Textbook description of R##\ddot{\text o}##mer's light speed

[brotherbobby]

TL;DR

A recent textbook (see below) has described the dutch astronomer R$\ddot{\text o}$mer's calculation of the velocity of light by alluding to the differences in the sighting of Jupiter's moon when the earth was $\textit{moving away}$ from the planet as against the earth $\textit{moving towards}$ the planet. I don't understand it. Why should light take longer or shorter times depending how the earth moved? I thought it was due to $\textit{where}$ the earth is, not how it was moving.

Statement of the problem : I copy and paste the clip from the text, underling in red the sentences where, in my opinion, the author(s) have gone wrong. I annonate to the right of the text in blue, assuming they are both readable. The text in question is titled Special and General Relativity (2026, Springer) by Boblest, Muller and Wunner.

I don't explain further, except for one point.

It is reported that R$\ddot{\text o}$mer found a time lag $\Delta t = 22\;\text{minutes}\,$ between the two sightings of the moon at different times of the year. Also, that he found the light's speed to measure to about $\boxed{c = 220,000\;\text{km/s}}$.

I can use my reasoning as in the image to see if it fits. If indeed the time discrepancy takes place due to where the earth is relative to Jupiter, then the "culprit" should be the diameter of the earth's orbit. It is this extra distance that light has to travel that explains why it appears that the moon Io's stays a longer time eclipsed "behind" Jupiter.

This should help me calculate. We know the radius of the earth's orbit around the sun $R_{\bigodot} = 1.5\times 10^8\;\text{km}$. The reported time lag $\Delta t = 22\; \text{minutes}\, = 22\times 60\; \text{seconds}\,$. Hence the speed of light $c = \dfrac{2R_{\bigodot}}{\Delta t}= \dfrac{2\times 1.5 \times 10^8\;\text{km}}{22\times 60\;\text{s}} = \boxed{227272\;\text{km/s}}$, which is close to what was obtained.

Thanks for your time. If indeed I am mistaken and the authors are correct, I'd like to know of their reasoning and the calculation. The only way out I can see is if light is assumed to move through ether. Indeed, according to Doppler's effect, the speed of light (or sound) will change if the observer moved towards or away from the source. Is this the author's reasoning? Even if it was, I don't see how it leads to the result of light speed.

 

[kuruman]

I think the book's description is inadequate. This wikipedia entry has a better explanation.

 

[brotherbobby]

I think it would be fair on my part to carry out the calculation assuming the Doppler's effect, fair to the authors of the text that is. Let's assume the following was known at the time of R$\ddot{\text o}$mer.

Data : The distance to Jupiter from earth, $d_J = 5.58\times 10^8\;\text{km}$. The velocity of the earth around the sun $v_{\bigodot} = 10^5\;\text{km/h} \cong 30\;\text{km/s}$.
The time lag observed by the astronomer was $\Delta t = 22\;\text{minutes}\; = 22\times 60\;\text{seconds}$
When the earth is moving towards Jupiter, the speed of light observed $c_T = c+v_{\bigodot}$ and when moving away, $c_A = c-v_{\bigodot}$.

Hence the time lag $\small{{\Delta t} = \dfrac{d_J}{c-v_{\bigodot}} - \dfrac{d_J}{c+v_{\bigodot} }=d_J\left( \dfrac{2v_{\bigodot}}{c^2-v^2_{\bigodot}}\right) = \dfrac{2v_{\bigodot}d_J}{c^2} \Rightarrow c = \left( \dfrac{2v_{\bigodot}d_J}{\Delta t} \right)^{\frac{1}{2}}},\;\text{since}\; c\gg v_{\bigodot}.$

This amounts to $c = \left( \dfrac {2\times 30\times 5.58\times 10^8}{22\times 60} \right)^{\frac{1}{2}} \cong \boxed{5000\;\text{km/s}}$. This can't be right, assuming I haven't made calculation errors.

 

[brotherbobby]

I think the book's description is inadequate. This wikipedia entry has a better explanation.

To call it "inadequate" is putting it mildly.
My calculation above (post #3) shows that if the authors had Doppler's effect in mind for the differing speeds of light when the earth was moving towards and away from Jupiter, they are wrong in a big way.
It's not about how the earth is moving. Rather it's about where the earth is, or so it seems.

 

[Ibix]

The Doppler effect is why the eclipse timings vary, because the distance to Jupiter changes between the beginning and end of the eclipse. You don't use the Doppler effect to calculate the speed of light though, not least because the radial velocity of Jupiter is zero at conjunction and opposition.

An idealised version of Rømer's experiment is to place a clock on Jupiter and watch it through a telescope. You observe that the clock takes 0.999s to tick 1s (numbers made up), from which you deduce that Jupiter is 0.001 light seconds closer now than it was 1s ago. You observe that the clock takes 0.998s to tick the next second, so Jupiter is 0.002 light seconds closer than 1s ago, 0.003 light seconds closer than two seconds ago. You keep cumulating the (signed!) time difference like this between opposition and conjunction to get a total time difference. You attribute this time to the extra distance, the diameter of Earth's orbit, and deduce that the diameter is 22 light minutes (or 16 light minutes, with more precise measurements).