Mass versus Radius for Main Sequence Stars and Brown Dwarfs

By Austin Gulotta, Muaz Burhanudin, Natchanon Kadkaew

Abstract

In this project, we are trying to calculate the radii of main sequence stars for different core

densities, and find the mass-radius relation of main sequence stars and brown dwarfs. We found,

from calculation and our sources, that the mass of a main-sequence star is independent of its

radius, meaning that a main-sequence star of a specific mass can have any radius. The radius of

the stars however is dependent on its core density. For brown dwarfs, the mass of a brown dwarf

is dependent on its radius. In our calculations, we use crude approximations of the value of the

constants that will be mentioned in this paper, and we also divided every equation that has

gravitational constant, G, with G to simplify the equation and allow the program that we write to

run faster. This paper covers the comparison between actual mass-relation of main sequence stars

and brown dwarfs observed and the analytical mass-relation of them.

Introduction

Brown dwarfs are substellar objects that are bigger than planets, but smaller than stars. They are

too big to be called planets, but also too light for nuclear reaction inside its core to occur, which

exclude them to be stars. Its mass range between 2 to 75 times mass of Jupiter. It is also known

as “failed stars” by astronomers. Studying brown dwarfs is important because, since they are

massive and hard to be detected (because they do not produce light), they may be the reason for

“missing mass” problem faced by cosmology. Main-sequence stars are hydrogen-burning stars

that fuse hydrogen atoms into helium atoms through nuclear fusion reactions in the core. This

type of stars is in hydrostatic equilibrium because the outward thermal pressure from the core is

balanced by the inward gravitational pressure from their own mass. Both of these types of stars

are polytropic stars. Polytropic stars are stars that their pressures are determined by their

densities, which follow the form:

𝑃 = 𝐾𝜌Γ

where P is the pressure, is the density, and K and are constants ( is called the polytropic

constant).

This equation is also often expressed as:

𝑃 = 𝐾𝜌1+1𝑛

where n is polytropic index, where it is a result from the solution of the dimensionless radius of

the star.

Brown dwarfs and main-sequence stars can be modelled using the polytropic model, where n = 3

for main-sequence stars, and n = 1.5 brown dwarfs. We use this model to calculate their radii

from their mass. However, this model isn’t very accurate. It is, in fact, a crude approximation.

This is because, different stars will have different compositions of elements that will contribute

to the errors when using the polytropic model.

Assumption and Numerical Method

We want to calculate masses and radii of main-sequence stars and brown dwarfs. This

can be done by using the equation of hydrostatic balance: 𝑑𝑃

𝑑𝑟= −

𝐺𝑀(𝑟)

𝑟2𝜌(𝑟)

where, 𝑑𝑀(𝑟)

𝑑𝑟= 4𝜋𝑟2𝜌(𝑟)

and equation of state of matter inside a star:

𝑃 = 𝐾𝜌1+1𝑛

The value of K is determined by the compositions and degeneracies of the stars, while the value

of n is determined by the dimensionless radius of the stars that is solved by using the Lane-

Emden equation. By determining the accurate approximation values of these two, we can then

calculate the masses or radii of stars given we know about the information of one of them, using

the details and specifics of polytropic stars. In this paper, we will try to predict the mass vs radius

relationship of stars. The calculations will then be compared with actual observations and makes

a conclusion based on that. In our calculation, we assume that the proportional constant K is the

same for all stars (we use K for sun), regardless of their masses. We also assume that the

isotropic index n is exactly 3 for all main-sequence stars and 1.5 for brown dwarfs. This will give

us large errors combined with the already large errors inherent in the polytropic model itself.

Nondimensionalize the equations in order to get two sets of equations with two desired

unknowns, namely M and R.

Equation of hydrostatic equilibrium (assuming hydrostatic equilibrium) 𝑑𝑃

𝑑𝑟= −𝐺𝑀/𝑟2 (1)

Continuity equation (assuming mass conversion) 𝑑𝑀

𝑑𝑟= 4𝜋𝑟2𝜌 ( 2)

Equation of state of polytrope (assuming polytropic form)

𝑃 = 𝐾𝜌Γ (3)

Consider the following nondimensionalization, where is related to the pressure and density, is

dimensionless distance, and is dimensionless mass. These are the dimensionless substitutions

that we will use.

𝑃 ≡ 𝑃𝑐𝜑ΓΓ−1

𝑃𝑐 = 𝐾𝜌𝑐Γ

𝜌 ≡ 𝜌𝑐𝜑1Γ−1

𝑟 ≡ 𝛼𝜉

𝑀 = 𝑀0𝜇

First divide both sides of (1) by , then differentiate with respect to r. Then make appropriate

substitutions based on previous assumptions and then multiply both sides by r and gather the

derivatives of P on one side.

