USAAAO 2024, II кръг — Short 2

2024 National Astronomy Competition · 10 т.

Внимание: Бележка към темата

All eight whole original problems, their native sections, lettered parts, three roman subparts of Long 1(f), full official worked keys and original marking/acceptance notes are retained. Printed scientific discrepancies are preserved in local source notes. The official key adds a Friedmann-variable clarification and a solar apparent magnitude assumption absent from the question booklet, and changes Long1(f)(iii) from a factor to a percentage; these source variants are explicitly attributed. A separate undated common archive constants/reference table is included with its own source identity; its use as the exact 2024 provided attachment is not established, and its constants are not substituted into the printed key. The native cover specifies blank paper rather than a designated answer sheet. April 13th 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 225 points.

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 April 13th. 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.

We acknowledge the following people for their contributions to this year’s exam:

Abhay Bestrapalli, Orion Foo, Hagan Hensley, David Lee, Sandesh Kalantre, Andrew Liu, Joe McCarty, Leo Yao

Common archive reference — Table of Physical Constants

This separate common USAAAO archive reference is undated in the catalogue. Its use as the exact 2024 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}

Source note: This undated common table prints G=6.673×10−11 N m2 kg−2G=6.673\times10^{-11}\ \mathrm{N\ m^2\ kg^{-2}}, ME=5.9742×1024 kgM_E=5.9742\times10^{24}\ \mathrm{kg} and RE=6.3781×106 mR_E=6.3781\times10^6\ \mathrm m. The 2024 eclipse key instead uses G=6.674×10−11G=6.674\times10^{-11}, Earth mass 5.976×1024 kg5.976\times10^{24}\ \mathrm{kg} and Earth radius 6.371×106 m6.371\times10^6\ \mathrm m. Both original sets of values are preserved separately; this reference is not an established exact dated 2024 attachment.

Условие

David the astronomy enthusiast loves looking at stars! Specifically, he particularly enjoys looking at stars on the ecliptic. One day, he is out stargazing at midnight (local solar time) and looks at the antisolar point (the point on the celestial sphere exactly opposite to the Sun). He notices a faint glow of magnitude 12 mag/arcsec212\ \mathrm{mag/arcsec^2}, and after some research he concludes that this is caused by a phenomenon known as gegenschien, where Solar System dust is lit up by the Sun and reflects some light back towards Earth. These particles are in an orbit of 2.06 AU around the Sun. Assuming the radii of these particles are around 1 cm and their albedo is 0.14, estimate the density of these particles. Express your answer in particles per square arcsecond.

Отговори

Решение

Теми и трудност

Оценена трудност: 3/5.

Светлина и астрономически измервания · Слънчева система · Звездни величини · Поток и светимост · Астероиди, комети и метеори

Предпоставки: Гимназиална подготовка, Разширена олимпиадна подготовка

Оригинал в Архива: second_exam_2024.pdf · официални решения: second_exam_2024-solutions.pdf · допълнителни данни: physical-constants.pdf