I have a rough understanding that clocks run slower the deeper you are in a gravitational field. There’s even a simple equation for the situation of a spherical nonrotating mass, which I will link to because I don’t know how to enter formulae: wikipedia for gravitational time dilation.
Now I can plug numbers into a formula with the best eighth graders, so to me it looks like tf/t0 for Earth’s surface (M = 5.972e24 kg, r = 6.371e6 m) is 1.000000001. So for every second on Earth, 1.000000001 seconds pass at infinity, assuming various things.
Standing on the Sun (at night, of course, for M=1.988e30 kg and r = 6.957e8 m) gives 1.000002. So for every second on the Sun, 1.000002 seconds pass at infinity. The Sun is a deeper gravity well, so time goes slower there than it does on Earth - if they were both isolated nonrotating masses, anyway.
But here’s a twist: if I check on tf/t0 for the Sun but at the radius of Earth’s orbit (so M = 1.988e30 kg but r = 1.49e11 m) I get 1.00000001. That’s seven zeros before the 1, not 8. So it appears that the time dilation that you feel on the surface of Earth, caused by Earth’s gravity alone, is less than the time dilation you would experience at Earth’s distance from the Sun, caused by the Sun’s gravity alone.
This seems odd to me, because the local acceleration due to gravity here is 9.8 m/s^2 due to Earth but only (looks up stuff) 5.9e-3 m/s^2 due to the Sun. So, naively, it would seem that it should be the other way around. Have I made an embarrassing math error, or is there something here that I’m missing?