USAAAO 2023, II кръг — Задача 1. Short 1

2023 National Astronomy Competition · 10 т.

Внимание: Бележка към темата

The separately attributed physical-constants.pdf is an undated common USAAAO archive reference; its identity as the table supplied for this exam is unproven. Printed question/key inconsistencies are preserved, including Medium 3 P5 radius and graph slope, Long 1 cosine denominator, and Long 2 exit-coordinate term and proper-motion magnitude.

Instructions (Please Read Carefully)

2023 National Astronomy Competition

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 240 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 1st. 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:

Wesley Antônio Machado Andrade de Aguiar, Erez Abrams, Lucas Pinheiro, Sahil Pontula, Joe McCarty, Hagan Hensley, Leo Yao, and Andrew Liu.

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

(10 points) White Dwarfs are stars in their last stage of life that are prevented from collapsing only by the electron degeneracy pressure. This pressure is an outward one exerted by the electrons inside the star, which are fermions subject to the Pauli exclusion principle. We can find its value by the following formula, which is derived from the theory of fermion gases:

pelectron=23u=ℏ25me(3π2)2/3n5/3p_{\mathrm{electron}}=\frac23u=\frac{\hbar^2}{5m_e}(3\pi^2)^{2/3}n^{5/3}

where nn is the number density of electrons in the star. This pressure balances the inward gravitational pressure, which is given by

pgrav=−Ω3V,Ω=−3GM25Rp_{\mathrm{grav}}=-\frac{\Omega}{3V},\quad\Omega=-\frac{3GM^2}{5R}

where Ω\Omega is the value of the total potential energy of the star.

(a) (8 points) If the star contains nuclei with atomic number ZZ and mass number AA, what is the density value of the white dwarf in function of its total mass MM, AA, ZZ, and other fundamental constants?

(b) (2 points) Find what is the value of kk in the relation M∝VkM\propto V^k, where VV is the volume of the star.

Решение

Оригинал в Архива: second_exam_2023-1.pdf · официални решения: second_exam_2023_sols-1.pdf · допълнителни данни: physical-constants.pdf