Problem 55 - An Introduction to Thermal Physics

Question

Heat capacities are normally positive, but there is an important class of exceptions: systems of particles held together by gravity, such as stars and star clusters. (a) Consider a system of just two particles, with identical masses, orbiting in circles about their center of mass. Show that the gravitational potential energy of this system is -2 times the total kinetic energy. (b) The conclusion of part (a) turns out to be true, at least on average, for any system of particles held together by mutual gravitational attraction: $$ \bar{U}_{\text {potential }}=-2 \bar{U}_{\text {kinetic }} $$ Here each \(\bar{U}\) refers to the total energy (of that type) for the entire system, averaged over some sufficiently long time period. This result is known as the virial theorem. (For a proof, see Carroll and Ostlie (1996), Section 2.4.) Suppose, then, that you add some energy to such a system and then wait for the system to equilibrate. Does the average total kinetic energy increase or decrease? Explain. (c) A star can be modeled as a gas of particles that interact with each other only gravitationally. According to the equipartition theorem, the average kinetic energy of the particles in such a star should be \(\frac{3}{2} k T,\) where \(T\) is the average temperature. Express the total energy of a star in terms of its average temperature, and calculate the heat capacity. Note the sign. (d) Use dimensional analysis to argue that a star of mass \(M\) and radius \(R\) should have a total potential energy of \(-G M^{2} / R,\) times some constant of order 1. (e) Estimate the average temperature of the sun, whose mass is \(2 \times 10^{30} \mathrm{kg}\) and whose radius is \(7 \times 10^{8} \mathrm{m}\). Assume, for simplicity, that the sun is made entirely of protons and electrons.

Step-by-Step Solution

Answer

A star's negative heat capacity implies kinetic energy increases when energy is added. The average temperature of the sun can be estimated using the derived expressions.

Step 1: Calculate Gravitational Potential Energy

Assume each particle has mass \(m\), and they are separated by a distance \(r\). Using the formula for gravitational potential energy, \(U = -\frac{Gm^2}{r}\), we calculate the potential energy of the system.

Step 2: Calculate Kinetic Energy per Particle

The centripetal force holding each particle in circular motion is provided by gravity. This force is \(\frac{Gm^2}{r^2}\). For circular motion, \(F = \frac{mv^2}{r}\). Equating the two gives:\[v^2 = \frac{Gm}{r}\]Thus, kinetic energy per particle is \(K = \frac{1}{2}mv^2 = \frac{1}{2}m\frac{Gm}{r}\).

Step 3: Equate Total Energy Calculations

The total kinetic energy is twice the kinetic energy of one particle: \(U_{\text{kinetic total}} = m\frac{Gm}{r}\). Using the potential energy expression:\[ U_{\text{potential }} = -2U_{\text{kinetic }} \].

Step 4: Interpret the Virial Theorem for Energy Change

Upon adding energy to the system, because \(\bar{U}_{\text {potential }}=-2\bar{U}_{\text {kinetic }}\), a system tends to redistribute its total energy such that potential energy becomes more negative. The kinetic energy must increase to maintain equilibrium, as potential energy dominates.

Step 5: Define Total Energy in Terms of Temperature

Using the equipartition theorem the kinetic energy is given by:\[U_{\text{kinetic}} = \frac{3}{2}NkT\]Thus, using the virial theorem, the total energy, \(U_{\text{total}} = U_{\text{kinetic}} + U_{\text{potential}}\), becomes:\[ U_{\text{total}} = U_{\text{kinetic}} - 2U_{\text{kinetic}} = -\frac{3}{2}NkT\]. This implies the star has negative total energy.

Step 6: Calculate Heat Capacity

Heat capacity is the derivative of energy concerning temperature. So:\[C = \frac{dU}{dT} = \frac{d}{dT}(-\frac{3}{2}NkT)\]\[C = -\frac{3}{2}Nk\]. The heat capacity is negative, indicating energy decreases as temperature increases.

Step 7: Dimensional Analysis of Gravitational Potential Energy

For a star with mass \(M\) and radius \(R\), the potential energy can be estimated from dimensional analysis:\[U = -\alpha \frac{GM^2}{R}\]Where \(\alpha\) is a constant of order 1 due to gravitational nature and particle distribution.

Step 8: Estimate Average Temperature of the Sun

Using the formula for average kinetic energy, we have:\[\frac{3}{2}kT = \frac{GM^2}{2R}\]Solve for \(T\) using the given values for Sun's mass \(M = 2\times10^{30}\,\text{kg}\), radius \(R=7\times10^{8}\,\text{m}\), gravitational constant \(G = 6.674\times10^{-11} \text{m}^3\text{/kg/s}^2\), and \(k \approx 1.38\times10^{-23}\text{J/K}\).

