Съдържание

Условие Basic equations of spherical trigonometry — reference diagram Решение

IOAA 2015, Теория, дълги задачи, английски — Задача 3

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IOAA 2015

International Olympiad on Astronomy and Astrophysics 2015, Central Java - Indonesia — Long Problems

Внимание: Бележка към темата

The revised official marking scheme is primary. Earlier image-key questions and marking weights differ and are labelled separately. The revised key leaves problem 3(d) blank; the earlier distance derivation adds an assumption absent from the final question. Printed algebraic inconsistencies are retained.

Instruction — Theoretical Round

Write your answer on the answer-sheet only. You can use more than one sheet for one problem. Please ask the organizer for the additional answer-sheets.
Different problems should be done on different sheets.
You can use your own calculator without program and graphic facility.
You can ask the organizer for rough paper.

Astronomical and Physical Constants — 20150729

Constant	Value
Astronomical unit (AU)	$1.4960\times10^{11}\ \mathrm m$
Light year (ly)	$9.4605\times10^{15}\ \mathrm m=63\,240\ \mathrm{au}$
Parsec (pc)	$3.0860\times10^{16}\ \mathrm m=206\,265\ \mathrm{au}$
Jansky (Jy)	$10^{-26}\ \mathrm{W\,m^{-2}\,Hz^{-1}}$
1 Sidereal year	365.2564 solar days
1 Tropical year	365.2422 solar days
1 Calendar year	365.2425 solar days
1 Sidereal day	$23^{\mathrm h}\ 56^{\mathrm m}\ 04^{\mathrm s}.091$
1 Solar day	$24^{\mathrm h}\ 03^{\mathrm m}\ 56^{\mathrm s}.555$ units of sidereal time
1 Å	$10^{-10}\ \mathrm m$
1 erg	$10^{-7}\ \mathrm J$
1 dyne	$10^{-5}\ \mathrm N$
Mass of Earth	$5.9736\times10^{24}\ \mathrm{kg}$
Mean radius of Earth	$6.371\times10^6\ \mathrm m$
Equatorial radius of Earth	$6.378\times10^6\ \mathrm m$
Mean velocity of Earth on its orbit	$29.783\ \mathrm{km\,s^{-1}}$
Mass of Moon	$7.3490\times10^{22}\ \mathrm{kg}$
Radius of Moon	$1.737\times10^6\ \mathrm m$
Mass of Jupiter	$1.89813\times10^{27}\ \mathrm{kg}$
Mean Earth – Moon distance	$3.844\times10^8\ \mathrm m$
Mass of Sun	$1.98892\times10^{30}\ \mathrm{kg}$
Radius of Sun	$6.96\times10^8\ \mathrm m$
Effective temperature of the Sun	5780 K
Luminosity of the Sun	$3.96\times10^{26}\ \mathrm{J\,s^{-1}}$
Solar constant	$1366\ \mathrm{W\,m^{-2}}$
Brightness of the Sun in V-band	−26.8 mag.
Absolute brightness of the Sun in V-band	4.75 mag.
Absolute bolometric brightness of Sun	4.72 mag.
Angular diameter of the Sun	30′
Speed of light in vacuum (c)	$2.9979\times10^8\ \mathrm{m\,s^{-1}}$
Gravitational constant (G)	$6.6738\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}$
Boltzmann constant (k)	$1.381\times10^{-23}\ \mathrm{m\,kg\,s^{-2}\,K^{-1}}$
Universal gas constant (R)	$8.31\ \mathrm{J\,K^{-1}\,mol^{-1}}$
Stefan–Boltzmann constant (σ)	$5.6704\times10^{-8}\ \mathrm{kg\,s^{-3}\,K^{-4}}$
Planck constant (h)	$6.6261\times10^{-34}\ \mathrm{J\,s}$
Wien's constant (b)	$2.8978\times10^{-3}\ \mathrm{m\,K}$
Hubble constant (H₀)	$70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$

Astronomical and Physical Constants — continuation

Constant	Value
electron charge (e)	$1.602\times10^{-19}\ \mathrm C$
Mass of hydrogen atom	$1.67\times10^{-27}\ \mathrm{kg}$
Current inclination of the ecliptic (ε)	23º 26.3′
Coordinates of the northern ecliptic pole for epoch 2000.0 (αE, δE)	$18^{\mathrm h}\ 00^{\mathrm m}\ 00^{\mathrm s},+66^{\circ}\ 33.6^{\prime}$
Coordinates of the northern galactic pole for epoch 2000.0 (αG, δG)	$12^{\mathrm h}\ 51^{\mathrm m},+27^{\circ}\ 08^{\prime}$

You can try to solve an equation $x=f(x)$ using iteration: $x_{n+1}=f(x_n)$ .

