USAAAO 2026, II кръг — Задача 1. A “gravitational atom”

2026 National Astronomy Competition · 10 т.

Внимание: Бележка към темата

Questions 3 and 9 refer to designated answer sheets, and question 6 to provided grid paper; these sheets were not obtained in the two native 2026-II PDFs or their two-row same-folder archive catalogue. The separately attributed physical-constants.pdf is an undated common USAAAO archive reference; its identity as the table supplied for this exam is unproven.

Instructions (Please Read Carefully)

2026 National Astronomy Competition

1 Instructions (Please Read Carefully)

The top 15 eligible scorers on the NAC will be invited to the training camp. A third round will be conducted during the camp to select the national team for the IOAA 2026. In order to qualify for camp, you must be a high school student with US citizenship or permanent residency.

The test must be completed within 2.5 hours (150 minutes) and the maximum number of points is 300 points. The point breakdown per problem is as follows.

ProblemP1P2P3P4P5P6P7P8P9Total
Points101515202040405585300

Please solve each problem on a blank piece of paper and mark the number of the problem at the top of the page. Some problems may require a designated answer sheet; these will be provided. 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 calculator (non-programmable and non-graphing) 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 20, 2026. 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. Students must be proctored during the entire duration of the exam.

This exam sheet and your answer sheets (including scratch papers) should be returned to your proctors once the exam ended. Your proctors should upload your answer sheets to the Google Form provided to them.

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

Srihari (Hari) Balaji, Feodor Yevtushenko, Ferdinand, Hagan Hensley, Joe McCarty, Lucas Pinheiro, Vincent Bian, Elizabeth Lee

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

  1. All work on this exam has been completed by me.
  2. I took this exam under the supervision of a proctor.
  3. I did not receive any external help beyond the materials provided.
  4. I will not discuss the contents of this exam with anyone until April 20, 2026.
  5. Failure to follow these rules will result in disqualification from the exam.

Student Signature: ____________________ Date: ____________________

Undated common archive reference · Table of Physical Constants

Table of Physical Constants

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 (C12C\vphantom{C}^{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_\infty1.0974×107 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
Undated common archive reference · Astrophysical Data and Conversion Factors

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_\odot3.85×1026 W3.85\times10^{26}\,\mathrm W
Mass of SunM⊙M_\odot1.989×1030 kg1.989\times10^{30}\,\mathrm{kg}
Radius of SunR⊙R_\odot6.96×108 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 N m−21\,\mathrm{N\,m^{-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

T = tera = 101210^{12}c = centi = 10−210^{-2}
G = giga = 10910^9m = milli = 10−310^{-3}
M = mega = 10610^6μ\mu = micro = 10−610^{-6}
k = kilo = 10310^3n = nano = 10−910^{-9}
p = pico = 10−1210^{-12}
f = femto = 10−1510^{-15}

Условие

Bohr’s semiclassical model of the hydrogen atom assumed that the ground state angular momentum of the electron is ℏ=h/(2π)\hbar=h/(2\pi). While this model is incorrect, the Bohr radius does give the correct lengthscale of the hydrogen atom. By analogy to the Bohr model, consider a bound system consisting of a neutron and an electron interacting purely gravitationally. If the orbital angular momentum is ℏ\hbar, what would be the radius of this “atom”?

Решение

Оригинал в Архива: usaaao2026.pdf · официални решения: usaaao2026_sol.pdf · допълнителни данни: physical-constants.pdf