USAAAO 2021, II кръг — Short 3

2021 National Astronomy Competition · 5 т.

Внимание: Бележка към темата

All ten whole original Short/Medium/Long problems, 30 lettered parts, 9 nested roman requests and 7 numbered satellite quantities, all original worked keys and marking notes are retained. Printed scientific discrepancies are preserved in attributed local source notes. Medium 3 original calls both masses m; its key introduces a distinct M for the station and assumes m<Mm\lt M. A separate undated common archive constants/reference table is fully included with its own source identity; its use as the exact 2021 provided attachment is not established, and its constants do not replace native exam values. The native cover specifies blank paper and a signed declaration. March 20th is the end of the examination period, not an established single sitting date.

1 Instructions (Please Read Carefully)

The top 5 eligible scorers on the NAC 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, Medium Questions and Long Questions.

The maximum number of points is 180.

The test must be completed within 2.5 hours (150 minutes).

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 is a written exam. Contestants can only use a scientific or graphing 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 20th. Receiving any external help during the exam is strictly prohibited. This means that the only allowed items are: a calculator, the provided table of constants, a pencil (or pen), an eraser, blank sheets of papers, and the exam. No books or notes are allowed during the exam. Exam is proctored and recorded. You are expected to have your video on at all times.

After reading the instructions, please make sure to sign, affirming that:

  1. All work on this exam is mine.
  2. I did not receive any external aids besides the materials provided.
  3. I am not allowed to discuss the test with others throughout the period of this examination.
  4. Failure to follow these rules will lead to disqualification from the exam.

Signed: ____________________

Common archive reference — Table of Physical Constants

This separate common USAAAO archive reference is undated in the catalogue. Its use as the exact 2021 exam attachment is not established.

QuantitySymbolValue
Electron rest massmem_e9.109×10−31 kg9.109\times10^{-31}\ \mathrm{kg}
Proton rest massMpM_p1.6726×10−27 kg1.6726\times10^{-27}\ \mathrm{kg}
Electronic chargeee1.6022×10−19 C1.6022\times10^{-19}\ \mathrm C
Speed of light in free spacecc2.9979×108 m s−12.9979\times10^8\ \mathrm{m\ s^{-1}}
Permeability of free spaceμ0\mu_04π×10−7 H m−14\pi\times10^{-7}\ \mathrm{H\ m^{-1}}
Permittivity of free spaceϵ0\epsilon_08.854×10−12 F m−18.854\times10^{-12}\ \mathrm{F\ m^{-1}}
Planck’s constanthh6.626×10−34 J s6.626\times10^{-34}\ \mathrm{J\ s}
Reduced Planck’s constantℏ=h/2π\hbar=h/2\pi1.0546×10−34 J s1.0546\times10^{-34}\ \mathrm{J\ s}
ℏc\hbar c197.33 MeV fm197.33\ \mathrm{MeV\ fm}
Boltzmann’s constantkBk_B1.3807×10−23 J K−11.3807\times10^{-23}\ \mathrm{J\ K^{-1}}
Gas constantR=kB/mH\mathcal R=k_B/m_H8.250×103 J kg−1 K−18.250\times10^3\ \mathrm{J\ kg^{-1}\ K^{-1}}
Molar gas constantRR8.315 J mol−1 K−18.315\ \mathrm{J\ mol^{-1}\ K^{-1}}
Avogadro’s numberNAN_A6.022×1023 mol−16.022\times10^{23}\ \mathrm{mol^{-1}}
Standard molar volume22.414×10−3 m3 mol−122.414\times10^{-3}\ \mathrm{m^3\ mol^{-1}}
Unified atomic mass unit (12C^{12}\mathrm C scale)uu931.5 MeV/c2=1.660538×10−27 kg931.5\ \mathrm{MeV}/c^2=1.660538\times10^{-27}\ \mathrm{kg}
Mass of hydrogen atommHm_H1.0078u=1.6735×10−27 kg1.0078u=1.6735\times10^{-27}\ \mathrm{kg}
Bohr magnetonμB\mu_B9.274×10−24 A m29.274\times10^{-24}\ \mathrm{A\ m^2} or J T−1\mathrm{J\ T^{-1}}
Nuclear magnetonμN\mu_N5.051×10−27 A m25.051\times10^{-27}\ \mathrm{A\ m^2} or J T−1\mathrm{J\ T^{-1}}
Proton magnetic momentμp\mu_p2.7928μN2.7928\mu_N
Neutron magnetic momentμn\mu_n−1.9130μN-1.9130\mu_N
Bohr radiusa0a_05.292×10−11 m5.292\times10^{-11}\ \mathrm 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)2.4263×10−12 m2.4263\times10^{-12}\ \mathrm m
Rydberg’s constantR∞R_{\infty}1.0974×107 m−11.0974\times10^7\ \mathrm{m^{-1}}
R∞hcR_{\infty}hc13.606 eV13.606\ \mathrm{eV}
Stefan-Boltzmann constantσ\sigma5.671×10−8 W m−2 K−45.671\times10^{-8}\ \mathrm{W\ m^{-2}\ K^{-4}}
Radiation density constanta=4σ/ca=4\sigma/c7.561×10−16 J m−3 K−47.561\times10^{-16}\ \mathrm{J\ m^{-3}\ K^{-4}}
Gravitational constantGG6.673×10−11 N m2 kg−26.673\times10^{-11}\ \mathrm{N\ m^2\ kg^{-2}}

Rest masses of some leptons and hadrons in MeV/c2\mathrm{MeV}/c^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±\mathrm K^{\pm} 493.7, K0\mathrm K^0 497.7, η\eta 547, D0\mathrm D^0 1865, D±\mathrm 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, Z0\mathrm Z^0 91.187×10391.187\times10^3, W±\mathrm W^{\pm} 80.41×10380.41\times10^3.

