USAAAO 2019, II кръг — Задача 5

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:

  1. All work on this exam is mine.
  2. I took this exam under a proctor’s supervision.
  3. I did not receive any external aids besides the materials provided.
  4. I am not allowed to discuss the test with others throughout the period of this examination.
  5. 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
ConstantSymbolValue
Electron rest massmem_e9.109×10−319.109\times10^{-31} kg
Proton rest massMpM_p1.6726×10−271.6726\times10^{-27} kg
Electronic chargeee1.6022×10−191.6022\times10^{-19} C
Speed of light in free spacecc2.9979×1082.9979\times10^8 m s−1^{-1}
Permeability of free spaceμ0\mu_04π×10−74\pi\times10^{-7} H m−1^{-1}
Permittivity of free spaceϵ0\epsilon_08.854×10−128.854\times10^{-12} F m−1^{-1}
Planck’s constanthh6.626×10−346.626\times10^{-34} J s
Reduced Planck’s constantℏ=h/2π\hbar=h/2\pi1.0546×10−341.0546\times10^{-34} J s
ℏc\hbar c197.33 MeV fm
Boltzmann’s constantkBk_B1.3807×10−231.3807\times10^{-23} J K−1^{-1}
Gas constantR=kB/mH\mathcal R=k_B/m_H8.250×1038.250\times10^3 J kg−1^{-1} K−1^{-1}
Molar gas constantRR8.315 J mol−1^{-1} K−1^{-1}
Avogadro’s numberNAN_A6.022×10236.022\times10^{23} mol−1^{-1}
Standard molar volume22.414×10−322.414\times10^{-3} m3^3 mol−1^{-1}
Unified atomic mass unit (12^{12}C scale)uu931.5931.5 MeV/c2=1.660538×10−27/c^2=1.660538\times10^{-27} kg
Mass of hydrogen atommHm_H1.0078u=1.6735×10−271.0078u=1.6735\times10^{-27} kg
Bohr magnetonμB\mu_B9.274×10−249.274\times10^{-24} A m2^2 or J T−1^{-1}
Nuclear magnetonμN\mu_N5.051×10−275.051\times10^{-27} A m2^2 or J T−1^{-1}
Proton magnetic momentμp\mu_p2.7928μN2.7928\mu_N
Neutron magnetic momentμn\mu_n−1.9130μN-1.9130\mu_N
Bohr radiusa0a_05.292×10−115.292\times10^{-11} m
Fine structure constantα=e2/(4πϵ0ℏc)\alpha=e^2/(4\pi\epsilon_0\hbar c)(137.04)−1(137.04)^{-1}
Compton wavelength of electronλC=h/(mec)\lambda_C=h/(m_ec)2.4263×10−122.4263\times10^{-12} m
Rydberg’s constantR∞R_\infty1.0974×1071.0974\times10^7 m−1^{-1}
R∞hcR_\infty hc13.606 eV
Stefan-Boltzmann constantσ\sigma5.671×10−85.671\times10^{-8} W m−2^{-2} K−4^{-4}
Radiation density constanta=4σ/ca=4\sigma/c7.561×10−167.561\times10^{-16} J m−3^{-3} K−4^{-4}
Gravitational constantGG6.673×10−116.673\times10^{-11} N m2^2 kg−2^{-2}

Rest masses of some leptons and hadrons in MeV/c2c^2:

e±e^{\pm} 0.5110, μ±\mu^{\pm} 105.66, τ±\tau^{\pm} 1777, π0\pi^0 134.98, π±\pi^{\pm} 139.57, K±K^{\pm} 493.7, K0K^0 497.7, η\eta 547, D0D^0 1865, D±D^{\pm} 1869, p 938.3, n 939.6, Λ0\Lambda^0 1115.7, Σ+\Sigma^+ 1189, Σ0\Sigma^0 1193, Σ−\Sigma^- 1197, Ξ0\Xi^0 1315, Ξ−\Xi^- 1321, Ω−\Omega^- 1672, Z0Z^0 91.187×10391.187\times10^3, W±W^{\pm} 80.41×10380.41\times10^3.