(1) ⇒1

𝜌

𝑑𝑃

𝑑𝑟= −

𝐺𝑀

𝑟2⇒𝑑

𝑑𝑟(1

𝜌

𝑑𝑃

𝑑𝑟) = −𝐺 (

𝑟2𝑑𝑀𝑑𝑟

− 2𝑀𝑟

𝑟4) = −

𝐺

𝑟2(𝑑𝑀

𝑑𝑟)

⏟ 4𝜋𝑟2ρ

+2𝐺𝑀

𝑟3⏟

−2𝑟∗1𝜌∗𝑑𝑃𝑑𝑟

= −4𝜋𝐺𝜌 −2

𝜌𝑟

𝑑𝑃

𝑑𝑟

⇒ 𝑟2𝑑

𝑑𝑟(1

𝜌

𝑑𝑃

𝑑𝑟) +

2𝑟

𝜌

𝑑𝑃

𝑑𝑟⏟ looks like product rule

=𝑑

𝑑𝑟(𝑟2

𝜌

𝑑𝑃

𝑑𝑟) = −4πr2Gρ

Now use the delightful substitutions from earlier.

⇒𝑑

𝑑(𝛼𝜉)((𝛼𝜉)2

𝜌𝑐𝜑1Γ−1

𝑑 (𝑃𝑐𝜑ΓΓ−1)

𝑑(𝛼𝜉)) = −4𝜋(𝛼𝜉)2𝐺 (𝜌𝑐𝜑

1Γ−1)

Gather constants onto one side and let 𝑛 ≡1

Γ−1

⇒𝑃𝑐

𝐺4𝜋𝛼2𝜌𝑐2𝑑

𝑑𝜉((

Γ

Γ − 1) 𝜉2

𝜑ΓΓ−1

𝜑1Γ−1

𝑑𝜑

𝑑𝜉) = −𝜑𝑛𝜉2

Set 𝑃𝑐 = 𝐾𝜌𝑐

Γ

Then set 𝛼2 =(𝑛+1)𝐾𝜌𝑐

1𝑛−1

4𝜋𝐺

We get 1

𝜉2𝑑

𝑑𝜉(𝜉2

𝑑𝜑

𝑑𝜉) + 𝜑𝑛 = 0 (3)

Now put all of this back into the original equations to get the characteristic mass as well as the dimensionless forms of the original two equations. Starting with (2), we have

𝑑𝑀

𝑑𝑟= 4𝜋𝑟2𝜌 ⇒

𝑀0𝛼

𝑑𝜇

𝑑𝜉= 4𝜋𝛼2𝜉2𝜌𝑐𝜑

𝑛 ⇒𝑑𝜇

𝑑𝜉=4𝜋𝛼3𝜌𝑐𝑀0

𝜉2𝜑𝑛

Set 𝑀0 ≡ 4𝜋𝛼3𝜌𝑐

Then the dimensionless form of (2) is (5). 𝑑𝜇

𝑑𝜉= 𝜉2𝜑𝑛 (5)

Now moving on to (1) we have

𝑑𝑃

𝑑𝑟= −

𝐺𝑀

𝑟2𝜌 ⇒

𝐾𝜌𝑐1+1𝑛

𝛼

𝑑(𝜑𝑛+1)

𝑑𝜉= −

𝐺4𝜋𝛼3𝜌𝑐2

𝛼2𝜇𝜑𝑛

𝜉2 ⇒

(𝑛 + 1)𝐾𝜌𝑐1+1𝑛

𝐺4𝜋𝜌𝑐2 𝜑𝑛

𝑑𝜑

𝑑𝜉= −𝛼2𝜑𝑛

𝜇

𝜉2

Notice that the constant on the left-hand side is exactly 𝛼2, and both sides have 𝜑𝑛 So (1) becomes (4)

⇒𝑑𝜑

𝑑𝜉= −

𝜇

𝜉2 (4)

So that we can see them together in a nice, compact form, here are our two heroes:

𝑑𝜑

𝑑𝜉= −

𝜇

𝜉2 (4)

𝑑𝜇

𝑑𝜉= 𝜉2𝜑𝑛 (5)

Rewrite (4) as two different differential equations. These are the two which will appear in the

code, and use (6) to find 1.

