by Julian Poli
Abstract
In this work various density models was explored to determine the pressure and temperature inside stars. This research aims to provide scientific reasoning for the behavior of stars and the relationships between their intensive and extensive properties. It was found that the linear density model is more accurate than the constant density model because, in the linear density model, the density decreases towards the star surface, which is a more realistic density profile.
1. Introduction
The sun is an essential component in our solar system that fosters life on our planet. By definition, the sun is a star that the Earth orbits that provides the energy necessary for supporting life on Earth. The sun’s surface temperature is approximately 6,000 Kelvin (K), while its inner core can reach scorching temperatures of 15 million Kelvin [1]. The sun also attracts our planet with a gravitational force, ensuring that the Earth does not escape its orbit and drift off into space. Since the Earth is dependent on the sun to sustain biological life, it is important and interesting to understand how and why the sun works the way it does.
Stars are composed of hot gases in which atoms and molecules are almost completely ionized in the interior. At a star’s core, outward pressure is generated by thermonuclear fusion between hydrogen atoms to form helium. This outward pressure is defined as thermal pressure. The magnitude of this pressure decreases as the distance from the star’s core becomes greater. The thermal pressure generated pushes outward while a gravitational force pushes inward, so these two forces balance out each other allowing the star to stabilize and have a certain size.
The thermal pressure and self-gravity balance each other in a state referred to as hydrostatic equilibrium. Since stars only have a finite amount of hydrogen atoms, the star won’t be able to hold up against gravity forever, and the star’s core will eventually collapse. Its outer layers will expand outward, classifying the star as a red giant. After this expansion, the star’s surface temperature will decrease significantly, which is expected since stars with the lowest temperatures are red in color. Once the thermal pressure in a star drops low enough, gravity will collapse the star and cause a new stage of thermonuclear reaction. The destiny of each star depends on its mass, which will define how exactly the star will end up its life: as a white dwarf, neutron star, or a black hole [1].
2. Theory
To understand the physics processes inside stars, it is crucial to understand how hydrostatic equilibrium works. We can formulate this equilibrium for a small element of mass dm within the star, see Fig. 1. The gravitational force on dm must be compensated by the difference in pressure in the star at any given point (the pressure increases towards the center of the star pushing dm upward). The balance of forces on dm can be written as:
where P(r) and g(r) are the pressure and free fall acceleration inside the star, and A is the area of the mass dm.
Figure 1: A small element of mass dm at distance r from the center of the star.
It is convenient to present mass dm as a product of the density of the star’s
interior at the location of the mass ρ(r) and the volume of the mass dV = Adr:
If we substitute dm from Eq.(2) into the balance condition (1) and cancel the
area A we can get the following differential equation for the pressure. This is the main equation we will try to analyze in this work.
The obtained equation establishes a relationship between the density and pressure inside stars. To solve Eq.(3) we will use the following logic: once a suitable model for the density distribution ρ(r) is introduced, it will be possible to derive the free fall acceleration g(r) and then integrate Eq.(3), and find the pressure as a function of r. Such solution can be formally written as the following integral:
where R is the star’s radius and P(R) the pressure at the star’s surface that can be put to zero P(R) = 0.
2.1 Gauss-Ostrogradskii Theorem
Newton’s law of universal gravitation states that every particle in the universe attracts every other particle with a force. This force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Such relationship is represented by:
where G is the gravitational constant, m1 and m2 are the two masses, and r is the distance between the two masses [2]. The universal law of gravity can be directly applied to any pair of point-like masses.
In the case when we are interested in the gravitational force on a point-like mass (in our case dm) exerted by a spherically-symmetric finite-size object (in our case a star), the Gauss-Ostrogradskii theorem1 should be used [3]. According to this theorem, it is useful to consider a closed surface (so called Gaussian surface). In the case of spherical symmetry it will be a spherical shell centered at the star’s center and with the radius equals to the distance from the star’s center to the point-like mass dm (so the point-mass dm lays on the Gaussian surface). In this case the Gaussian surface encloses certain amount of star mass Mencl(r) that is a function of the distance r and can be used to calculate the gravitational force on dm as:
To find the mass enclosed by the Gaussian surface we need to integrate the star’s mass density multiplied by the volume starting from the star’s center up to the point r where dm is located. Thus the mass enclosed will depend on the density profile as well:
where ρ(r) is the star’s density and dV represents the volume. Now we can express the free fall acceleration needed for Eq.(3) as a function of the density. The free fall acceleration equals to
With a clear relationship between pressure and density, the next step entails finding a suitable model for density. For this article, a constant density model and a linear density model will be employed.
