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USAAAO 2019, II кръг — Задача 7

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Всички материали от USAAAO 2019

2019 National Astronomy Olympiad (NAO) · 15 т.

Instructions (Please Read Carefully)

(Start the exam after reading the instructions carefully and affirming that the contestant understood the instructions given.)

The top 5 eligible scorers on the NAO will be invited to represent USA at the next IOAA. In order to qualify for the national team, you must be a high school student with US citizenship or permanent residency.

This exam consists of 3 parts: Short Questions (7 points each), Medium Questions (15 points each) and Long Questions (25 points each). The maximum score is 153 points.

The test must be completed within 2.5 hours (150 minutes). The proctor should mark the start and end time of the exam on the front page.

Please solve each problem on a blank piece of paper and mark the number of the problem at the top of the page. The contestant’s full name in capital letters should appear at the top of each solution page. If the contestant uses scratch papers, those should be labeled with the contestant’s name as well and marked as ”scratch paper” at the top of the page. Scratch paper will not be graded. Partial credit will be available given that correct and legible work was displayed in the solution. This exam document, solution, and all the scratch paper used should be turned in to the proctor at the end of the exam.

This is a written exam. Contestants can only use a scientific calculator for this exam. A table of physical constants will be provided. Discussing the problems with other people is strictly prohibited in any way until the end of the examination period on March 8th. Receiving any external help during the exam is strictly prohibited. This means that the only allowed items are: a scientific calculator, the provided table of constants, a pencil (or pen), an eraser, blank sheets of papers, and the exam. No books, notes, laptops, mobile phones, or any other devices are allowed while taking the exam.

After reading the instructions, please make sure to sign at the bottom of this page, affirming that:

All work on this exam is mine.
I took this exam under a proctor’s supervision.
I did not receive any external aids besides the materials provided.
I am not allowed to discuss the test with others throughout the period of this examination.
Failure to follow these rules will lead to disqualification from the exam.

To be completed by the proctor:

Full name (First, Last): ______

Position (e.g. Physics Teacher at High School x): ______

Email address: ______

Student began exam at (hh:mm): ______

Student submitted the exam at (hh:mm): ______

Signature: ______ Date (mm/dd/yy): ______

To be completed by the student:

Last name: ______

First name: ______ Middle name: ______

Date of birth (mm/dd/yy): ______ Email address: ______

Signature: ______ Date (mm/dd/yy): ______

Table of Physical Constants

Constant	Symbol	Value
Electron rest mass	$m_e$	$9.109\times10^{-31}$ kg
Proton rest mass	$M_p$	$1.6726\times10^{-27}$ kg
Electronic charge	$e$	$1.6022\times10^{-19}$ C
Speed of light in free space	$c$	$2.9979\times10^8$ m s $^{-1}$
Permeability of free space	$\mu_0$	$4\pi\times10^{-7}$ H m $^{-1}$
Permittivity of free space	$\epsilon_0$	$8.854\times10^{-12}$ F m $^{-1}$
Planck’s constant	$h$	$6.626\times10^{-34}$ J s
Reduced Planck’s constant	$\hbar=h/2\pi$	$1.0546\times10^{-34}$ J s
	$\hbar c$	197.33 MeV fm
Boltzmann’s constant	$k_B$	$1.3807\times10^{-23}$ J K $^{-1}$
Gas constant	$\mathcal R=k_B/m_H$	$8.250\times10^3$ J kg $^{-1}$ K $^{-1}$
Molar gas constant	$R$	8.315 J mol $^{-1}$ K $^{-1}$
Avogadro’s number	$N_A$	$6.022\times10^{23}$ mol $^{-1}$
Standard molar volume		$22.414\times10^{-3}$ m $^3$ mol $^{-1}$
Unified atomic mass unit ( $^{12}$ C scale)	$u$	$931.5$ MeV $/c^2=1.660538\times10^{-27}$ kg
Mass of hydrogen atom	$m_H$	$1.0078u=1.6735\times10^{-27}$ kg
Bohr magneton	$\mu_B$	$9.274\times10^{-24}$ A m $^2$ or J T $^{-1}$
Nuclear magneton	$\mu_N$	$5.051\times10^{-27}$ A m $^2$ or J T $^{-1}$
Proton magnetic moment	$\mu_p$	$2.7928\mu_N$
Neutron magnetic moment	$\mu_n$	$-1.9130\mu_N$
Bohr radius	$a_0$	$5.292\times10^{-11}$ m
Fine structure constant	$\alpha=e^2/(4\pi\epsilon_0\hbar c)$	$(137.04)^{-1}$
Compton wavelength of electron	$\lambda_C=h/(m_ec)$	$2.4263\times10^{-12}$ m
Rydberg’s constant	$R_\infty$	$1.0974\times10^7$ m $^{-1}$
	$R_\infty hc$	13.606 eV
Stefan-Boltzmann constant	$\sigma$	$5.671\times10^{-8}$ W m $^{-2}$ K $^{-4}$
Radiation density constant	$a=4\sigma/c$	$7.561\times10^{-16}$ J m $^{-3}$ K $^{-4}$
Gravitational constant	$G$	$6.673\times10^{-11}$ N m $^2$ kg $^{-2}$