𝑑𝜑

𝑑𝜉= 𝑧 (7)

𝑑𝑧

𝑑𝜉= −(

2𝑧

𝜉+ 𝜑𝑛) (8)

Use Runge-Kutta-4 method to calculate and for main-sequence stars and brown dwarfs.

This is the convergence plot. Clearly, we could not get it to cooperate, so we moved through

many values of h until we found a suitable one. We will use ℎ = 1.9 × 10 − 4, because it gave

us a numerically found 𝜉! = 3.141531 for the 𝑛 = 1 polytrope, whereas the analytical 𝜉1 = 𝜋.

The error is 1.96 × 10−5.

With that value of h, we used our code to find the dimensionless value of \xi and \mu for each

polytrope.

For Brown Dwarfs:

𝜉1.5 = 3.65363.

𝜇1.5 = 2.71454.

For Main Sequence Stars:

𝜉3 = 6.89155.

𝜉3 = 2.01582.

These numbers will be used to calculate radii and masses of main-sequence stars and brown

dwarfs. We will use the equations:

𝑅 = 𝛼𝜉1 = [(𝑛 + 1)𝐾

4𝜋𝐺]

12

𝜌𝑐

1−𝑛2𝑛 𝜉1(9)

And

𝑀 = 𝑀0𝜇1 = 4𝜋 [(𝑛 + 1)𝐾𝜌𝑐

1𝑛−1

4𝜋𝐺]

32

𝜌𝑐𝜇1 (10)

Now, we can see that both equations (9) and (10) have ρc. We can find the relationship between

R and M. First, write ρc in terms of R:

(

𝑅

[(𝑛 + 1)𝐾4𝜋𝐺

]

12𝜉1)

2𝑛1−𝑛

= 𝜌𝑐

Substitute this into (10), and we have:

4𝜋 [(𝑛+1)𝐾

4𝜋𝐺]

3

2(

𝑅

[(𝑛+1)𝐾

4𝜋𝐺]

12𝜉1

)

3−𝑛

1−𝑛

𝜇1 = 𝑀 (11)

Notice that everything except R in equation (10) is a constant. Simplify (11) to get:

𝑀 = 𝐶𝑅3−𝑛

1−𝑛

where

𝐶 ≡

(4𝜋 [(𝑛 + 1)𝐾4𝜋𝐺 ]

32𝜇1)

([(𝑛 + 1)𝐾4𝜋𝐺 ]

12𝜉1)

3−𝑛1−𝑛

=4𝜋 [

(𝑛 + 1)𝐾4𝜋𝐺 ]

𝑛𝑛−1

𝜇1

𝜉1

3−𝑛1−𝑛

From equation (12), we can see that if the polytropic index n is 3, which is the case for main-

sequence stars like the Sun, the equation reduces to 𝑀 = 𝐶. This indicates that for main-

sequence stars, radius is independent of mass. Whereas brown dwarfs, whose n is 1.5, the

equation reduces to 𝑀 =𝐶

𝑅3.

φ vs ξ for different n indexes

n=1.5:

n=3.0:

To compare, this the graph of radius vs mass of main sequence stars and brown dwarfs that have

been discovered:

This data can be seen in one of the links given in the references section. (Johnston)

We will now compare these values from observations to our analytical data from the model. To

find our model, we used the code to scale dimensionless mass and radius for each polytrope

according to the equations above, noting that the most significant scaling factor is the core

density (and as a result, the core pressure) of the star.

Analytical graph of mass vs radius for main sequence stars:

Analytical graph of mass vs radius for brown dwarfs:

Summary of the Results

Based on the analytical data shown, we can conclude that, for main sequence stars, the mass is

constant for any value of radius, which means the mass of the stars does not depends on the

radius of the stars.The mass only has a dependence on the core density of the star.

For brown dwarfs, the mass is related to the radii of the dwarfs. As the radius of the dwarfs

increases, their mass decreases.

In actual results, as plotted, for the main sequence stars, we can see that with the same value of

mass, the radius of the stars can be vary wildly. There is a significant amount of clustering near

where m=1, which is moderately expected. For the brown dwarf, as the radius increases, the

mass decreases, and there is a large cluster of stars which decrease their mass very rapidly as

radius increases, near 1 Jupiter mass.

Discussion

Based on the analytical data plotted, for the brown dwarfs, we can observe that as the radius of

the dwarf increase, the mass decreases. Comparing with the actual data, we also observe the

same pattern. As the radius approaching zero, the value of the mass increases rapidly and give us

the high slope.

