USAAAO 2022, II кръг — Задача 6. The speed of light

• (a) December 1671: 42 hours, 28 minutes, and 21 seconds; June 1672: 42 hours, 28 minutes, and 40 seconds.

• (c) $4.8 \times 10^{8}$ m/s

  About 60% larger than the true value, per the official solution.

• (d) $c = \frac{2\pi p}{(1\ \mathrm{year})\Delta p} a = \frac{a}{5\mathrm{m}12\mathrm{s}}$

(a) In December 1671, the observed interval between successive eclipses is 42 hours, 28 minutes, and 21 seconds.

In June 1672, the observed interval between successive eclipses is 42 hours, 28 minutes, and 40 seconds.

Notice that the periods differ by 19 seconds.

(b) Since light has a finite speed, we don’t observe the eclipses of Io until some time after they actually happen: there is a light delay time. When the distance between Jupiter and Earth is decreasing, the light delay time should be decreasing with each eclipse, so the time between eclipses seems to be shorter. Similarly, when the distance between Jupiter and Earth is increasing, the time between eclipses appears to be longer.

This is very similar to the modern concept of the Doppler effect, though Rømer didn’t call it that at the time.

(c) Let the true orbital period be $p$, and the radial velocity of Jupiter relative to Earth be $v$. If the speed of light is $c$, then the delay time between successive orbits will change by $\frac{pv}{c}$. Therefore the observed orbital period is longer than the true orbital period by $\Delta p = \frac{pv}{c}$. The speed of light is thus

Jupiter’s orbital velocity is somewhat slower than Earth’s, and it is also almost entirely tangential rather than radial. So we make the simplifying assumption that the radial velocity of Jupiter relative to Earth at first and second quadrature is just the Earth’s orbital velocity (towards Earth at first quadrature and away from Earth at second quadrature.) So $v$ is just the Earth’s orbital velocity, which from Kepler’s third law is

The true orbital period of Io is roughly the average of the two measured periods, so $p =$ 42:28:30.5, and $\Delta p = 9.5$ sec. Plugging in these values to the above expression for the speed of light gives

Today we know that this is about 60% larger than the true value, which is not bad given all the approximations that were made.

(d) We could try to carry out the same derivation as in the previous part, but leave $a$ as an unknown constant. This gives

But this expression depends on $G$ and $M_{\mathrm{sun}}$, which Rømer would not have known at the time. Rømer did not know any of $a$, $G$ and $M_{\mathrm{sun}}$ individually, but he did know that the Earth’s orbital period

is one year.

So we can instead write

This is equivalent to saying that it takes 5 minutes and 12 seconds for light to travel from the Sun to the Earth. It can be argued that Rømer was not really measuring the speed of light, but instead just the light travel time from the Earth to the Sun. (The first accurate determination of the Earth-Sun distance was made a year later by Cassini.)