IOAA 2023, Отборен тур — Задача 6. Saros

Group Competition

Указания
  • For this round, the participants will be grouped by random selection into 5-person teams, each representing a named asteroid. Each participant in the group will be from a different country. The selection will take place near the beginning of the IOAA, so that the team members can get to know each other.
  • Please remember the name and number of your asteroid, as this will also be used to identify your place during the Planetarium and Observation rounds.
  • The group competition consists of several tasks, which you will receive in a sealed envelope. Each team works together at one table under the supervision of the guides to solve the tasks. You are not allowed to communicate with participants from other teams during the round.
  • Dedicated answer sheets are provided for writing your answers. Enter the final answers into the appropriate boxes in the answer sheet (marked A).
  • Open the envelopes on the START signal given by the judges. Time is measured from this moment; the winning team will be the one which finishes in the shortest total time, after any time penalties (for example for incorrect or missing answers) are applied. The time penalties are explained in each task.
  • When you have solved all the problems, hand your answer sheets to the guide, who will note the total time.
  • The maximum time available for the round is 90 minutes. After this time any remaining answer sheets should be handed in.
  • The completed answer sheets will then be marked by the jury, who will apply time penalties as appropriate. The winning team will be announced at the closing ceremony.
  • Everything you need will be provided on the table (calculator, office supplies, geometrical instruments, paper, table of constants).
  • For one of the tasks, all the screens in the room will simultaneously display a video at a specified time (the video will be repeated several times).
Table of Constants
Fundamental constants
Speed of light in vacuumcc==2.998×108 m s−12.998 \times 10^{8}\ \mathrm{m\,s^{-1}}
Planck constanthh==6.626×10−34 J s6.626 \times 10^{-34}\ \mathrm{J\,s}
Boltzmann constantkBk_B==1.381×10−23 J K−11.381 \times 10^{-23}\ \mathrm{J\,K^{-1}}
Stefan-Boltzmann constantσ\sigma==5.670×10−8 W m−2 K−45.670 \times 10^{-8}\ \mathrm{W\,m^{-2}\,K^{-4}}
Elementary chargeee==1.602×10−19 C1.602 \times 10^{-19}\ \mathrm{C}
Universal gravitational constantGG==6.674×10−11 N m2 kg−26.674 \times 10^{-11}\ \mathrm{N\,m^{2}\,kg^{-2}}
Universal electric constantϵ0\epsilon_0==8.854×10−12 m−3kg−1s4A28.854 \times 10^{-12}\ \mathrm{m^{-3}kg^{-1}s^{4}A^{2}}
Universal gas constantRR==8.315 J mol−1 K−18.315\ \mathrm{J\,mol^{-1}\,K^{-1}}
Avogadro constantNAN_\mathrm{A}==6.022×1023 mol−16.022 \times 10^{23}\ \mathrm{mol^{-1}}
Wien’s displacement constantb=λmTb = \lambda_m T==2.898×10−3 m K2.898 \times 10^{-3}\ \mathrm{m\,K}
Mass of electronmem_\mathrm{e}==9.109×10−31 kg9.109 \times 10^{-31}\ \mathrm{kg}
Mass of protonmpm_\mathrm{p}==1.673×10−27 kg1.673 \times 10^{-27}\ \mathrm{kg}
Mass of neutronmnm_\mathrm{n}==1.675×10−27 kg1.675 \times 10^{-27}\ \mathrm{kg}
Mass of Helium nucleusmHem_\mathrm{He}==6.645×10−27 kg6.645 \times 10^{-27}\ \mathrm{kg}
Atomic mass unit (a.m.u., Dalton)==1.661×10−27 kg1.661 \times 10^{-27}\ \mathrm{kg}
Astronomical data
Hubble constantH0H_0==70 km s−1 Mpc−170\ \mathrm{km\,s^{-1}\,Mpc^{-1}}
North Ecliptic Pole (J2000.0J2000.0)(αE, δE)(\alpha_\mathrm{E},\ \delta_\mathrm{E})(18h00m00s,+66∘33′39′′)(18^\mathrm{h}00^\mathrm{m}00^\mathrm{s}, +66^{\circ}33'39'')
North Galactic Pole (J2000.0J2000.0)(αG, δG)(\alpha_\mathrm{G},\ \delta_\mathrm{G})(12h51m26s,+27∘07′42′′)(12^\mathrm{h}51^\mathrm{m}26^\mathrm{s}, +27^{\circ}07'42'')
1 jansky1 Jy==10−26 W m−2 Hz−110^{-26}\ \mathrm{W\,m^{-2}\,Hz^{-1}}
1 parsec1 pc==3.086×1016 m3.086 \times 10^{16}\ \mathrm{m}
206 265 au
3.262 ly
1 astronomical unit (au)1 au==1.496×1011 m1.496 \times 10^{11}\ \mathrm{m}
1 sidereal dayTSDT_\mathrm{SD}==23.93444 h23.93444\ \mathrm{h}
23h56m04s23^\mathrm{h}56^\mathrm{m}04^\mathrm{s}
1 tropical year==365.2422 solar days
1 sidereal year==365.2564 solar days

Approximations

(1+x)n≈1+nx(1+x)^n \approx 1 + nx

(1+x)(1+y)≈1+x+y(1+x)(1+y) \approx 1 + x + y if x≪1x \ll 1 and y≪1y \ll 1

Gauss’s formulae

Spherical law of cosines: cos⁡a=cos⁡bcos⁡c+sin⁡bsin⁡ccos⁡A\cos a = \cos b \cos c + \sin b \sin c \cos A

