OWAO 2024, Теория — Задача 6. Jupiter in Sirius

• a) Taurus

• b) At about 6.0ʰ civil time, at altitude +69°

• c) From its rise at 22.5ʰ until sunrise at about 6ʰ: about 7.5ʰ in total

• d) 96 °

  About 96° west of the Sun; the spherical-law-of-cosines estimate in the official solution gives ≈ 94°.

• e) 38 ″

• f) About December 10, 2024 (the actual opposition is on December 7)

a) The analysis of the equatorial coordinates makes it clear that Jupiter is located on the ecliptic approximately at $43^{\mathrm{m}} \approx 11^{\circ}$ west of the summer solstice point ($\alpha = 6^{\mathrm{h}},\ \ \delta = +23^{\circ}\,26'$). Since the summer solstice point is located in the Taurus constellation, almost on the border with the Gemini constellation, Jupiter is also located in Taurus.

b) The altitude of Jupiter’s upper culmination is

On the day of the Northern hemisphere autumnal equinox, September 22, the solar and sidereal times are equal. On September 16, 6 days before the equinox, the sidereal time is $\Delta t = 3^{\mathrm{m}}\,56^{\mathrm{s}} \times 6 \approx 0.4^{\mathrm{h}}$ behind solar time. Thus, the upper culmination of Jupiter occurs at $\alpha + \Delta t \approx 5.7^{\mathrm{h}}$ local solar time. That is

or $3.0^{\mathrm{h}} + 3^{\mathrm{h}} = 6.0^{\mathrm{h}}$ civil time.

c) Next, we determine the hour angle at the moment of Jupiter’s set:

Thus, Jupiter crosses the horizon at $6.0^{\mathrm{h}} \pm 7.5^{\mathrm{h}}$ civil time. It rises at $22.5^{\mathrm{h}}$ and sets at $13.5^{\mathrm{h}}$. Here the difference in the duration of the solar and sidereal days can be neglected due to the reasonable accuracy of calculations.

The Sun interferes with observations. Near the equinox, the declination of the Sun changes at a rate of about $1^{\circ} \cdot \sin\varepsilon \approx 0.4^{\circ}$ per day. On September 16, $\delta_\odot = 0.4^{\circ} \times 6 \approx +2.4^{\circ}$.

On this day, the center of the Sun crosses the horizon at $12^{\mathrm{h}} + x$ after the local noon (see Figure 5), where

In addition, the atmospheric refraction ($r = 35'$) and the angular radius of the Sun ($\rho = 16'$) extend the daylength by

both at sunrise and sunset. In total, we have a correction $\Delta = 9.2^{\mathrm{m}} + 4.8^{\mathrm{m}} = 14^{\mathrm{m}}$.

The Sun rises at

and sets at

Thus, Jupiter can be observed approximately since its rise at $22.5^{\mathrm{h}}$ until its upper culmination and the sunrise at approximately $6^{\mathrm{h}}$ in the morning, $7.5^{\mathrm{h}}$ in total.

d) The elongation of Jupiter is the angular distance between the Sun and Jupiter as observed from the Earth. It can be estimated as the difference of the ecliptic longitudes of the Sun (near the autumnal equinox) and Jupiter (near the summer solstice):

It is also possible to estimate the elongation of Jupiter as the angular distance between the Sun and Jupiter for an observer on the Earth. The equatorial coordinates of Jupiter are given in the table. The equatorial coordinates of the Sun we can estimate based on the date of observation. The declination of the Sun was calculated earlier ($\delta_\odot = 2.4^{\circ}$). The right ascension near the autumnal equinox can be estimated approximately using the formula

The elongation of Jupiter can be estimated using the spherical law of cosines

e) Jupiter is near the quadrature, so the distance to the planet can be estimated by the Pythagorean theorem: $d = \sqrt{5.2^2 - 1.0^2} \approx 5.1$ au. The angular size of Jupiter is

f) Since Jupiter is located to the west of the Sun, the Earth “catches up” with it in its orbital motion. For the opposition to occur, the Earth has to shift relatively to the Sun–Jupiter direction by

This shift is associated with the relative motion of the planets and with the synodic period of Jupiter relatively to the Earth.

The synodic period of Jupiter

so the opposition will come in

That is about December 10, 2024. The actual opposition of Jupiter will occur on December 7.

Marking Scheme:

1. Question (a) — 2 pt.

• Question (b) 2. Altitude — 2 pt. 3. Civil time — 3 pt.

4. Question (c) — 4 pt.
5. Question (d) — 3 pt.
6. Question (e) — 3 pt.
7. Question (f) — 3 pt.