USAAAO 2020, I кръг — Задача 24

USAAAO National Astronomy Olympiad First Round 02/10/2020-02/16/2020 · 10 февруари 2020 г. · 1 т.

Указания

This exam contains 8 pages and 30 questions. Each question is worth 1 point, so there are 30 points total. Students MUST take the exam between 02/10/2020-02/16/2020. Proctors must scan students’ responses by 11:59 pm PST on 02/16/2020.

PHYSICAL AND ASTRONOMICAL CONSTANTS
cSpeed of light in vacuo2.998×108m s−12.998 \times 10^{8} \mathrm{m\ s^{-1}}
eElementary charge1.602×10−19 C1.602 \times 10^{-19}\ \mathrm{C}
mn\mathrm{m}_nNeutron rest mass1.675×10−27 kg1.675 \times 10^{-27}\ \mathrm{kg}
mp\mathrm{m}_pProton rest mass1.673×10−27 kg1.673 \times 10^{-27}\ \mathrm{kg}
me\mathrm{m}_eElectron rest mass9.110×10−31 kg9.110 \times 10^{-31}\ \mathrm{kg}
hPlanck’s constant6.626×10−34 J s6.626 \times 10^{-34}\ \mathrm{J\ s}
H0\mathrm{H}_0Hubble’s constant70 (km/s)/Mpc
kBoltzmann’s constant1.381×10−23 J K−11.381 \times 10^{-23}\ \mathrm{J\ K^{-1}}
bWien’s constant2.898×10−3 m K2.898 \times 10^{-3}\ \mathrm{m\ K}
GGravitational constant6.673×10−11 N m2 kg−26.673 \times 10^{-11}\ \mathrm{N\ m^{2}\ kg^{-2}}
σ\sigmaStefan-Boltzmann constant5.670×10−8 J m−2 K−4 s−15.670 \times 10^{-8}\ \mathrm{J\ m^{-2}\ K^{-4}\ s^{-1}}
c1\mathrm{c}_1First Radiation Constant (=2πhc2)(= 2\pi hc^2)3.742×10−16 J m2 s−13.742 \times 10^{-16}\ \mathrm{J\ m^{2}\ s^{-1}}
c2\mathrm{c}_2Second Radiation Constant (=hc/k)(= hc/k)1.439×10−2 m K1.439 \times 10^{-2}\ \mathrm{m\ K}
NA\mathrm{N}_AAvogadro constant6.022×1023 mol−16.022 \times 10^{23}\ \mathrm{mol^{-1}}
RGas constant8.314 J K−1 mol−18.314\ \mathrm{J\ K^{-1}\ mol^{-1}}
a0\mathrm{a}_0Bohr radius5.292×10−11 m5.292 \times 10^{-11}\ \mathrm{m}
μB\mu_BBohr magneton9.274×10−24 J T−19.274 \times 10^{-24}\ \mathrm{J\ T^{-1}}
M⊙\mathrm{M}_{\odot}Solar mass1.989×1030 kg1.989 \times 10^{30}\ \mathrm{kg}
R⊙\mathrm{R}_{\odot}Solar radius6.96×108 m6.96 \times 10^{8}\ \mathrm{m}
L⊙\mathrm{L}_{\odot}Solar luminosity3.827×1026 J s−13.827 \times 10^{26}\ \mathrm{J\ s^{-1}}
T⊙\mathrm{T}_{\odot}Solar temperature5770 K
M⊕\mathrm{M}_{\oplus}Earth mass5.976×1024 kg5.976 \times 10^{24}\ \mathrm{kg}
R⊕\mathrm{R}_{\oplus}Mean Earth radius6.371×106 m6.371 \times 10^{6}\ \mathrm{m}
I⊕\mathrm{I}_{\oplus}Earth moment of Inertia8.04×1037 kg m28.04 \times 10^{37}\ \mathrm{kg}\ m^{2}
R☾Mean Moon radius1.737×106 m1.737 \times 10^{6}\ \mathrm{m}
M♃Mean Jupiter mass1.9×1027 kg1.9 \times 10^{27}\ \mathrm{kg}
R♃Mean Jupiter radius7.1492×107 m7.1492 \times 10^{7}\ \mathrm{m}
a♃Mean orbital radius of Jupiter5.2 AU
a☾Mean semimajor axis Moon orbit3.84399×108 m3.84399 \times 10^{8}\ \mathrm{m}
1 light year9.461×1015 m9.461 \times 10^{15}\ \mathrm{m}
1 AUAstronomical Unit1.496×1011 m1.496 \times 10^{11}\ \mathrm{m}
1 pcParsec3.086×1016 m3.086 \times 10^{16}\ \mathrm{m}
1 year3.156×107 s3.156 \times 10^{7}\ \mathrm{s}
1 erg1×10−7 J1 \times 10^{-7}\ \mathrm{J}

ENERGY CONVERSION : 1 joule (J) = 6.2415×10186.2415 \times 10^{18} electronvolts (eV)

Условие

The information below applies to the following two questions:

In a particular compact binary system consisting of a black hole and a main sequence star, the black hole has a mass of 23.2MSun23.2M_{Sun} and the main sequence star has mass of 15.6MSun15.6M_{Sun}. The two stars are separated by 1 AU.

We assume the accretion disk around the black hole is in a steady state (i.e. does not change with time) and that it has a perfectly circular cross section. We further assume that the mass transfer rate is constant. At a distance r1r_1 from the black hole, the temperature of the disk is T1T_1. What is the temperature T2T_2 of the disk at a distance r2=2r1r_2 = 2r_1? Hint: consider the Stefan-Boltzmann law.

A. 2.0

B. 1.4

C. 0.71

D. 0.60

E. 0.50

Решение

Оригинал в Архива: 2020_usaaao_first_round.pdf · официални решения: usaaao_first_round_answer_key_2020.pdf