Spherical law of sines: sin⁡Asin⁡a=sin⁡Bsin⁡b=sin⁡Csin⁡c\dfrac{\sin A}{\sin a} = \dfrac{\sin B}{\sin b} = \dfrac{\sin C}{\sin c}

next page ⟹\Longrightarrow

The Sun
Solar luminosityL⊙L_\odot==3.826×1026 W3.826 \times 10^{26}\ \mathrm{W}
Apparent angular diameter of Sunθ⊙\theta_\odot==32’
Effective temperature of SunTeff,⊙T_{\mathrm{eff},\odot}==5778 K5778\ \mathrm{K}
Apparent visual magnitude==−26.75-26.75
Absolute visual magnitude==+4.82+4.82
Apparent bolometric magnitude==−26.83-26.83
Absolute bolometric magnitude==+4.74+4.74
Distance of the Sun from the Galactic centre≈\approx8 kpc8\ kpc
The Earth and Moon
Obliquity of the ecliptic (Earth)ϵ\epsilon==23.5′23.5'
Platonic year (period of precession of Earth’s axis)==25 765 years
Apparent visual magnitude of full Moon==−12.74-12.74
Apparent angular diameter of MoonθL\theta_\mathrm{L}==31′31'
Inclination of the lunar orbit to the ecliptic==05∘08′43′′05^{\circ}08'43''
Inclination of the lunar equator to its orbital plane==6.687∘6.687^{\circ}
Lunar sidereal monthTSLT_\mathrm{SL}==27.321661 d27.321661\ \mathrm{d}
655.71986 h655.71986\ \mathrm{h}
Synodic month==29.530589d29.530589d
Tropical month==27.321582d27.321582d
Anomalistic month==27.554550d27.554550d
Draconic month==27.212221d27.212221d

The Solar System

ObjectMean radius [km]Mass [kg]Semimajor axis [au]Eccentricity
Sun695 7001.988×10301.988 \times 10^{30}——
Mercury2 4403.301×10233.301 \times 10^{23}0.3870.206
Venus6 0524.867×10244.867 \times 10^{24}0.7230.007
Earth6 3785.972×10245.972 \times 10^{24}1.0000.016 710
Moon1 7377.346×10227.346 \times 10^{22}3.844×1053.844 \times 10^{5} km0.054 900 (range 0.026−0.0770.026 - 0.077)
Mars3 3906.417×10236.417 \times 10^{23}1.5240.093
Jupiter69 9111.898×10271.898 \times 10^{27}5.2030.048
Saturn58 2325.683×10265.683 \times 10^{26}9.5370.054
Uranus25 3628.681×10258.681 \times 10^{25}19.1890.047
Neptune24 6221.024×10261.024 \times 10^{26}30.0700.009

Условие

Use the following table of lunar eclipses from the last 25 years to predict when the next lunar eclipse clearly visible from Poland (50∘N50^{\circ}\mathrm{N}, 19∘E19^{\circ}\mathrm{E}) will occur. Give the date and predicted hour on the answer sheet.

Penalties: for a missing answer +10 minutes, eclipse with weak visibility +1 minute

DateTime UTTypeJD
1991 Dec 2110:34:00Partial2448602.940
1992 Jun 1504:57:57Partial2448788.707
1992 Dec 0923:45:05Total2448966.49
1993 Jun 0413:01:26Total2449143.042
1993 Nov 2906:27:06Total2449320.769
1994 May 2503:31:20Partial2449497.647
1995 Apr 1512:19:04Partial2449823.013
1996 Apr 0400:10:47Total2450177.508
1996 Sep 2702:55:24Total2450353.622
1997 Mar 2404:40:28Partial2450531.694
1997 Sep 1618:47:42Total2450708.283
1999 Jul 2811:34:46Partial2451387.983
2000 Jan 2104:44:34Total2451564.698
2000 Jul 1613:56:39Total2451742.081
2001 Jan 0920:21:40Total2451919.349
2001 Jul 0514:56:23Partial2452096.115
2003 May 1603:41:13Total2452775.653
2003 Nov 0901:19:38Total2452952.556
2004 May 0420:31:17Total2453130.345
2004 Oct 2803:05:11Total2453306.628
2005 Oct 1712:04:27Partial2453661.003
2006 Sep 0718:52:25Partial2453986.286
2007 Mar 0323:21:59Total2454163.474
2007 Aug 2810:38:27Total2454340.943
2008 Feb 2103:27:09Total2454517.644
2008 Aug 1621:11:12Partial2454695.383
2009 Dec 3119:23:46Partial2455197.308
2010 Jun 2611:39:34Partial2455373.986
2010 Dec 2108:18:04Total2455551.846
2011 Jun 1520:13:43Total2455728.343
DateTime UTTypeJD
2011 Dec 1014:32:56Total2455906.106
2012 Jun 0411:04:20Partial2456082.961
2013 Apr 2520:08:38Partial2456408.34
2014 Apr 1507:46:48Total2456762.824
2014 Oct 0810:55:44Total2456938.956
2015 Apr 0412:01:24Total2457117.001
2015 Sep 2802:48:17Total2457293.617
2017 Aug 0718:21:38Partial2457983.265
2018 Jan 3113:31:00Total2458150.063
2018 Jul 2720:22:54Total2458327.349
2019 Jan 2105:13:27Total2458504.717
2019 Jul 1621:31:55Partial2458681.397
2021 May 2611:19:53Total2459360.972
2021 Nov 1909:04:06Partial2459537.878
2022 May 1604:12:42Total2459715.676
2022 Nov 0811:00:22Total2459891.958

Решение

Внимание: Непълно решение

В архива няма официално решение на тази задача.

Оригинал в Архива: team.pdf