QuarkChargeI3I_3SSCCBBTTMass (GeV/c2\mathrm{GeV}/c^2)
u+23+\frac2312\frac120000∼0.003\sim0.003
d−13-\frac13−12-\frac120000∼0.006\sim0.006
c+23+\frac2300+100∼1.25\sim1.25
s−13-\frac130-1000∼0.11\sim0.11
t+23+\frac230000+1174.3
b−13-\frac13000-104.2
Common archive reference — Astrophysical Data and identities

Astrophysical Data

QuantitySymbolValue
1 astronomical unitAU1.496×1011 m1.496\times10^{11}\ \mathrm m
1 parsecpc3.086×1016 m3.086\times10^{16}\ \mathrm m
Luminosity of SunL⊙L_{\odot}3.85×1026 W3.85\times10^{26}\ \mathrm W
Mass of SunM⊙M_{\odot}1.989×1030 kg1.989\times10^{30}\ \mathrm{kg}
Radius of SunR⊙R_{\odot}6.96×108 m6.96\times10^8\ \mathrm m
Mass of EarthMEM_E5.9742×1024 kg5.9742\times10^{24}\ \mathrm{kg}
Radius of EarthRER_E6.3781×106 m6.3781\times10^6\ \mathrm m

Other data and conversion factors

QuantitySymbolValue
1 ångstromÅ10−10 m10^{-10}\ \mathrm m
1 fermifm10−15 m10^{-15}\ \mathrm m
1 barnb10−28 m210^{-28}\ \mathrm{m^2}
1 pascalPa1 Nm−21\ \mathrm{Nm^{-2}}
1 standard atmosphere1.0132×105 Pa1.0132\times10^5\ \mathrm{Pa}
Standard acceleration due to gravitygg9.807 m s−29.807\ \mathrm{m\ s^{-2}}
1 electron volteV1.6022×10−19 J1.6022\times10^{-19}\ \mathrm J
eV/hc\mathrm{eV}/hc8.065×105 m−18.065\times10^5\ \mathrm{m^{-1}}
eV/kB\mathrm{eV}/k_B1.1604×104 K1.1604\times10^4\ \mathrm K
Wavelength of 1 eV photon1.2399×10−6 m1.2399\times10^{-6}\ \mathrm m

Trigonometrical identities

sin⁡(θ+ϕ)=sin⁡(θ)cos⁡(ϕ)+cos⁡(θ)sin⁡(ϕ)\sin(\theta+\phi)=\sin(\theta)\cos(\phi)+\cos(\theta)\sin(\phi)
cos⁡(θ+ϕ)=cos⁡(θ)cos⁡(ϕ)−sin⁡(θ)sin⁡(ϕ)\cos(\theta+\phi)=\cos(\theta)\cos(\phi)-\sin(\theta)\sin(\phi)
sin⁡α+sin⁡β=2sin⁡12(α+β)cos⁡12(α−β)\sin\alpha+\sin\beta=2\sin\frac12(\alpha+\beta)\cos\frac12(\alpha-\beta)
cos⁡α+cos⁡β=2cos⁡12(α+β)cos⁡12(α−β)\cos\alpha+\cos\beta=2\cos\frac12(\alpha+\beta)\cos\frac12(\alpha-\beta)
cos⁡α−cos⁡β=2sin⁡12(α+β)sin⁡12(β−α)\cos\alpha-\cos\beta=2\sin\frac12(\alpha+\beta)\sin\frac12(\beta-\alpha)

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

PrefixNameFactor
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}

Условие

Galaxies are very hard to spot, even those that are nearest to us. For instance, Andromeda, despite having an apparent magnitude of 3.44, appears very “dim” in the sky. This is because its light is very spread out, since its solid angle in the sky is so large (around 3 times that of the Sun!).

Hence, it is often useful to use the surface magnitude of a galaxy, defined as the magnitude that a certain solid angle of that galaxy has. It is usually measured in mag/arcmin2\mathit{mag/arcmin^2}.

Show that, in a non expanding universe, the surface magnitude is independent of the distance to the galaxy.

Отговори

Решение

Теми и трудност

Оценена трудност: 2/5.

Светлина и астрономически измервания · Звездни величини · Ъглов размер

Предпоставки: Гимназиална подготовка, Разширена олимпиадна подготовка

Оригинал в Архива: usaaao_second_exam_2021_final.pdf · официални решения: usaaao_second_exam_2021_solutions.pdf · допълнителни данни: physical-constants.pdf