Съдържание

USAPhO 2019, Полуфинал — Задача 5. Question B2

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от USAPhO 2019

USA Physics Olympiad Exam — Part A / Part B · 25 т.

Внимание: Бележка към темата

Original question scan and matching revised key have been read in full. Printed sign/variable/wording defects in A1, A2, A3 and B1 are retained and documented; key alternatives and all answer sheets are preserved. The scanned B2(c) typo remains source faithful, with revised-key wording difference explicitly recorded.

Important Instructions for the Exam Supervisor — DO NOT DISTRIBUTE THIS PAGE

This examination consists of two parts. Part A has three questions and is allowed 90 minutes. Part B also has three questions and is allowed 90 minutes.
Divide the exam paper into 4 parts: the instructions (pages 2–3), Part A (pages 4–6), Part B (pages 8–10), and answer sheets for one of the questions in Part A (pages 12–13). The exam should be printed single-sided to facilitate dividing the test and scanning the answer sheets.
Provide students with the instructions for the competition (pages 2–3). Students can keep the pages for both parts of the exam, as they contain a reference list of physical constants.
Provide students with blank sheets of paper as scratch paper. Students are not allowed to bring their own papers.
Then provide students with Part A and the associated answer sheets, 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 solutions to Part A along with the answer sheets and questions.
Students are allowed a 10 to 15 minute break between Parts A and B. Then allow 90 minutes to complete Part B. Do not let students go back to Part A.
At the end of the exam, the supervisor must collect all papers, including the questions, the instructions, and any scratch paper used by the students. Students may not take the exam questions. The examination questions may be returned to the students after April 8th, 2019.
Students are allowed calculators, but they may not use symbolic math, programming, or graphical features of these calculators. Calculators may not be shared and their memory must be cleared of data and programs. Cell phones, PDAs, or cameras may not be used during the exam or while the exam papers are present. Students may not use any tables, books, or collections of formulas.

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

JiaJia Dong, Abijith Krishnan, Brian Skinner, and Kevin Zhou.

Instructions for the Student

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

At the beginning of the exam, you shall be provided with the instruction sheets, blank papers (both for your answers and scratch work), and the exam packet.
Work on Part A first. You have 90 minutes to complete three problems. Each question is worth 25 points, but they are not necessarily of the same difficulty. Do not look at Part B during this time.
After you have completed Part A you may take a break. You may consider checking your answers to Part A with the remaining time as you will not be allowed to return to Part A once you start Part B.
Then 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.
Show your work and reasoning. Partial credit will be given if you make your reasoning clear. Do not write on the back of any page. Further guidance is given on the next page.
A hand-held calculator may be used. Its memory must be cleared of data and programs. You may use only the basic functions found on a simple scientific calculator. Calculators may not be shared. Cell phones, PDAs or cameras may not be used during the exam or while the exam papers are present. You may not use any tables, books, or collections of formulas.
In order to maintain exam security, do not communicate any information about the questions of this exam, or their solutions until after April 8th, 2019.

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

$g$	$9.8\ \mathrm{N/kg}$
$k=1/(4\pi\varepsilon_0)$	$8.99\times10^9\ \mathrm{N\cdot m^2/C^2}$
$c$	$3.00\times10^8\ \mathrm{m/s}$
$N_A$	$6.02\times10^{23}\ (\mathrm{mol})^{-1}$
$\sigma$	$5.67\times10^{-8}\ \mathrm{J/(s\cdot m^2\cdot K^4)}$
$1\ \mathrm{eV}$	$1.602\times10^{-19}\ \mathrm J$
$m_e$	$9.109\times10^{-31}\ \mathrm{kg}=0.511\ \mathrm{MeV}/c^2$
$\sin\theta$	$\approx\theta-\frac16\theta^3$ for $\vert \theta\vert \ll1$
$G$	$6.67\times10^{-11}\ \mathrm{N\cdot m^2/kg^2}$
$k_m=\mu_0/(4\pi)$	$10^{-7}\ \mathrm{T\cdot m/A}$
$k_B$	$1.38\times10^{-23}\ \mathrm{J/K}$
$R=N_Ak_B$	$8.31\ \mathrm{J/(mol\cdot K)}$
$e$	$1.602\times10^{-19}\ \mathrm C$
$h$	$6.63\times10^{-34}\ \mathrm{J\cdot s}=4.14\times10^{-15}\ \mathrm{eV\cdot s}$
$(1+x)^n$	$\approx1+nx$ for $\vert x\vert \ll1$
$\cos\theta$	$\approx1-\frac12\theta^2$ for $\vert \theta\vert \ll1$

Instructions for the Student — Further Guidance

Following is some further guidance on formatting your solutions. Start each question on a new sheet of paper. Put your AAPT ID number, your proctor’s AAPT ID number, the problem number and the page number and total number of pages for this problem, in the upper right hand corner of each page. As an example, the second page of your solution to B3 might look as follows.

student AAPT #29485 proctor AAPT #1038 B3: 2/2

Remember to also write the AAPT ID numbers on the provided answer sheets. Write single- sided to facilitate scanning. You may use either pencil or pen, but in either case, make sure to write sufficiently clearly so your work will be legible after scanning. To preserve anonymity of grading, do not write your name on any sheet.

End of Instructions for the Student

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

Important Instructions for the Exam Supervisor — DO NOT DISTRIBUTE THIS PAGE

Divide the exam paper into 4 parts: the instructions (pages 2–3), Part A (pages 4–12), Part B (pages 14–23), and answer sheets for one of the questions in Part A (pages 25–26). The exam should be printed single-sided to facilitate dividing the test and scanning the answer sheets.

