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Условие Отговори Решение

USAPhO 2024, Полуфинал — Задача 2. Question A2 · Stellar Stability

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Всички материали от USAPhO 2024

AAPT AIP UNITED STATES PHYSICS TEAM 2024 — USA Physics Olympiad Exam · 25 т.

Important Instructions for the Exam Supervisor

DO NOT DISTRIBUTE THIS PAGE

This examination has two parts. Each part has three questions and lasts for 90 minutes.
For each student, print out one copy of the exam and one copy of the answer sheets. Print everything single-sided, and do not staple anything. Divide the exam into the instructions (pages 2–3), Part A questions (pages 4–7), and Part B questions (pages 8–10). Page 11 is a graph page, in case students wish to use it as an additional page for problem B3.
Begin by giving students the instructions and all of the answer sheets. Let the students read the instructions and fill out their information on the answer sheets. They can keep the instructions for both parts of the exam. Also give students blank sheets of paper to use as scratch paper throughout the exam.
Students may bring calculators, but they may not use symbolic math, programming, or graphing features of these calculators. Calculators may not be shared, and their memory must be cleared of data and programs. Cell phones or other electronics may not be used during the exam or while the exam papers are present. Students may not use books or other references.
To start the exam, collect signed Honesty Policy, give students the Part A questions, and allow 90 minutes to complete Part A. Do not give students Part B during this time, even if they finish with time remaining. At the end of the 90 minutes, collect the Part A questions and answer sheets.
Then give students a 5 to 10 minute break. Then give them the Part B questions, and allow 90 minutes to complete Part B. Do not let students go back to Part A.
At the end of the exam, collect everything, including the questions, the instructions, the answer sheets, and the scratch paper. Give them the Honor Code Certification and collect signed Codes. Students may not keep the exam questions.
After the exam, sort each student’s answer sheets by page number. Scan every answer sheet, including blank ones. Everybody may discuss the questions after April 3rd. Until April 19th, hold on to all of the answer sheets in the event that your scans are lost or illegible.

We acknowledge the following people for their contributions to this year’s exam (in alphabetical order):

Tengiz Bibilashvili, Kellan Colburn, Natalie LeBaron, Rishab Parthasarathy, Elena Yudovina, and Kevin Zhou.

Instructions for the Student

You should receive these instructions, the reference table on the next page, answer sheets, and blank paper for scratch work. Read this page carefully before the exam begins.
You may use a calculator, but its memory must be cleared of data and programs, and you may not use symbolic math, programming, or graphing features. Calculators may not be shared. Cell phones or other electronics may not be used during the exam or while the exam papers are present. You may not use books or other outside references.
When the exam begins, your proctor will give you the questions for Part A. You will have 90 minutes to complete three problems. Each question is worth 25 points, but they are not necessarily of the same difficulty. If you finish all of the questions, you may check your work, but you may not look at Part B during this time.
After 90 minutes, your proctor will collect the questions and answer sheets for Part A. You may then take a short break.
Then you will work on Part B. You have 90 minutes to complete three problems. Each question is worth 25 points. Do not look at Part A during this time. When the exam ends, you must return all papers to the proctor, including the exam questions.
Do not discuss the questions of this exam, or their solutions, until April 4. Violations of this rule may result in disqualification.

Below are instructions for writing your solutions.

All of your solutions must be written on the official answer sheets. Nothing outside these answer sheets will be graded. Before the exam begins, write your name, student AAPT number, and proctor AAPT number as directed on the answer sheets.
There are several answer sheets per problem. If you run out of space for a problem, you may use the extra answer sheets, which are at the end of the answer sheet packet. To ensure this work is graded, you must indicate, at the bottom of your last answer sheet for that problem, that you are using these extra answer sheets.
Only write within the frame of each answer sheet. To simplify grading, we recommend drawing a box around your final answer for each subpart. You should organize your work linearly and briefly explain your reasoning, which will help you earn partial credit. You may use either pencil or pen, but sure to write clearly so your work will be legible after scanning.

