stars, such as dwarf stars, which is
understandable since such small stars out number larger stars in star counts.
Let us consider exoplanetary systems around K & G type stars. Drawing from the
existing data from exoplanets.org, we can filter the exoplanetary data with the
following criteria
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Parameter Range Remarks
Mstar 0.8 < Mstar < 1.2 Solar masses
Teff[k] 5300 < T Surface temperature (K)
DIST(pc) Dist < 35 pc Distance in parsecs
MSINI variable Mass of planet(Jupiter)
Table 2: Criteria used to extract Exoplanetary data for K & G type stars
Here are the results from the search of the data set (as of October 18, 2021):
Mass Less than or Equal
(Jupiter Mass)
Number
< 12 72
< 6 60
< 3 50
< 1.5 36
< 0.75 26
< 0.375 16
< 0.1875 14
< 0.094 13
< 0.047 5
Table 3: Star Count of K & G Star Exoplanets Systems
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Let us plot this data and fit a power law to model the distribution of planet sizes:
0 2 4 6 8 10 12 14
0
10
20
30
40
50
60
70
80 f(x) = 27.6299198349271 x^0.447521049149662
R² = 0.955180638249987
K & G Type Stars with Planets
less than * (Mass Jupiter)
Number
Fig. 2: A Power law model for Mass – Number distribution of exoplanets
A power law estimate of the number of exoplanets within 35 pc of our sun is
y = 27.63 x 0.4475
An Earth-sized planet would have a x value of 1/318 Jupiter masses, that is x =
0.0031, from which we arrive at
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y = 27.63 (0.0031) 0.4475 = 2.08 ~ 2
Ceteris paribus, based on this phenomenological model we predict there may be 2
Earth-sized planets within 35 pc of our solar system.
Let us define an Earth-sized planet as a planet with a mass less than two earth
masses. To date only one K or G exoplanetary system GJ 9827 within 35 pc has
been found to have an Earth-sized planet (GJ 9827c, mass = 1.92 MassEarth) in orbit.
When you add our solar system to the total you arrive at two systems with a total
of three Earth-sized planets. An analysis of the three planets (GJ 9827c, Earth and
Venus) yields a combined mass of 3.74 earth masses or an average mass of 1.25
earth masses.
It is interesting to note that if we set x = 0.0063 (2 earth masses), y = 2.84 earth
masses, which is within 3.4 % of the measured combined mass of the two planets
GJ 9827c and Earth. A priori, our phenomenological model result fits the data
well.
Adding Our Solar System into the Count
If we add the Jupiter sized planets in our solar system into the count (Jupiter,
Saturn, Neptune and Uranus) the power law fit becomes
7
y = 31.40 x 0.4057
with a R2 of 0.9611. An Earth-sized planet would have a x value of 1/318 Jupiter
masses, that is x = 0.0031 from which we arrive at
y = 31.40(0.0031)0.4057 = 3.03
which is what we see (GJ 9827c, Earth and Venus).
Discussion
There are an estimated 947 K-type stars within 35 pc of the sun. 3 There are an
estimated 512 G-type stars within 35 pc of the sun. 4 This provides for a count of
1459 K & G type stars within 35 pc of the sun. This provides for an estimate of
2.08/ 1459 = 0.14 %
that a K or G type star within 35 pc of the sun will have an earth-sized planet in
orbit around it. There is an estimated 4,385 stars within 35 pc of the sun.5 It
appears that 38 % of these stars are K & G type stars.
8
Extrapolating to the Milky Way as a whole, with perhaps 100 x 109 star, we may
estimate the number N of Earth like planets around K & G type stars as
NEarths = 100 x 109 stars x 0.38 (K & G type starts) x 0.0014 = 5.32 x 107 planets
If we use just raw numbers of 2 per 4,385 stars within 35 pc of the sun we may
estimate the number Nraw of Earth like planets
Nraw = (2/4385) 100 x 109 stars = 4.56 x 107 planets
An upper bound would be 3/4385 (if we include GJ 9827c, Earth and Venus) from
which we get
Nraw = (3/4385) 100 x 109 stars = 6.84 x 107 planets
This provides for the following estimates
Model Number
K & G model 5.32 x 107
Lower bound 4.56 x 107
Upper bound 6.84 x 107
(5.57+ 1.27/- 0.25) x 107
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Table 4: Number Earth-sized planets around K & G Type stars in Milky Way
It is estimated that there are
(5.57+ 1.27/- 0.25) x 107
Earth-sized planets in the Milky Way.
If perhaps 1/5 of these Earth like planets are within the habitable zone of their K &
G type stars, then we may estimate that there
Nhabitable = 0.2 x 5.57 x 107 = 1.11 x 107 planets
habitable Earth-sized planets in the Milky Way.
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References:
[1] Bruskiewich, P. The Two Parameter Turbulence Model and the Formation of
Jupiter, Researchgate, 2020. Bruskiewich, P. Wu, T., Zhang, C., Using the
Bifurcation Algorithm to Define the Kuiper Belt and Kuiper Cliff, Researchgate
2020, Bruskiewich P., The Two Parameter Turbulence Model and Planetary
Structure for the Terrestrial Planets, Researchgate, July 2021
[2] A paper will shortly be forthcoming on this phenomenon.
[3] Refer to http://www.solstation.com/stars3/100-ks.htm
[4] Refer to http://www.solstation.com/stars3/100-gs.htm
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[5] Stars within 100 light years (30.7 pc) (from http://www.solstation.com)
Type Est. Number
M ~2,000
K 947
G 512
F 303
A 76
B 4
3,842
Scaled to 35 pc the total is 4,385 stars. Around 38 % of these stars are K or G
type.
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