Basic equations of spherical trigonometry

\sin a\sin B=\sin b\sin A

\sin a\cos B=\cos b\sin c-\sin b\cos c\cos A,

\cos a=\cos b\cos c+\sin b\sin c\cos A.

Rayleigh-Jeans formula (long wavelength approximation of Planck’s Law):

B_\nu(T)=\frac{2kT\nu^2}{c^2}

Условие

Suppose a static spherical star consists of $N$ neutral particles with radius $R$ (see Figure 3).

Sphere of radius R centred at the origin of x, y, z axes, with a dotted equator; a dashed radius R makes polar angle θ with the z axis and its projection onto the x-y plane makes azimuthal angle ϕ. — Figure 3

with $0 \le \theta \le \pi$ , $0 \le \phi \le 2\pi$ , satisfying the following equation of states

PV = N k \,\frac{T_R - T_0}{\ln(T_R/T_0)} \qquad (1)

where $P$ and $V$ are the pressure inside the star and the volume of the star respectively, $k$ is the Boltzmann constant. $T_R$ and $T_0$ are the temperatures at the surface $r = R$ and the temperature at the center $r = 0$ respectively. Assume that $T_R \le T_0$ .

a. Simplify the stellar equation of state (1) if $\Delta T = T_R - T_0 \approx 0$ (this is called ideal star)
(Hint: Use the approximation $\ln(1 + x) \approx x$ for small $x$ )

Suppose the star undergoes a quasi-static process, in which it may slightly contract or expand, such that the above stellar equation of state (1) still holds.

The star satisfies first law of thermodynamics

Q = \Delta M c^2 + W \qquad (2)

where $Q$ , $M$ , and $W$ are heat, mass of the star, and work respectively, while $c$ is the light speed in the vacuum and $\Delta M \equiv M_{\mathrm{final}} - M_{\mathrm{initial}}$ .

In the following we assume $T_0$ to be constant, while $T_R \equiv T$ varies.

b. Find the heat capacity of the star at constant volume $C_v$ in term of $M$ and at constant pressure $C_p$ expressed in $C_v$ and $T$ (Hint: Use the approximation $(1 + x)^n \approx 1 + nx$ for small $x$ )

Assuming that $C_v$ is constant and the gas undergo the isobaric process so the star produces the heat and radiates it outside to the space.

c. Find the heat produced by the isobaric process if the initial temperature and the final temperature are $T_i$ and $T_f$ , respectively.

d. Suppose there is an observer far away from the star. Using information from part c., estimate the distance of the observer from the star.

For the next parts, assume the star is the Sun.

e. If the sunlight is monochromatic with frequency $5 \times 10^{14}$ Hz, estimate the number of photons radiated by the Sun per second.

f. Calculate the heat capacity $C_v$ of the Sun assuming its surface temperature varies from 5500 K to 6000 K in one second.

Basic equations of spherical trigonometry — reference diagram

Spherical triangle with vertices A,B,C and opposite sides a,b,c.

Решение

Внимание: Непълно решение

The revised official key leaves final problem 3(d) blank (15 points). The earlier image edition gives a distance derivation only after adding a 0.1% temperature-error assumption absent from the final question; it is retained separately, not presented as a complete final-edition answer.

Revised marking scheme — a. (15 т.)

Defining $\Delta T=T_R-T_0$ and $\Delta T\approx0$ , we have

PV=Nk\frac{\Delta T}{\ln(1+\Delta T/T_0)}\qquad (3)

using $\ln(1+\Delta T/T_0)\approx\Delta T/T_0$ , we then obtain

PV=NkT_0\qquad (4)

Revised marking scheme — b. (30 т.)

The internal energy of the star is $U=Mc^2$ ( $U(T)=M(T)c^2$ for ideal star). Thus, the constant volume heat capacity of the star has the form

C_v=\left(\frac{\Delta Q}{\Delta T}\right)_V=\left(\frac{\Delta M}{\Delta T}\right)_Vc^2

for small $\Delta T$ . Then, using first law of thermodynamics, the constant pressure heat capacity of the star is

C_p=\left(\frac{\Delta Q}{\Delta T}\right)_P=\left(\frac{\Delta M}{\Delta T}\right)_Vc^2+P\frac{\Delta V}{\Delta T}=C_v+P\frac{\Delta V}{\Delta T}

for small $\Delta T$ . Defining $\Delta T=T_2-T_1$ , then

P\Delta V=Nk\left(\frac{T_1-T_0+\Delta T}{\ln((T_1+\Delta T)/T_0)}-\frac{T_1-T_0}{\ln(T_1/T_0)}\right)

Using the approximation

\ln((T_1+\Delta T)/T_0)\approx\ln\left(\frac{T_1}{T_0}\right)+\frac{\Delta T}{T_1}

