As of now, Proxima b is the only confirmed planet (or dwarf planet if it didn't clear its orbit - sorry, couldn't resist) around Proxima Centauri, the nearest-known star to the Sun about 4.2 ly away. From that distance, the Sun looks just like an average star in the night sky. Now if the Sun became a red giant, at 256 times its current diameter (if I recall correctly), what would it look like from Proxima b's night sky? On a good telescope, Betelgeuse is recognizable as more than a dot, its surface is visible, that would surely be the case for a red giant Sun too. What would it take to recognize the Sun's surface from the Proxima system? Would the Sun perhaps even cast a visible shadow?
The Sun will not become a red giant for approximately 7 billion years, so there is little chance that Proxima Cen will be anywhere near it.
[EDIT: To answer a comment, the current helocentric radial velocity of Proxima Cen is -22 km/s. The Sun's escape velocity at the distance of Proxima Cen is about 0.1 km/s. So the Sun and Proxima Cen are totally unbound. With a relative speed of 22 km/s (at least - I haven't calculated the additional tangential velocity difference) they will be at least 100 pc apart within 7 million years, let alone 7 billion years.]
The Sun goes through TWO red giant phases. In the first, hydrogen shell-burning phase, the Sun will achieve a peak absolute V magnitude of around -3 at the tip of the first ascent red giant branch. It wil have a temperature of around 2700 K, be 256 times its present size and 2800 times more luminous according to Schroeder & Connon Smith (2008).
From a distance of 0.77 pc (the current distance of Proxima Cen) it would have an apparent magnitude of about -8.5 (about 40 times brighter than Venus).
Its apparent size in the sky would be 1.54 arcseconds, so would have a disc that was resolvable using a small telescope at a good site and would be orange/red in colour.
According to Schroeder & Connon Smith's model, the asymptotic giant branch phase of the Sun, when it has an inert carbon/oxygen core and He + H-burning shells will be slightly smaller and less luminous.
The exact values for both these phases are somewhat dependent on the details of the mass loss model used for the strong stellar winds expected.
$\begingroup$ I thought since both the Sun and Proxima Cen are extremely close on a galactic scale, they wouldn't get too much apart. The Alpha Cen system is "just" 4.5 ly closer to the galactic center which is close to nothing on a galactic scale. And they may even be gravitationally influenced by the Sun and vice versa, so why would the Alpha Cen system be so far away in 7 billion years? $\endgroup$ Oct 30, 2020 at 14:50
$\begingroup$ @John see edit for additional info. $\endgroup$– ProfRobOct 30, 2020 at 16:43
$\begingroup$ Thank you. Interesting they have so different velocities. I can't accept your answer anymore because my account changed in the meantime. $\endgroup$ Oct 30, 2020 at 17:02
$\begingroup$ @John a 10-20 km/s velocity difference is typical between stars in the solar neighbourhood. i.e. The stars around us are the stars around us for a very short period of time (Galactically speaking) and are then replaced by a different set. $\endgroup$– ProfRobOct 30, 2020 at 17:42
$\begingroup$ Can we predict the nearest known star to the Sun in say a million years? $\endgroup$ Oct 30, 2020 at 17:50
The sun, when it becomes a red giant, will be more 100 times more luminous than it is now, but that only gives it an absolute magnitude of about -1, and an apparent magnitude at 4 light-years of about -5: similar to Venus. It is possible for this to cast a very faint shadow, but you need a very dark place to see it.
The sun will still be much too small to appear as a disc, it will be about 1 arcsecond. (And Betelgeuse is not visible as a disc, even in a "good telescope" you need special techniques, such as interferometry, to see its surface)
But it would be bright! Unfortunately(!) the sun won't become a red giant for billions of years, and by the time it does, Proxima will have moved away and could be on the other side of the galaxy, and the sun is unlikely to be visible at all.
$\begingroup$ I thought that because the Sun and Proxima 're so close they would never go too far from each other, even in billions of years? As for Betelgeuse, it can appear like this: sueddeutsche.de/image/sz.1.4747331/704x396?v=1587835481 $\endgroup$ Oct 26, 2020 at 17:25
$\begingroup$ And why will it be "just" a 100 times more luminous if it became 256 times larger? Is it due to its loss of mass during becoming a red giant? $\endgroup$ Oct 26, 2020 at 17:27
1$\begingroup$ You might want to go and accept Rob answer which has actual calculations. $\endgroup$– James KOct 26, 2020 at 19:41
$\begingroup$ @JamesK Why did you delete your answering comment? It would also have been informative to any reader who comes by. $\endgroup$ Oct 30, 2020 at 14:52