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is the energy associated with an object due to its position in a gravitational field. It represents the potential for work done by gravity. In the context of particles orbiting each other, the potential energy is given by the formula: \( U = -\frac{Gm^2}{r} \), where \( G \) is the gravitational constant, \( m \) is the mass of each particle, and \( r \) is the separation between them.

This formula shows why potential energy is negative; it's due to the nature of gravitational attraction, which means energy is needed to separate the masses further apart. The result from the exercise reveals that for a gravitational system, potential energy ties directly to kinetic energy. In this case, it's equal to \(-2\) times the total kinetic energy, as explained by the Virial Theorem.

• Associated with gravitational attraction.
• Negative because it requires work to separate the objects.
• Directly related to kinetic energy through the Virial Theorem.

Equipartition Theorem

The equipartition theorem provides a way to relate temperature and energy at a microscopic level. It states that each degree of freedom in a system contributes \( \frac{1}{2}kT \) (where \( k \) is Boltzmann's constant and \( T \) is the temperature in Kelvin) to the system's total energy.

For stars, modeled as gases where particles interact only through gravity, the average kinetic energy per particle can be calculated as \( \frac{3}{2}kT \). This underscores the connection between a star's temperature and its internal kinetic energy partitioned across movements in space. The theorem helps in determining total energy and other thermodynamic properties in astrophysical contexts.

Heat Capacity

Heat capacity is a measure of the total energy a system requires to change its temperature by a given amount. It describes how much heat energy gets stored as a system's temperature changes. In the case of gravitational systems like stars, the heat capacity is unusual because it can be negative.

Typically calculated as \( C = \frac{dU}{dT} \), where \( U \) is the internal energy and \( T \) is temperature, for a gravitational system, the heat capacity derived from the Virial Theorem turns out to be negative. This is because, as a star's temperature increases, its kinetic energy increases, while the total energy, surprisingly, decreases. It's a peculiar behavior intrinsic to self-gravitating systems.

• Indicates how much energy is required to change system temperature.
• For stars, derived heat capacity often negative.
• Shows unconventional thermodynamic behavior.

Dimensional Analysis

Dimensional analysis is a mathematical technique used to predict broad parameters of physical systems based on unit consistency. When applied to gravitational potential energy of a star, it helps estimate forms of potential energy, confirming our results through reasoning not tethered to exact constants.

By considering a star's mass \( M \) and radius \( R \), dimensional analysis leads to the conclusion that potential energy is proportional to \(-G\frac{M^2}{R}\). This provides insight into characteristics of complex systems such as stars, using simple ideas and fundamental dimensions like mass, length, and time.

• A tool for validating physical relationships any system.
• Utilizes fundamental units to estimate forms of physical quantities.
• Helpful in astrophysics for broad concepts like gravitational binding energy.

Kinetic Energy

Kinetic energy is the energy of motion and is critical in systems held by gravitational forces. It connects dynamically with gravitational potential energy in these systems.

For a system of particles in gravitational orbits, their total kinetic energy can be found from the Virial Theorem, which links it to gravitational energies. Per particle, kinetic energy is given by \( K = \frac{1}{2}mv^2 \). In gravity-bound systems, stable movements result from the balance between kinetic and potential energies, maintaining a certain ratio. This interplay explains why bodies like stars maintain equilibrium even as internal energies adjust.

• Energy associated with motion of particles.
• Essential for maintaining self-gravitating systems.
• Balances gravitational forces to stabilize systems.

Thermal Physics

Thermal physics deals with heat and temperature and their connection to energy and matter. It merges concepts from macroscopic and microscopic perspectives, integrating thermodynamics, kinetic theory, and statistical mechanics.

In the exercise, thermal physics principles clarify how heat capacity, thermodynamic behavior, and temperature can be understood in self-gravitating systems like stars. These systems are intriguing because energy distributions don't follow simplistic rules due to their nature. Insights into stellar heat capacities and behaviors derive from these thermal physics principles, showing stars deviate from conventional systems in how they store and release energy.

Stellar Structure

Stellar structure studies the equilibrium and physical conditions inside stars, helping us understand these celestial giants' life cycles and energy processes.

Stars, like the sun, are complex systems capable of maintaining stability over long periods through intricate balances of forces. Gravitational contraction drives energy generation, while the Virial Theorem shows how stars counteract these forces. Modeling stars as gravitational gases allows us to use tools like dimensional analysis and thermal physics to calculate core temperatures, energy dissipation, and structural stability. These mechanisms help explain why stars have long lifetimes and vary in brightness and composition.

• Involves understanding equilibria and processes within stars.
• Helps explain stellar stability and lifespan.
• Uses theoretical tools to estimate temperatures and structural impacts.