Fundamental Constants

$g = 9.8\ \mathrm{N/kg}$ $\qquad$ $G = 6.67 \times 10^{-11}\ \mathrm{N \cdot m^2/kg^2}$

$k = 1/4\pi\epsilon_0 = 8.99 \times 10^{9}\ \mathrm{N \cdot m^2/C^2}$ $\qquad$ $k_{\mathrm{m}} = \mu_0/4\pi = 10^{-7}\ \mathrm{T \cdot m/A}$

$c = 3.00 \times 10^{8}\ \mathrm{m/s}$ $\qquad$ $k_{\mathrm{B}} = 1.38 \times 10^{-23}\ \mathrm{J/K}$

$N_{\mathrm{A}} = 6.02 \times 10^{23}\ (\mathrm{mol})^{-1}$ $\qquad$ $R = N_{\mathrm{A}} k_{\mathrm{B}} = 8.31\ \mathrm{J/(mol \cdot K)}$

$\sigma = 5.67 \times 10^{-8}\ \mathrm{J/(s \cdot m^2 \cdot K^4)}$ $\qquad$ $e = 1.602 \times 10^{-19}\ \mathrm{C}$

$1\ \mathrm{eV} = 1.602 \times 10^{-19}\ \mathrm{J}$ $\qquad$ $h = 6.63 \times 10^{-34}\ \mathrm{J \cdot s} = 4.14 \times 10^{-15}\ \mathrm{eV \cdot s}$

$m_e = 9.109 \times 10^{-31}\ \mathrm{kg} = 0.511\ \mathrm{MeV/c^2}$

Useful Approximations

(1+x)^n \approx 1 + nx + n(n-1)x^2/2 \text{ for } |nx| \ll 1

e^x \approx 1 + x + x^2/2 + x^3/6 \text{ for } |x| \ll 1

\sin\theta \approx \theta - \theta^3/6 \text{ for } |\theta| \ll 1

\cos\theta \approx 1 - \theta^2/2 \text{ for } |\theta| \ll 1

You may use this sheet for both parts of the exam.

End of Instructions for the Student

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

Условие

A star in hydrostatic equilibrium has inward gravitational forces balanced by pressure gradients. Though the material in a star is not simply an ideal gas, in many cases its pressure $P$ and density $\rho$ are simply related by $P = K\rho^\gamma$ for constants $K$ and $\gamma$ . Throughout this problem, assume the star is spherically symmetric, its mass is conserved, and relativistic effects can be neglected.

a. A thin shell of the star at radius $r_0$ has density $\rho_0$ and thickness $\Delta r$ , and experiences an inward gravitational field of magnitude $g_0$ .

a. i. What is the pressure difference $\Delta P_0$ across the shell in equilibrium?

a. ii. Suppose the entire star expands uniformly by a factor $1 + x$ , so that the shell now has radius $r = r_0(1+x)$ . In terms of $\Delta P_0$ , $x$ , and $\gamma$ , what is the new pressure difference across it?

a. iii. By considering the forces on the shell, write an expression for $d^2r/dt^2$ valid when $x$ is small, in terms of $g_0$ , $\gamma$ , and $x$ . For what values of $\gamma$ will the star be stable?

Next, we consider some simple models of stars, where $\gamma$ can be computed.

b. In a giant star, the pressure is $P = P_{\mathrm{gas}} + P_{\mathrm{rad}}$ , where $P_{\mathrm{gas}}$ is due to the gas, which obeys the ideal gas law, and $P_{\mathrm{rad}} \propto T^4$ is due to blackbody radiation. In the star’s “radiation zone”, $P_{\mathrm{rad}}$ is much larger than $P_{\mathrm{gas}}$ , but the two have a constant ratio. In this case, what is the value of $\gamma$ ?

c. A white dwarf is composed of electrons and nuclei. The electrons provide the outward pressure, while the nuclei cancel the electrons’ charge, and are responsible for most of the mass density. Consider a region of a white dwarf where the number density of electrons is $n_e$ .

c. i. The electrons obey the Heisenberg uncertainty principle, $\Delta p\,\Delta x \gtrsim \hbar$ , where $\Delta x$ is the spacing between them, and $\Delta p$ is the typical momentum that quantum mechanics implies they must have. Find a rough estimate for $\Delta p$ in terms of $n_e$ and $\hbar$ .

c. ii. Using this result, find $\gamma$ for a white dwarf.

c. iii. A white dwarf has total mass $M$ , radius $R$ , and a relatively uniform density of order $\rho \sim M/R^3$ . The radius is related to the mass by $R \propto M^n$ for a constant $n$ . Find the value of $n$ .