\left(1+\frac{\Delta T}{T_1\ln(T_1/T_0)}\right)^{-1}\approx1-\frac{\Delta T}{T_1\ln(T_1/T_0)}

then we have

P\frac{\Delta V}{\Delta T}=\frac{Nk}{\ln(T/T_0)}\left(1-\frac{(T-T_0)/T}{\ln(T/T_0)}\right)

where $T_1\equiv T$ . Finally, we obtain

C_p=\left(\frac{\Delta Q}{\Delta T}\right)_P=\left(\frac{\Delta M}{\Delta T}\right)_Vc^2+P\frac{\Delta V}{\Delta T}=C_v+\frac{Nk}{\ln(T/T_0)}\left(1-\frac{(T-T_0)/T}{\ln(T/T_0)}\right)

Revised marking scheme — c. (20 т.)

Since $C_v$ is constant, the heat produced by the star is given by

Q_H=C_v(T_f-T_i)+P\Delta V

=C_v(T_i-T_f)+Nk\left(\frac{T_f-T_0}{\ln(T_f/T_0)}-\frac{T_i-T_0}{\ln(T_i/T_0)}\right)

Revised marking scheme — d. (15 т.)

Part	Points
d.	15

Revised marking scheme — e. (10 т.)

Energy per second radiated by the Sun

L_\odot=Nh\nu

where $N$ is the number of photon. Thus

N=\frac{L_\odot}{h\nu}=\frac{3.96\times10^{26}}{6.6261\times10^{-34}\times5\times10^{14}}=1.195\times10^{45}\ \text{photons}

Revised marking scheme — f. (10 т.)

Energy per second radiated by the Sun is proportional to mass defect of the Sun

L_\odot=\Delta Mc^2

Thus,

C_v=\frac{\Delta Mc^2}{\Delta T}=\frac{L_\odot}{\Delta T}=\frac{3.96\times10^{26}}{6000-5500}\ \mathrm{J/K}=7.92\times10^{23}\ \mathrm{J/K}

Earlier image edition — b. Isothermal work (15 т.)

Find the work of the star when it expands from $V_1$ to $V_2$ in isothermal process where $T_R$ and $T_0$ are constants.

If the gas undergoes isothermal process where $T_R$ and $T_0$ are constant, then the work has the form

W=\int_{V_1}^{V_2}P\,dV=Nk\frac{T_R-T_0}{\ln(T_R/T_0)}\ln\left(\frac{V_2}{V_1}\right)

Earlier image edition — e. Distance with 0.1% temperature error (15 т.)

Suppose there is an observer far away from the star. Related to point d., estimate the distance of the observer if the observer has 0.1% error in measuring the effective temperature around the star.

The error 0.1%, namely

\frac{T_f-T_{\mathrm{eff}}}{T_f}=\frac1{1000}

then $T_{\mathrm{eff}}=\frac{999}{1000}T_f$ . From Stefan law of black body radiation

\frac{Q_H}{4\pi r_0^2}=\sigma T_{\mathrm{eff}}^4=\sigma\left(\frac{999}{1000}\right)^4T_f^4

where $r_0$ is the observer's distance which is given by

r_0=\left(\frac{Q_H}{4\pi\sigma}\right)^{1/2}\left(\frac{1000}{999}\right)^2T_f^{-2}

Earlier image edition — marking weights

Part	Points
a.	10
b. Isothermal work	15
c. Heat capacities	25
d. Isobaric heat	20
e. Distance	15
f. Photons	7
g. Heat capacity	8

Earlier image edition — original page 7

Complete original earlier-edition key page 7 of 10, preserving draft questions, diagrams and marking weights.

Earlier image edition — original page 8

Complete original earlier-edition key page 8 of 10, preserving draft questions, diagrams and marking weights.

Earlier image edition — original page 9

Complete original earlier-edition key page 9 of 10, preserving draft questions, diagrams and marking weights.

Earlier image edition — original page 10

Complete original earlier-edition key page 10 of 10, preserving draft questions, diagrams and marking weights.

Оригинал в Архива: Long_problem_Final_EN.pdf · официални решения: Long Problem Solution-Marking Scheme [Revised] - 310715 - 13.58.pdf · допълнителни данни: long_sol_1.png · допълнителни данни: long_sol_2.png · допълнителни данни: long_sol_3.png · допълнителни данни: long_sol_4.png · допълнителни данни: long_sol_5.png · допълнителни данни: long_sol_6.png · допълнителни данни: long_sol_7.png · допълнителни данни: long_sol_8.png · допълнителни данни: long_sol_9.png · допълнителни данни: long_sol_10.png · допълнителни данни: 9th IOAA Constants Table 20150729.pdf · допълнителни данни: constants.pdf · допълнителни данни: instructions.docx