Условие

Stellar Black Box

Scientists have recently detected a new star, the MAR-Kappa. The star is almost a perfect blackbody, and its measured light spectrum is shown below.

Complete native measured stellar spectrum with all absorption dips, intensity multiplier10^-15 and outer ticks0to1.2, wavelength nm ticks0to800 and full axis labels.

The total measured light intensity from MAR-Kappa is $I=1.12\times10^{-8}\ \mathrm{W/m^2}$ . The mass of MAR-Kappa is estimated to be $3.5\times10^{30}\ \mathrm{kg}$ . It is stationary relative to the sun. You may find the Stefan-Boltzmann law useful, which states the power emitted by a blackbody with area $A$ is $\sigma AT^4$ .

a. The spectrum of wavelengths $\lambda$ emitted from a blackbody only depends on $h$ , $c$ , $k_B$ , $\lambda$ , and $T$ . Given that the sun has a surface temperature of $5778\ \mathrm K$ and peak emission at $500\ \mathrm{nm}$ , what is the approximate surface temperature of MAR-Kappa?

b. The “lines” in the spectrum result from atoms in the star absorbing specific wavelengths of the emitted light. One contribution to the width of the spectral lines is the Doppler shift associated with the thermal motion of the atoms in the star. The spectral line at $\lambda=389\ \mathrm{nm}$ is due to helium. Estimate to within an order of magnitude the thermal broadening $\Delta\lambda$ of this line. The mass of a helium atom is $6.65\times10^{-27}\ \mathrm{kg}$ .

c. Over the course of a year, MAR-Kappa appears to oscillate between two positions in the background night sky, which are an angular distance of $1.6\times10^{-6}\ \mathrm{rad}$ apart. How far away is MAR-Kappa? Assume that MAR-Kappa lies in the same plane as the Earth's orbit, which is circular with radius $1.5\times10^{11}\ \mathrm m$ .

d. What is the radius of MAR-Kappa?

Over the course of some time, you observe that the star's intensity periodically dips from $I$ to $(1-10^{-5})I$ and then rises back to $I$ , with period $t$ . One possible explanation for this observation is that an exoplanet is orbiting the star and blocking the starlight for some time.

e. Estimate the exoplanet's radius, assuming that it is much closer to the star than to the Earth.

f. Assume the exoplanet is a blackbody with uniform temperature in a circular orbit around the star. What must $t$ be so that the planet has a temperature of $250\ \mathrm K$ ? (If this were true, and the planet had an appropriate atmosphere, the temperature would increase enough to support life.)

Решение

Note: because a range of answers were accepted for part (a), credit was given for later subparts if they were consistent with the value given in part (a).

a.

By dimensional analysis, we have two energy scales – $hc/\lambda$ and $k_BT$ . Then, we have that $hc/\lambda_{\max}\propto k_BT$ . The diagram has a peak around $360\ \mathrm{nm}$ , so, because $\lambda\propto T^{-1}$ , we see a peak around $8030\ \mathrm K$ . We accepted any answer within $\pm500\ \mathrm K$ of $8000\ \mathrm K$ .

b.

The rms velocity of the atoms in the star, in the direction towards Earth, is about $\sqrt{k_BT/m}=4080\ \mathrm{m/s}$ . The red shift is thus $v/c\approx1.4\times10^{-5}$ , and so $\Delta\lambda\approx5.3\times10^{-3}\ \mathrm{nm}$ . This thermal broadening is very insignificant.

c.

This is a parallax effect, as shown in the below diagram:

Complete original parallax ray diagram, two Earth circles/labels, MAR-Kappa filled star/label, both crossing rays and full Background Night Sky label.

The angle Earth makes with MAR-Kappa is the angular shift it makes in the night sky. Then we have that $D=R_{ES}/(\theta/2)=1.9\times10^{17}\ \mathrm m$ .

d.

The luminosity (total power given off by the star) is given by

4\pi D^2I=L=4\pi(1.9\times10^{17}\ \mathrm m)^2\cdot(1.12\times10^{-8}\ \mathrm{W/m^2})=5.1\times10^{27}\ \mathrm W,

with $D$ the distance from Earth to MAR-Kappa. The radius of the star is given by $\sqrt{L/(4\pi\sigma T^4)}=1.3\times10^9\ \mathrm m$ .

e.

The planet is sufficiently close to the star that we can just take the ratio of areas. Then, $R_p^2/R_s^2=10^{-5}$ , so $R_p=10^{-2.5}R_s=4.1\times10^6\ \mathrm m$ .

f.

Suppose the planet is $r$ away from the star. Then, the power from the star absorbed by the planet is

\frac{L(\pi R_p^2)}{4\pi r^2}=\frac{LR_p^2}{4r^2}.

The planet radiates heat with power $4\pi R_p^2\sigma T_p^4$ . Setting the two equal gives $L/(16\pi r^2)=\sigma T_p^4$ . Then,

r=\frac1{4T_p^2}\sqrt{\frac L{\pi\sigma}}=6.75\times10^{11}\ \mathrm m.

Recall that for a circular orbit,

v=\sqrt{\frac{M_SG}r},

\frac{2\pi r}t=\sqrt{\frac{M_SG}r}.

Therefore,

t=2\pi\sqrt{\frac{r^3}{M_SG}}=7.3\ \mathrm{yr}.

Оригинал в Архива: 2019_USAPhO.pdf · официални решения: 2019_USAPHO_revised-solutions.pdf