USAAAO 2021, I кръг — Задача 11

USAAAO 2021 - First Round · 30 януари 2021 г.

PHYSICAL AND ASTRONOMICAL CONSTANTS
cSpeed of light in vacuum2.998×108m s−12.998 \times 10^{8}\mathrm{m\ s^{-1}}
eElementary charge1.602×10−19 C1.602 \times 10^{-19}\ \mathrm{C}
mn_nNeutron rest mass1.675×10−27 kg1.675 \times 10^{-27}\ \mathrm{kg}
mp_pProton rest mass1.6725×10−27 kg1.6725 \times 10^{-27}\ \mathrm{kg}
me_eElectron rest mass9.110×10−31 kg9.110 \times 10^{-31}\ \mathrm{kg}
mHe_{He}Helium-4 rest mass6.644×10−27 kg6.644 \times 10^{-27}\ \mathrm{kg}
hPlanck’s constant6.626×10−34 J s6.626 \times 10^{-34}\ \mathrm{J\ s}
H0_0Hubble’s constant70 (km/s)/Mpc
kB_BBoltzmann’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_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_2Second Radiation Constant (=hc/k)(= hc/k)1.439×10−2 m K1.439 \times 10^{-2}\ \mathrm{m\ K}
NA_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_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⊙_\odotSolar mass1.989×1030 kg1.989 \times 10^{30}\ \mathrm{kg}
R⊙_\odotSolar radius6.96×108 m6.96 \times 10^{8}\ \mathrm{m}
L⊙_\odotSolar luminosity3.827×1026 J s−13.827 \times 10^{26}\ \mathrm{J\ s^{-1}}
T⊙_\odotSolar temperature5770 K
M⊕_\oplusEarth mass5.976×1024 kg5.976 \times 10^{24}\ \mathrm{kg}
R⊕_\oplusMean Earth radius6.371×106 m6.371 \times 10^{6}\ \mathrm{m}
I⊕_\oplusEarth 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 sidereal day86164 s
1 erg1×10−7 J1 \times 10^{-7}\ \mathrm{J}
1 bar105 N m−210^{5}\ \mathrm{N}\ m^{-2}

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

Условие

Let T⊙,CT_{\odot,C} and T⊙,ST_{\odot,S} be the temperatures at the core and the surface of the sun, respectively. Similarly, let TA,CT_{A,C} and TA,ST_{A,S} be the temperatures at the core and surface of the red giant Arcturus, and let TS,CT_{S,C} and TS,ST_{S,S} be the temperatures at the core and surface of the white dwarf Sirius B. Which of the following inequalities is true?

(a) T⊙,CT⊙,S<TA,CTA,S<TS,CTS,S\frac{T_{\odot,C}}{T_{\odot,S}} < \frac{T_{A,C}}{T_{A,S}} < \frac{T_{S,C}}{T_{S,S}}

(b) T⊙,CT⊙,S<TS,CTS,S<TA,CTA,S\frac{T_{\odot,C}}{T_{\odot,S}} < \frac{T_{S,C}}{T_{S,S}} < \frac{T_{A,C}}{T_{A,S}}

(c) TA,CTA,S<T⊙,CT⊙,S<TS,CTS,S\frac{T_{A,C}}{T_{A,S}} < \frac{T_{\odot,C}}{T_{\odot,S}} < \frac{T_{S,C}}{T_{S,S}}

(d) TS,CTS,S<T⊙,CT⊙,S<TA,CTA,S\frac{T_{S,C}}{T_{S,S}} < \frac{T_{\odot,C}}{T_{\odot,S}} < \frac{T_{A,C}}{T_{A,S}}

(e) TS,CTS,S<TA,CTA,S<T⊙,CT⊙,S\frac{T_{S,C}}{T_{S,S}} < \frac{T_{A,C}}{T_{A,S}} < \frac{T_{\odot,C}}{T_{\odot,S}}

Отговори

Решение

Оригинал в Архива: usaaao2021_exam1.pdf · официални решения: usaaao2021_exam1_solutions.pdf