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I started Stellarium on my computer and pressed F6 to bring up the "Location" window. Then I changed the planet to "Uranus", and marvelled at the view of the many rings and many moons from the planet's "surface" For convenience I clicked the buttons to remove the ground and the atmosphere. then I found and clicked on Saturn. It ...

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You would need to find the distance from one to the other. That is a mathematical exercise in using the cosine rule. You then use magnitude formulae. For convenience you could calculate the absolute magnitude of each galaxy: $$M_{\text{abs}} = m_{\text{app}} -5(\log_{10}(d_{\text{parsec}}) -1)$$ And then the same formula, rearranged to find the apparent ...

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Leveraging Pierre Paquette's excellent answer and reference to Hilton and Mallama, the magnitude of Saturn can be estimated by: $$V = 5 \log_{10} (rd) - 8.95 - 3.7\times10^{-4} \alpha + 6.16\times10^{-4} \alpha^2$$ Here, $r\approx9.5$ AU is the distance from Saturn to the Sun, $d$ is the distance from Saturn to the observer, and $\alpha$ is the angle of ...

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A magnitude is defined as $-2.5\log_{10}$ of a flux. But flux scales as the inverse square of a distance. $$-2.5\log_{10}\left(\frac{k}{d^2}\right) = 5\log_{10}(d) -2.5\log_{10}(k)$$

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This appears to be a question directed at an answer to a previous question by the OP. https://astronomy.stackexchange.com/a/47296/26216 The 5log_10(...) appears to just be a multiplier the author of the paper cited in the aforementioned answer found to describe the visible magnitude of Neptune. Sorry, but that's probably the best answer you'll get... There ...

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Two stars in a binary system are at the same distance from Earth. If they have similar spectral types then the difference in their magnitudes tells us the ratio of their luminosities. $$\Delta m = m_1 - m_2 = -2.5\log_{10}\left(\frac{L_1}{L_2}\right) \ .$$

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The brightness of a Solar System object, seen in reflected light, depends on how far it is from the Sun, $d_s$, and how far away it is from the observer, $d_o$, (and the angles between them). Both dependencies are "inverse square laws": $${\rm brightness} \propto \left(\frac{1}{d_s^2}\right)\left(\frac{1}{d_o^2}\right)\ .$$ Both Uranus and Neptune ...

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