2.2 Constant Density Model
The constant density model, or average density model, assumes that the density in the star will stay constant throughout the enclosed surface. Since density is
assumed to be constant, all that needs to be done is to integrate the volume under the Gaussian surface to find the enclosed mass. First, let’s find the average density:
where Vstar = 4/3πR3 represents the volume of the star, with R representing the radius of the star. With the density and mass of the star being known, the mass enclosed under the Gaussian surface can be calculated as:
Now that the mass enclosed is known, this can be substituted into Eq.(6) to get the gravitational acceleration as a function of r. We obtained that for this density model the free fall acceleration inside the star is a linear function of the distance r
where g0 = GMstar/R2 stands for the free fall acceleration at the star’s surface.
Now that functions for density and gravity are known, these functions can be substituted into Eq.(3) to find a function for the pressure. The resulting function is:
With this model, as the distance between some point r and the radius R gets closer, the pressure will go to 0 (we put P(R) = 0). Conversely, as the distance between some point r and the radius R increases, the pressure will increase, and the pressure will be equal to the pressure found at the star’s core when r is equal to 0, which would be:
However, such a choice of the density profile is too simplistic since in reality density is not constant throughout a star. However, the same steps can be replicated to derive a more realistic star’s pressure when a better model is picked. In the next section we will consider a linear density model.
2.3 Linear Density Model
The linear density model assumes that the density will decrease at a constant rate of change as the distance between a point in the star, r, gets closer to the radius R. This linear dependency will be represented as:
where A and B are some constants.
Let’s find these constants. The integral of the density function over the volume of the star should give the total mass of the star:
where 4πR3/3 represents the volume of the star and (A + 3BR/4) are the coefficients for the linear density model, which using Eq.(16) can be expressed through the average density of the star:
Assuming that when the point r is on the star’s surface (r = R) the density will be equal to zero,
and using relation (17) we can find the constants A and B, and rewrite the density profile as:
where ρ0 represents the average density value, r represents some point within the star, and R is the radius of the star.
With the new density substituted into Eq.(7), the mass enclosed can be found as:
Using Eq.(8) we can determine the free fall acceleration as a function of the distance r:
where g0 = GMstar/R2 represents the free fall acceleration at the surface of the star. We notice that the gravitational acceleration decreases as r gets closer to the core of the star.
Finally, knowing the functions for density and gravity with respect to r, the the pressure inside a star can be derived. Substituting Eq.(19) and Eq.(21) into Eq.(4) results in:
where the pressure decreases as the distance gets closer to the center. The central pressure is:
which is about 10/3 ≈ 3.3 times greater than the pressure we got from the constant density model. The linear density model provides a more realistic approach for calculation of the pressure inside a star at any given point, since
the density of the star decreases as the distance between the star’s core increases. Using the constant and linear density model, it is now possible to determine the temperature of a star with respect to r.
2.5 Ideal Gas Law
To estimate the temperature inside stars we will use the ideal gas law that can
provide a clear relationship between the local values of the pressure, concentration, and temperature inside a star. The ideal gas law states that:
where P represents the pressure, n represents the concentration of particles within the star, k is the Boltzmann constant, and T is the temperature in Kelvin [4]. Using the mass of the star’s particle m0 (mostly hydrogen and helium), the concentration can be expressed through the mass density ρ:
and the ideal gas law can be modified as:
where ρ represents the average density and m0 represents the average mass of one atom. For our density models the central temperatures are actually quite close to each other:
where the coefficients are 0.5 (constant) and 0.42 (linear).
3. Results
We can apply the obtained formulas to typical stars in our Universe. The corresponding numerical results are presented in the table below. It has to be mentioned that the obtained numbers are not in a good agreement with the accepted values. This is due to our choice of the density models, which are not quite realistic. We picked these models as the first approach to a more serious consideration.
Table 1: Central pressure and temperature of various stars calculated using the linear density model. The parameters for each star type (such as its radius and mass) were taken from [5].
4. Conclusion
Throughout this research project, a constant and linear density model for a star were formulated, and using the balance equation and the ideal gas law, the relationship between pressure, density, and temperature were drawn. Additionally, it was concluded that the linear density model proved to be the more accurate model. One of the main conclusions that can be made from this research is that the knowledge of the density distribution is a key ingredient of the theory and this knowledge is missing. We can assume various density models, but none of them are realistic. Future work is required to derive a realistic pressure and temperature inside stars.
References
[1] R. Kippenhahn, A. Weigert, Stellar Structure and Evolution. Springer; 2nd edition, 2012.
[2] J.R. Taylor, Classical Mechanics. UniversityScienceBooks, 2004.
[3] V. I. Arnold, Mathematical Methods of Classical Mechanics. Springer; 2nd edition, 1989.
[4] S.J. Blundell, K.M. Blundell, Concepts in Thermal Physics. Oxford University Press; 2nd edition, 2009.
[5] Star Fact, https://www.star-facts.com/types-of-stars/
