IOAA 2008, Анализ на данни — Задача 1. Virgo Cluster

The 2nd International Olympiad on Astronomy and Astrophysics — Practical Competition: Data Analysis · 21 август 2008 г. · 100 т.

Please read this carefully:
  1. Every student receives problem sheets in English and/or in his/her native language, answer sheets, millimeter block papers, and scratch sheets.
  2. The time available is five hours for the data analysis and observation competitions. There are three data analysis problems, and two observation problems.
  3. Use only the materials provided.
  4. Fill in the boxes at the top of each sheet of paper with your country code and your student code.
  5. Use only the front side of answer sheets. Write only inside the boxed area.
  6. Begin answering each question on a separate sheet.
  7. Numerical results should be written with as many digits as are appropriate.
  8. Write on the answer sheets and the millimeter block papers whatever you consider is required for the solution of each question. Please express your answer primarily in term of equations, numbers, figures, and plots. If necessary provide your answers with concise text. Full credit will be given to correct answer with detailed steps for each question. Underline your final result.
  9. At the end of the exam place the answer sheets and the millimeter block papers inside the envelope and leave everything on your desk.
Astronomical and Physical Constants
QuantityValue
Astronomical unit (AU)149,597,870.691 km
Light year (ly)9.4605×10179.4605 \times 10^{17} cm = 63,240 AU
Parsec (pc)3.0860×10183.0860 \times 10^{18} cm = 206,265 AU
Sidereal year365.2564 days
Tropical year365.2422 days
Gregorian year365.2425 days
Sidereal month27.3217 days
Synodic month29.5306 days
Mean sidereal day23h56m4s.09123^{\mathrm{h}}56^{\mathrm{m}}4^{\mathrm{s}}.091 of mean solar time
Mean solar day24h3m56s.55524^{\mathrm{h}}3^{\mathrm{m}}56^{\mathrm{s}}.555 of sidereal time
Mean distance, Earth to Moon384,399 km
Earth mass (M⊕\mathfrak{M}_{\oplus})5.9736×10275.9736 \times 10^{27} g
Earth's mean radius6,371.0 km
Earth's mean velocity in orbit29.783 km/s
Moon’s mass (𝔐☽)7.3490×10257.3490 \times 10^{25} g
Moon’s mean radius1,738 km
Sun mass (M⊙\mathfrak{M}_{\odot})1.9891×10331.9891 \times 10^{33} g
Mean Earth radius6.3710×1066.3710 \times 10^{6} cm
Sun radius (R⊙R_{\odot})6.96×10106.96 \times 10^{10} cm
Sun luminosity (L⊙L_{\odot})3.96×10333.96 \times 10^{33} erg s−1^{-1}
Sun effective temperature (Teff⊙T_{eff\odot})5 800 °K
Sun apparent magnitude (m⊙m_{\odot})-26.8
Sun bolometric magnitude (mbol⊙m_{bol\odot})-26.79
Sun absolute magnitude (M⊙M_{\odot})4.82
Sun absolute bolometric magnitude (Mbol⊙M_{bol\odot})4.72
Speed of light (cc)2.9979×10102.9979 \times 10^{10} cm/s
Gravitational constant (GG)6.6726×10−86.6726 \times 10^{-8} dyne cm2^{2} g−2^{-2}
Boltzmann constant (kk)1.3807×10−161.3807 \times 10^{-16} erg. K−1^{-1}
Stefan-Boltzmann constant (σ\sigma)5.6705×10−55.6705 \times 10^{-5} erg cm−2^{-2} K−4^{-4} s−1^{-1}
Planck constant (hh)6.6261×10−276.6261 \times 10^{-27} erg s
Electron charge (ee)1.602×10−191.602 \times 10^{-19} C = 4.803×10−104.803 \times 10^{-10} esu
Electron mass (mem_e)5.48579903×10−45.48579903 \times 10^{-4} amu
Proton mass (mpm_p)1.007276470 amu
Neutron mass (mnm_n)1.008664904 amu
Deuterium nucleus mass (mdm_d)2.013553214 amu
Hydrogen mass1.00794 amu
Helium mass4.002603 amu
Carbon mass12.01070 amu

Conversion table

1 Å0.1 nm
1 barn10−2810^{-28} m2^{2}
1 G10−410^{-4} T
1 erg10−710^{-7} J = 1 dyne cm
1 esu3.3356×10−103.3356 \times 10^{-10} C
1 amu (atomic mass unit)1.6606×10−241.6606 \times 10^{-24} g
1 atm (atmosphere)101,325 Pa = 1.01325 bar
1 dyne10−510^{-5} N
Points

300 points for 3 problems, 100 points for each problem

Условие

The Virgo cluster of galaxies is the nearest large cluster which extends over nearly 10 degrees across the sky and contains a number of bright galaxies. It is very interesting to find the distance to Virgo and to deduce certain cosmological information from it. The table below provides the distance estimates using various distance indicators (listed in the left column). The right column lists the mean distance did_i ± the standard deviation sis_i.

iiDistance IndicatorVirgo Distance (Mpc
1Cepheids14.9±1.214.9 \pm 1.2
2Novae21.1±3.921.1 \pm 3.9
3Planetary Nebulae15.2±1.115.2 \pm 1.1
4Globular Cluster18.8±3.818.8 \pm 3.8
5Surface Brightness Fluctuation15.9±0.915.9 \pm 0.9
6Tully-Fisher relation15.8±1.515.8 \pm 1.5
7Faber-Jackson relation16.8±2.416.8 \pm 2.4
8Type Ia Supernovae19.4±5.019.4 \pm 5.0

1. By applying a weighted mean, compute the average distance (which can be taken as an estimate to the distance to Virgo)

davg=∑idisi2∑i1si2d_{avg} = \frac{\sum_i \frac{d_i}{s_i^2}}{\sum_i \frac{1}{s_i^2}}

where the sum runs over the eight distance indicator used.

2. What is the uncertainty (rms) (in unit of Mpc) in that estimate?

3. Spectra of the galaxies in Virgo indicate an average recession velocity of 1136 km/sec for the cluster. Can you estimate the Hubble constant H0H_0 and its uncertainty (rms)?

4. What is the Hubble Time (age of the universe) using the value of Hubble constant you found and the uncertainty (rms)?

Отговори

Решение

Оригинал в Архива: 2008_Data_Analysis.pdf · официални решения: dataan_sol.doc