Отговори

Решение

a. i. The mass of the shell is $m = \rho_0 A_0\,\Delta r$ , where $A_0 = 4\pi r_0^2$ is its surface area. The total inward gravitational force is $g_0 m$ , and the total outward pressure force is $A_0\,\Delta P_0$ . Equating them yields

\Delta P_0 = g_0\rho_0\,\Delta r.

a. ii. The density scales as $\rho \propto 1/(1+x)^3$ , so the pressure everywhere in the star is scaled by the factor $\rho^\gamma \propto 1/(1+x)^{3\gamma}$ . Thus, the new pressure difference is $\Delta P_0/(1+x)^{3\gamma}$ .

a. iii. The radial form of Newton’s second law for the shell is

m\frac{d^2r}{dt^2} = F_{\mathrm{pr}} - F_{\mathrm{gr}}

where the outward pressure force is

F_{\mathrm{pr}} = A\,\Delta P = A_0\,\Delta P_0\,\frac{(1+x)^2}{(1+x)^{3\gamma}}

since area is proportional to $r^2$ , and the inward gravitational force is

F_{\mathrm{gr}} = gm = \frac{g_0 m}{(1+x)^2}

since gravity obeys an inverse-square law. Applying part 1(a) and simplifying gives

\frac{d^2r}{dt^2} = g_0\left(\frac{(1+x)^2}{(1+x)^{3\gamma}} - \frac{1}{(1+x)^2}\right) \approx g_0(4-3\gamma)x.

The star is stable when this is a restoring force, which requires $\gamma > 4/3$ .

b. The density $\rho$ of the star is due to the gas, and the ideal gas law states that $P_{\mathrm{gas}} \propto \rho T$ . On the other hand, if the two pressure contributions have a fixed ratio, then $P_{\mathrm{gas}} \propto P_{\mathrm{rad}} \propto T^4$ . Combining these results gives $\rho \propto T^3$ and $P \propto T^4$ , so that $\gamma = 4/3$ .

c. i. The spacing between electrons is $\Delta x \sim n_e^{-1/3}$ , so $\Delta p \sim \hbar\, n_e^{1/3}$ .

c. ii. Since the white dwarf is electrically neutral, the density of nuclei is proportional to the density of electrons, so the mass density $\rho$ is proportional to $n_e$ . As for the pressure, basic kinetic theory shows that it scales as the product of the number density $n_e$ , the momentum $\Delta p \sim n_e^{1/3}$ , and the typical speed $\Delta v \sim \Delta p/m \sim n_e^{1/3}$ . Thus, $P \propto n_e^{5/3}$ , so $\gamma = 5/3$ .

c. iii. The pressure at the center is proportional to $G$ and otherwise depends only on $M$ and $R$ . Thus, by dimensional analysis, $P \sim GM^2/R^4$ . (This argument assumes the density is relatively uniform. By contrast, a typical star has a very large, low-density envelope surrounding a dense core of radius $R_{\mathrm{core}} \ll R$ , so the correct estimate would be $P \sim GM^2/R_{\mathrm{core}}^4$ . That is, the assumption of relatively uniform density just means that in the dimensional analysis, the only relevant length is the total radius $R$ .)

We just showed that the electrons provide pressure $P \propto \rho^{5/3} \sim M^{5/3}/R^5$ . Equating these two yields $R \propto M^{-1/3}$ , so $n = -1/3$ . Perhaps surprisingly, more massive white dwarfs are smaller.

Оригинал в Архива: USAPhO_Exam_24.pdf · официални решения: 2024-USAPhO-Exam_solutions.pdf