IOAA 2009, Теория — Задача 15

3rd International Olympiad on Astronomy and Astrophysics — Theoretical Competition · 10 т.

Внимание: Бележка към темата

The problem PDF's constants table has missing equation glyphs; the intact matching table and logarithm formula were recovered from original proceedings PDF 41, explicitly attributed. The full 147-page proceedings file and hash are retained, with a declared reading scope of 41 and47–67. Printed solution inconsistencies are retained and annotated: Q10 intermediate omits field factor/unit; Q12 negative sign and 3.21/3.19 mismatch; Q15 signed −1.5 and referenced but absent Figures 1/2; Q16(d) denominator minus sign; Q17 final m−5.2 versus preceding m+5.2. No missing figure or correction was invented.

Please read these instructions carefully
  1. Each student will receive problem sheets in English and/or in his/her native language.
  2. The available time for answering theoretical problems is 5 hours. You will have 15 short problems (Theoretical Part 1, Problem 1 to 15), and 2 long problems (Theoretical Part 2, Problem 16 and 17).
  3. Use only the pen that has been provided on your desk.
  4. Do Not use the back side of your writing sheets. Write only inside the boxed area.
  5. Yellow scratch papers are not considered in marking.
  6. Begin answering each problem in separate sheet.
  7. Fill in the boxes at the top of each sheet of your paper with your "country name", your "student code", "problem number", and total number of pages which is used to answer to that problem.
  8. Write the final answer for each problem in the box, labeled "Answer Sheet".
  9. Starting and the end of the exam will be announced by ringing a bell.
  10. The final answer in each question part must be accompanied by units, which should be in SI or appropriate units. 20% of the marks available for that part will be deducted for a correct answer without units.
  11. The required numerical accuracy for the final answer depends on the number of significant figures given in the data values in the problem. 20% of the marks available for the final answer in each question part will be deducted for answers without required accuracy as given in the problem. Use the constant values exactly as given in the table of constants.
  12. At the end of the exam put all papers, including scratch papers, inside the envelope and leave everything on your desk.
Table of Constants

The problem PDF's constants table has missing embedded equation glyphs. This table and useful formula are transcribed from the matching intact table on original PDF page 41 of IOAA2009_final.pdf (printed page 36), retaining the full proceedings source and its original page numbering.

(All constants are in SI)

ParameterSymbolValue
Gravitational constantGG6.67×10−11 N m2 kg−26.67\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}
Plank constanthh6.63×10−34 J s6.63\times10^{-34}\ \mathrm{J\,s}
ℏ\hbar1.05×10−34 J s1.05\times10^{-34}\ \mathrm{J\,s}
Speed of lightcc3.00×108 m s−13.00\times10^8\ \mathrm{m\,s^{-1}}
Solar MassM⊙M_\odot1.99×1030 kg1.99\times10^{30}\ \mathrm{kg}
Solar radiusR⊙R_\odot6.96×108 m6.96\times10^8\ \mathrm m
Solar luminosityL⊙L_\odot3.83×1026 W3.83\times10^{26}\ \mathrm W
Apparent solar magnitude (V)m⊙m_\odot−26.8-26.8
Solar constantb⊙b_\odot1.37×103 W m−21.37\times10^3\ \mathrm{W\,m^{-2}}
Mass of the EarthM⊕M_\oplus5.98×1024 kg5.98\times10^{24}\ \mathrm{kg}
Radius of the EarthR⊕R_\oplus6.38×106 m6.38\times10^6\ \mathrm m
Mean density of the Earthρ⊕\rho_\oplus5×103 kg m−35\times10^3\ \mathrm{kg\,m^{-3}}
Gravitational acceleration at sea levelgg9.81 m s−29.81\ \mathrm{m\,s^{-2}}
Tropical year365.24 days
Sidereal year365.26 days
Sidereal day86164 s
Inclination of the equator with respect to the eclipticϵ\epsilon23.5∘23.5^\circ
Parsecpc3.09×1016 m3.09\times10^{16}\ \mathrm m
Light yearly9.46×1015 m9.46\times10^{15}\ \mathrm m
Astronomical UnitAU1.50×1011 m1.50\times10^{11}\ \mathrm m
Solar distance from the center of the GalaxyRR8×103 pc8\times10^3\ \mathrm{pc}
Hubble constantHH75 km s−1 Mpc−175\ \mathrm{km\,s^{-1}\,Mpc^{-1}}
Mass of electronmem_e9.11×10−31 kg9.11\times10^{-31}\ \mathrm{kg}
Mass of protonmpm_p1.67×10−27 kg1.67\times10^{-27}\ \mathrm{kg}
Central wavelength of V-bandλ\lambda550 nm550\ \mathrm{nm}
Refraction of star light at horizon34′34'
π\pi3.1416

Useful mathematical formula:

ln⁡(1+x)≃xfor x→0.\ln(1+x)\simeq x\quad\text{for }x\to0.

Условие

An eclipsing close binary system consists of two giant stars with the same sizes. As a result of mutual gravitational force, stars are deformed from perfect sphere to the prolate spheroid with a=2ba=2b, where aa and bb are semi-major and semi-minor axes (the major axes are always co-linear). The inclination of the orbital plane to the plane of sky is 90∘90^\circ. Calculate the amplitude of light variation in magnitude (Δm\Delta m) as a result of the orbital motion of two stars. Ignore temperature variation due to tidal deformation and limb darkening on the surface of the stars.

Hint: A prolate spheroid is a geometrical shape made by rotating of an ellipse around its major axis, like rugby ball or melon.

Отговори

Решение

Теми и трудност

Трудност: неоценена.

Звезди · Светлина и астрономически измервания · Двойни звезди · Звездни величини

Предпоставки: Разширена олимпиадна подготовка

Оригинал в Архива: 2009_Theory.pdf · официални решения: IOAA2009_final.pdf