Rest masses of some leptons and hadrons in MeV/ $c^2$ :

$e^{\pm}$ 0.5110, $\mu^{\pm}$ 105.66, $\tau^{\pm}$ 1777, $\pi^0$ 134.98, $\pi^{\pm}$ 139.57, $K^{\pm}$ 493.7, $K^0$ 497.7, $\eta$ 547, $D^0$ 1865, $D^{\pm}$ 1869, p 938.3, n 939.6, $\Lambda^0$ 1115.7, $\Sigma^+$ 1189, $\Sigma^0$ 1193, $\Sigma^-$ 1197, $\Xi^0$ 1315, $\Xi^-$ 1321, $\Omega^-$ 1672, $Z^0$ $91.187\times10^3$ , $W^{\pm}$ $80.41\times10^3$ .

Quark	Charge	$I_3$	$S$	$C$	$B$	$T$	Mass (GeV/ $c^2$ )
u	$+2/3$	$+1/2$	0	0	0	0	$\sim0.003$
d	$-1/3$	$-1/2$	0	0	0	0	$\sim0.006$
c	$+2/3$	0	0	+1	0	0	$\sim1.25$
s	$-1/3$	0	−1	0	0	0	$\sim0.11$
t	$+2/3$	0	0	0	0	+1	174.3
b	$-1/3$	0	0	0	−1	0	4.2

Astrophysical Data — Other data and conversion factors — Trigonometrical identities — Prefixes

Astrophysical Data

Quantity	Symbol	Value
Astronomical unit	AU	$1.496\times10^{11}$ m
Parsec	pc	$3.086\times10^{16}$ m
Luminosity of Sun	$L_\odot$	$3.85\times10^{26}$ W
Mass of Sun	$M_\odot$	$1.989\times10^{30}$ kg
Radius of Sun	$R_\odot$	$6.96\times10^8$ m
Mass of Earth	$M_E$	$5.9742\times10^{24}$ kg
Radius of Earth	$R_E$	$6.3781\times10^6$ m

Other data and conversion factors

Quantity	Symbol	Value
1 ångstrom	Å	$10^{-10}$ m
1 fermi	fm	$10^{-15}$ m
1 barn	b	$10^{-28}$ m $^2$
1 pascal	Pa	1 N m $^{-2}$
1 standard atmosphere		$1.0132\times10^5$ Pa
Standard acceleration due to gravity	$g$	9.807 m s $^{-2}$
1 electron volt	eV	$1.6022\times10^{-19}$ J
	eV/ $hc$	$8.065\times10^5$ m $^{-1}$
	eV/ $k_B$	$1.1604\times10^4$ K
Wavelength of 1 eV photon		$1.2399\times10^{-6}$ m

Trigonometrical identities

\sin(\theta+\phi)=\sin\theta\cos\phi+\cos\theta\sin\phi

\cos(\theta+\phi)=\cos\theta\cos\phi-\sin\theta\sin\phi

\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}

\cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}

\cos\alpha-\cos\beta=2\sin\frac{\alpha+\beta}{2}\sin\frac{\beta-\alpha}{2}

In a triangle ABC, $a/\sin A=b/\sin B=c/\sin C$

and $a^2=b^2+c^2-2bc\cos A$ .

Prefixes

Symbol	Prefix	Factor
T	tera	$10^{12}$
G	giga	$10^9$
M	mega	$10^6$
k	kilo	$10^3$
c	centi	$10^{-2}$
m	milli	$10^{-3}$
$\mu$	micro	$10^{-6}$
n	nano	$10^{-9}$
p	pico	$10^{-12}$
f	femto	$10^{-15}$

Archive constants sheet

The archive retains this general constants sheet alongside the USAAAO collection. The visible sheet does not identify a year or an issuing body. Its values differ from some constants explicitly used in the 2019 solutions; the archive does not establish that this was the exact sheet given to the 2019 contestants.