QuarkChargeI3I_3SSCCBBTTMass (GeV/c2c^2)
u+2/3+2/3+1/2+1/20000∼0.003\sim0.003
d−1/3-1/3−1/2-1/20000∼0.006\sim0.006
c+2/3+2/300+100∼1.25\sim1.25
s−1/3-1/30−1000∼0.11\sim0.11
t+2/3+2/30000+1174.3
b−1/3-1/3000−104.2
Astrophysical Data — Other data and conversion factors — Trigonometrical identities — Prefixes

Astrophysical Data

QuantitySymbolValue
Astronomical unitAU1.496×10111.496\times10^{11} m
Parsecpc3.086×10163.086\times10^{16} m
Luminosity of SunL⊙L_\odot3.85×10263.85\times10^{26} W
Mass of SunM⊙M_\odot1.989×10301.989\times10^{30} kg
Radius of SunR⊙R_\odot6.96×1086.96\times10^8 m
Mass of EarthMEM_E5.9742×10245.9742\times10^{24} kg
Radius of EarthRER_E6.3781×1066.3781\times10^6 m

Other data and conversion factors

QuantitySymbolValue
1 ångstromÅ10−1010^{-10} m
1 fermifm10−1510^{-15} m
1 barnb10−2810^{-28} m2^2
1 pascalPa1 N m−2^{-2}
1 standard atmosphere1.0132×1051.0132\times10^5 Pa
Standard acceleration due to gravitygg9.807 m s−2^{-2}
1 electron volteV1.6022×10−191.6022\times10^{-19} J
eV/hchc8.065×1058.065\times10^5 m−1^{-1}
eV/kBk_B1.1604×1041.1604\times10^4 K
Wavelength of 1 eV photon1.2399×10−61.2399\times10^{-6} m

Trigonometrical identities

sin⁡(θ+ϕ)=sin⁡θcos⁡ϕ+cos⁡θsin⁡ϕ\sin(\theta+\phi)=\sin\theta\cos\phi+\cos\theta\sin\phi
cos⁡(θ+ϕ)=cos⁡θcos⁡ϕ−sin⁡θsin⁡ϕ\cos(\theta+\phi)=\cos\theta\cos\phi-\sin\theta\sin\phi
sin⁡α+sin⁡β=2sin⁡α+β2cos⁡α−β2\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}
cos⁡α+cos⁡β=2cos⁡α+β2cos⁡α−β2\cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}
cos⁡α−cos⁡β=2sin⁡α+β2sin⁡β−α2\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⁡Ca/\sin A=b/\sin B=c/\sin C

and a2=b2+c2−2bccos⁡Aa^2=b^2+c^2-2bc\cos A.

Prefixes

SymbolPrefixFactor
Ttera101210^{12}
Ggiga10910^9
Mmega10610^6
kkilo10310^3
ccenti10−210^{-2}
mmilli10−310^{-3}
μ\mumicro10−610^{-6}
nnano10−910^{-9}
ppico10−1210^{-12}
ffemto10−1510^{-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

An alien spaceship from the planet Kepler 62f is in search of a rocky planet for a remote base. They’re attracted to Earth because of a fortunate coincidence: its axis of rotation points directly at their home planet. That means they can have uninterrupted communication with home by planting fixed transmitters on The North Pole. But first, they need to find out if Earth’s axis will always point in the same direction or if it undergoes precession. They can’t know without years of observation, so they hope that we, its now-extinct intelligence, have left behind the answer.

While orbiting Earth, they see a few remarkable structures, including the Hoover Dam in Nevada. Zooming in on the dam, a colorful plaza with peculiar markings on its floor catches their attention. Descending on the plaza, they realize the markings are a map of the sky when the dam was built, left to indicate the date to posterity. Figure 1 is an overhead architectural map of this plaza. The center-point depicts the north ecliptic pole, and the large circle represents the path of the Earth’s axis throughout its counter-clockwise procession. As they interpret the map, they’re dismayed to realize that their star has not been and will not be Earth’s north star for very long.

Scanned architectural drawing of the Hoover Dam plaza floor with concentric circular bands and hand-written dimensions; a red line runs from the label “Kepler-62f” at the upper left through the central point to a small circle labelled “Polaris” at the lower right, and a red arc at the center is marked “θ = 167.6°”.
Figure 1: Overhead architectural plan of the Hoover Dam plaza depicting Polaris as north star

For the purpose of this question, assume that the Earth’s axial tilt is a constant i=23.5∘i = 23.5^{\circ} and its axis precesses at a constant rate.

a) Using the values on the map, and knowing that the aliens used carbon-aging to determine that the dam is 12,000 years old, find all possible values for the period of Earth’s axial precession.

b) Using the most optimistic answer (longest period) from part (a), calculate how many arcseconds the Earth’s axis precesses each day. Use the period you calculate here in the next two sections.

c) If they hadn’t been lucky enough to come across the star map and decided to build a radio interferometer to observe the movement of the celestial pole over the course of 30 days instead, how many kilometers would the baseline of their telescope array have to be, assuming it operated at a 20cm wavelength?

d) As a last resort, to keep Earth’s axis fixed, the aliens decide to counter the forces that cause the Earth’s precession by building giant nuclear thrusters on the Earth’s surface. Assume Earth’s precession is caused by external forces alone and calculate the average force (in kN) that a strategically positioned thruster would have to exert to counter them.

Отговори

Решение

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