For the main sequence stars, if we look at the plotted graph that compares the actual data and

analytical data, we can see that both of them have different pattern. However, the value of K that

we use in the calculation is the value of K for the sun. If we look at the plot, for radius of main

sequence stars that are close to one solar radius, the mass and radius values of the real stars are

following a similar pattern as our analytical data. With this, we can say that the mass-radius

relation of the actual data is nearly the same as the one that we get analytically, for stars whose

radii are close to 1 solar mass.

We made many assumptions in our calculation, which caused our results to be very erroneous.

Further testing is required if we want to make the model more versatile in calculating radii of

main-sequence stars and brown dwarfs of different masses. The constant K is observed to be

very different for different masses, but we assumed that it is the same for every star and brown

dwarf. To make this model more accurate, we suggest using different values of K.

We also take less number of actual stars to compare with the analytical results considering the

number of stars existing. This result can be more accurate and the mass-radius relation will be

more obvious if more number of stars are taken when comparing the actual and analytical data.

For example, most the main-sequence stars we selected gathered near each other in the graph,

with a few outliers. If more data is available, the trend should be more clear, and we should be

able to reasonably plot a horizontal constant trendline.

The data that we get is also affected by the equation of the polytropic model itself. The

polytropic model used to find the mass-relation of main sequence stars and brown dwarfs is just

a crude approximation, which will definitely gives us some of the error that we get.

The Code

In the program, 𝜑 → 𝑝, 𝜉 → 𝑥, 𝜇 → 𝑚.

Table of mass and radius of main sequence stars in solar mass and solar radius

Name Mass (times of mass of the Sun) Radius (times radius of the Sun)

Alcor 1.84 1.846

Alcyone 5.9 9.3

Castor(Alpha Gem Aa) 2.76 2.4

Castor(Alpha Gem Ba) 2.98 3.3

Delta Aquarii 2 2.4

Delta Doradus 1.85 2.1

Epsilon Bootis 4.6 33

Epsilon Herculis 2.6 2.72

Epsilon Microscopii 2.18 2.2

Epsilon Pavonis 2.2 1.74

Achernar 6.7 83.22

Albireo B 3.7 2.59

Beta Canis Minoris 3.5 3.5

Beta Chamaeleontis 5 2.84

Delta^2 Chamaeleontis 5 3.9

Delta Arae 3.56 3.12

Beta Crucis 16 8.4

Beta Librae 3.5 4.9

Epsilon Apodis 6.15 3.9

Epsilon Capriconi 7.6 4.8

Epsilon Cassiopeiae 9.2 6

Alpha Caeli 1.48 1.3

Beta Caeli 1.32 1.3

Beta Coronae Borealis A 2.09 2.63

Beta Coronae Borealis B 1.4 1.56

CC Andromedrae 1.98 3.04

Delta Cygni 2.93 5.13

Delta Equulei 1.192 1.3

Mu Ceti 1.6 1.7

Sigma Ceti 1.21 1.5

Chi Cancri 1.07 1.387

Delta Velorum 2.43 2.79

ADS 16402 A 1.16 1.23

AK Pic 1.03 1.22

AB Andromedae Primary 1.04 1.03

AB Andromedae Secondary 1.04 1.03

Chi Herculis 1.054 1.71

c UMa A 1.213 2.6

Delta Tucanae 2.99 2.7

FL Lyrae A 1.218 1.283

FL Lyrae B 0.958 0.963

Gamma Columbae 5.7 4.8

HAT-P-5 1.16 1.167

AA Tauri 0.76 1.81

AB Doradus A 0.86 0.96

Chi Draconis 1.029 1.2

Epsilon Eridani 0.82 0.735

Epsilon Indi A 0.762 0.732

Epsilon Indi Ba 0.066 0.08

Epsilon Indi Bb 0.047 0.08

Eta Cas A 0.972 1.0386

Eta Cas B 0.57 0.66

Gamma Ceti A 1.88 1.9

Alpha Centauri A 1.1 1.2234

Reference

Johnston, Robert, Wm. List of Brown Dwarfs. Johnstonsarchive.

http://www.johnstonsarchive.net/astro/browndwarflist.html

Polytropes. Princeton. https://www.astro.princeton.edu/~gk/A403/polytrop.pdf

Polytropes. Astrorug. http://www.astro.rug.nl/~peletier/chapter4.pdf

Main Sequence Stars. CSIRO.

http://www.atnf.csiro.au/outreach/education/senior/astrophysics/stellarevolution_mainsequence.h

tml

The list of main-sequence stars was acquired from Wikipedia. Papers which report these values

are cited on Wikipedia.