IAO 2003, Практически тур, Групи A и B — Задача 7. Измерване на скоростта на светлината.

7.1. Speed of light

Let $A$ be Saturn’s orbital radius, $B$ be Saltis’s orbital radius and $c$ be the velocity of light. It takes light the time $A/c$ to reach the Sun from Saturn. At opposition it takes light $(A-B)/c$ to reach Saltis from Saturn, and at conjunction $(A+B)/c$. This means that the difference in the timings from the tabulated values will be $A/c-(A-B)/c=B/c$ at opposition (i.e. the eclipse will be seen earlier) and $A/c-(A+B)/c=-B/c$ at conjunction (i.e. the eclipse will be observed later than tabulated values).

Looking at the differences in Table 1, one sees that the spread in value is greater than the estimated timing error of 0.03 pinit. Thus the spread is mainly due to Saturn not being exactly at opposition/conjunction at the time of eclipse. Thus the extreme value is the best approximation, 6.23 pinit. Since the three opposition values are ranging from 6.12–6.23 pinit, the true opposition difference is probably not larger than 6.30 pinit. Thus a sensible estimated error is ~0.1 pinit. We ignore the conjunction values since they are further from true conjunction, probably due to difficulties in observing Saturn near the Sun.

With a time difference $B/c=6.23\pm0.1$ pinit, and $B$ defined to $10^9$ seter, we readily compute

Table 1. Eclipses of Titan by Saturn

Source note: this is the printed official table and its subsequent calculation. Several printed differences disagree with direct subtraction of the first two columns (18.50−12.28=6.22, 444.70−450.85=−6.15, 615.43−621.52=−6.09). The original key’s values are retained.

7.2. Earth’s orbital radius

The difference between the largest and smallest distance between Saltis and Earth is equal to 2 Earth orbital radii. From Fig. 1 we estimate the time difference to 5.3 pinit. With $c$ estimated as above, we get $5.3c=8.5\times10^8$ seter. Thus $1\ \mathrm{AU}=4.3\cdot10^8$ seter.

7.3. Units

$1\ \mathrm{AU}=149.6\cdot10^6\ \mathrm{km}=4.3\cdot10^8$ seter gives 1 seter = 348 meter. With $c=2.998\cdot10^8\ \mathrm{m/s}=1.61\cdot10^8$ seter/pinit we have 1 pinit = 187 seconds.

7.4. Saltis’s orbital period

From Fig. 1 we estimate the interval between oppositions to $2.3\cdot10^5$ pinit = 1.4 yr. Since the period of Earth is 1 yr, we use the formula for the synodic period $(1/T=1/P_1-1/P_2)$ to get the period of 3.7 yr. Alternatively, one may use the orbital radius of Saltis $a=348\cdot10^9\ \mathrm m=2.33\ \mathrm{AU}$ and Kepler’s law, $p^2=a^3$, to get $p=a^{3/2}=2.33^{3/2}\ \mathrm{yr}=3.6\ \mathrm{yr}$.