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USAAAO 2019, II кръг — Задача 2

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Всички материали от USAAAO 2019

2019 National Astronomy Olympiad (NAO) · 7 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

On the night of January 21st, 2019, there was a total lunar eclipse during a supermoon. At the time, the moon was close to perigee, at a distance of 351837 km from the earth, which was $1.4721\times10^8$ km from the sun. The gamma ( $\gamma$ ) of a lunar eclipse refers to the closest distance between the center of the moon and the center of the shadow, expressed as a fraction of the earth’s radius. For this eclipse, $\gamma = 0.3684$ . Given this information, find the closest estimate for the duration of totality of the eclipse.

Отговори

Решение

Note: constants required are the semimajor axis of the moon’s orbit (384399 km), the semimajor axis of the earth’s orbit ( $1.4960\times10^8$ km), the radius of the sun (695700 km), the radius of the earth (6371 km), the radius of the moon (1737 km), the mass of the earth ( $5.972\times10^{24}$ kg), the mass of the sun ( $1.989\times10^{30}$ kg), and $G$ .

Solution: First, calculate the radius of the shadow cast by earth. The size of the umbra at a distance of 351837 km is $r_{umbra} = 6371 - \frac{695700-6371}{1.4721\times10^8}\times351837 = 4723$ km. However, the distance travelled by the moon through the umbra is far less than twice the umbral radius. Firstly, the moon does not pass through the center of the shadow; secondly, totality occurs when the moon is completely within the shadow, not when the center of the moon is in the shadow. So, the distance travelled by the moon during totality is $d = 2\times\sqrt{(r_{umbra}-1737)^2-(0.3684\times6371)^2} = 3693$ km. The velocity of the moon relative to the earth can be calculated using the vis-viva equation: $v_{moon} = \sqrt{G\times5.972\times10^{24}\times\left(\frac{2}{351837000}-\frac{1}{384399000}\right)} = 1.108$ km/s. However, the earth is also moving around the sun, so the umbra is also moving relative to the earth. The velocity of the earth is $v_{earth} = \sqrt{G\times1.989\times10^{30}\times\left(\frac{2}{1.4721\times10^{11}}-\frac{1}{1.4960\times10^{11}}\right)} = 30.259$ km/s. Therefore, the velocity of the umbra is $v_{umbra} = \frac{v_{earth}}{1.4721\times10^8}\times351837 = 0.072$ km/s. The earth and the moon orbit in the same direction, so the velocity of the moon relative to the umbra is $v = v_{moon} - v_{umbra} = 1.036$ km/s. Finally, the duration of totality is $t_{totality} = \frac{d}{v} = 59.426$ minutes.

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