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If I were standing on the surface of Phobos looking at Mars, how big would it appear in the sky? Would it be like one hand at arm's length? Or take up like a third of the sky?

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  • $\begingroup$ Roughly like the Earth from the ISS. Hopefully you will get more precise answer, too. $\endgroup$ – peterh Nov 30 '20 at 19:57
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    $\begingroup$ At first you need the distance from Phobos to Mars and the diameter of Mars. Then you could calculate the visual angle for Mars and compare this angle to the visual image of our Moon and to one hand at arm's length. So try the first steps. You will get some help here for the next steps. $\endgroup$ – Uwe Nov 30 '20 at 22:23
  • $\begingroup$ Some answers to the question are to be found here. $\endgroup$ – Uwe Nov 30 '20 at 23:35
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We can start by noticing that Mars, Phobos, and the horizon forms a right-angled triangle.

By trigonometry, the angular radius ($\alpha$) is then given by:

$$\alpha = \sin^{-1}\left(\frac{r_{parent}}{r_{orbit}}\right)$$

The angular diameter is twice that. Inserting the average radius of Mars of 3,389.5km, and the semi-major axis of Phobos of 9,376km, we have $2\alpha = 42.4°$.

The orbit of Phobos is not perfectly circular though, so at periapsis (9,234km) and apoapsis (9,518km), the corresponding angles are $43.1°$ and $41.7°$ (More than a fifth but less than a fourth of the sky).

By comparison, the Moon as viewed from Earth is about $0.52°$ in diameter.

A hand is not quite a constant unit of measurement, but it's about 10°, or 20° with fingers spread. So ~4 hands wide.

Mars-Phobos triangle

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doing the math, I get about 40 degrees diameter, or about two stretched out hands: https://www.timeanddate.com/astronomy/measuring-the-sky-by-hand.html#:~:text=There%20are%20360%C2%B0%20in,is%20further%20divided%20into%2060%22.&text=This%20sphere%20is%20360%C2%B0,horizon%20and%20cannot%20be%20seen.

the math has two parts. If you want to know how much of mars you can see, you need the horizon formula https://sites.math.washington.edu/~conroy/m120-general/horizon.pdf

but if you just want to know the angle of mars in the sky, it is a simple tangent formula: tan theta = mars radius/ distance to center of mars

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    $\begingroup$ Could you show us the math? Formatting help is available. $\endgroup$ – Organic Marble Dec 3 '20 at 18:48
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    $\begingroup$ temporary and reversible -1 to encourage this to become a useful answer i.e. one that shows the equations used and how the result was obtained, unless of course you were actually on Phobos and used the "outstretched hands" method described your link, in which case you'll have to add some supporting information on how you got to Phobos! $\endgroup$ – uhoh Dec 4 '20 at 12:35
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    $\begingroup$ Yes, please show us the calculations. While the linked reference provides a good empirical tool, it does not include any of the mathematical calculations needed to answer this question; and we do not want to search for data to edit the answer when you seem to already have them. $\endgroup$ – Oscar Lanzi Dec 6 '20 at 12:58

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