Условие

a) Mass-Radius Relation Stellar physics often involves guessing the equation of state for stars, which is typically a relation between the pressure $P$ and the density $\rho$ . A family of such guesses are known as polytopes and go as follows-

P\ =\ K\ \rho^{\gamma} \qquad (1)

where $K$ is a constant and the exponent $\gamma$ is fixed to match a certain pressure and core temperature of a star. Given this, show that one can obtain a crude power-law scaling between the mass $M$ of a polytopic star and its radius $R$ of the form $M \propto R^{\alpha}$ . Find the exponent $\alpha$ for polytopic stars (justify all steps in your argument). Also, indicate the exponent $\gamma$ for which the mass is independent of the radius $R$ . Bonus: Why is this case interesting?

b) Black Holes as Blackbodies The mass radius relation for ideal non-rotating, uncharged black holes is known from relativity to be

R = \frac{2GM}{c^2} \qquad (2)

Moreover, Stephen Hawking showed that a black hole behaves like a blackbody, where its temperature (known as the Hawking temperature) is given by

T = \frac{\hbar c^3}{8\pi k_BGM} \qquad (3)

Given this information, show that the lifetime of a black hole (justify this phrase!) $t^*$ scales with its mass $M$ as

t^* \propto M^{\beta} \qquad (4)

where you should find the exponent $\beta$

c) Minimal Black Holes Using the information of the previous part, and Wien’s displacement law, estimate the smallest possible mass of a black hole. State any possible flaws with this estimate.

Отговори

Решение

Solution:

(a) To maintain hydrostatic equilibrium in a star, we need the pressure to balance the gravitational pull. This translates to the radial force balance equation

\frac{dP}{dr} = -\frac{GM\rho}{r^2} \qquad (5)

(Consider the forces on a thin shell of gas in the stellar interior to obtain this). Now to obtain a crude estimate, we do the following

\frac{dP}{dr} \approx \frac{\Delta P}{\Delta R} = \frac{-P}{R} \approx -\frac{GM\rho}{R^2} \qquad (6)

where $P$ is the pressure in the core of the star and $R$ is the radius of the star. Thus, we obtain the scaling

P \sim \frac{M\rho}{R} \sim \rho^2R^2 \qquad (7)

Using (1), we obtain then

\rho \sim R^{\frac{2}{\gamma-2}} \qquad (8)

Thus, we finally arrive at

M \sim R^{\frac{3\gamma-4}{\gamma-2}} \qquad (9)

\implies \alpha = \frac{3\gamma-4}{\gamma-2} \qquad (10)

Thus, if $\gamma = 4/3$ , the mass of the polytopic star is independent of the radius $R$ . This is an interesting case, since this has connections to the equation of state for a relativistic degenerate fermion gas, which is used to model white dwarves. This independence of the mass and radius can be thought of as a crude way to understand the Chandrashekar limit.

(b) We have that

R = \frac{2GM}{c^2} \qquad (11)

T = \frac{\hbar c^3}{8\pi k_BGM} \qquad (12)

Using the assumption that a black hole is a blackbody, we apply Stefan’s law to find the power of radiation emitted by a black hole

\frac{dE}{dt} = 4\pi R^2\sigma T^4 = \frac{\hbar^4c^8\sigma}{256\pi^3G^2k_B^4}\frac{1}{M^2} \qquad (13)

Since this energy has to arise from somewhere, we take this to emerge from the mass-energy of a black hole, giving us

\frac{dE}{dt} = -c^2\frac{dM}{dt} \propto \frac{1}{M^2} \qquad (14)

Thus, we get

M^2dM = -Kdt \qquad (15)

where $K$ is a constant of proportionality involving the fundamental constants and $\sigma$ . Integrating the above equation, we see that the blackhole loses mass and its lifetime scales with its mass as follows

t^* \propto M^3 \qquad (16)

\implies \beta = 3 \qquad (17)

(c) Under the assumption that a black hole is a blackbody, we are justified in thinking that its spectrum has the characteristic Planckian spectrum with the Wien’s law peak for the emitted photon being given by:

\lambda = \frac{b}{T}, \qquad b\ =\ 2.9\times10^{-3}\mathrm{m\ K} \qquad (18)

If we estimate that one photon (or any $\mathcal{O}(1)$ photons, as required to preserve momentum) having this wavelength carries away the entire mass energy of the black hole, we estimate that

Mc^2 \approx \frac{2\pi\hbar c}{\lambda} \qquad (19)

\implies M^2 \approx \left(\frac{\hbar c}{2\sqrt{Gk_Bb}}\right)^2 \qquad (20)

M \approx \left(\frac{\hbar c}{2\sqrt{Gk_Bb}}\right) \approx 9.68\times10^{-9}\ \mathrm{kg} \qquad (21)

M \approx 4.84\times10^{-39}M_{\odot} \qquad (22)

There are several problems with this method of estimation. Firstly radiation is a fully quantum process, so our assumption of radiating a single photon is not really correct. Moreover, we use Wien’s law in its wavelength form, to obtain energy but we really need it in the frequency form to actually obtain the peak of the energy of the photon spectrum. This is an order of magnitude estimation, so any reasonably justified argument should be given full credit.

Оригинал в Архива: nao_2019.pdf · официални решения: nao_2019_solution.pdf · допълнителни данни: physical-constants.pdf