Astronomy question, how much larger in diameter would Jupiter have to be to be observed as an orb from Earth?

Assume the earth and Jupiter are on the same side of the sun, both in circular orbits, in the same plane. The moon and the sun appear just about the same size in the sky, and since we're familiar with what those look like, we'll try to make Jupiter appear the same size as the sun. I'll use the equatorial radii of the sun and Jupiter, and I'll use the semimajor axes of the orbits of the earth and Jupiter as their distances from the sun so I don't have to deal with how their orbits actually align. I'm getting the numbers from Wikipedia and I'm not going to deal with significant figures. Hopefully that covers all the bases. I had a thermo professor in undergrad who graded exams with a stamp that said "Assumptions -1" wherever you forgot any.

  • Sun radius: r_s = 696342 km
  • Jupiter radius: r_j = 71492 km
  • Jupiter distance from the sun: R_sj = 778547200 km
  • Earth distance from sun: R_se = 149598261 km
  • Earth/Jupiter distance: D_ej = R_sj - R_se = 628948939 km
  • Modified Jupiter radius: r_j*
  • Modified Jupiter distance: D_ej*

So how big is the sun? Two legs of a triangle are formed by the line from your eyes to the center of the sun (huge leg) and the radius of the sun (tiny leg). The hypotenuse is the line from your eyes to the edge of the sun. The angle that makes at your eyes is atan(r_s/R_se) = 0.267 degrees. Since that was based on the radius, the angular diameter will be twice that: 0.533 degrees. That's how big we want Jupiter to be. There are two ways to make that happen: make it bigger or make it closer, like you said. For two things to appear the same size, the ratio of their diameter to their distance from you have to be the same.

We want (r_s) / (R_se) = tan(0.267 deg) = (r_j) / (D_sj*) = (r_j*) / (D_sj). Actually, any any combination of r_j* and D_ej* will work as long as they're in that ratio.

Keep it the same size and move it: tan(0.267 deg) = (r_j) / (D_sj*), solve for D_ej*.
D_ej* = r_j/tan(0.267) = 15341424 km, or about 1/41 of the original distance.

Keep it the same distance and grow it: tan(0.267 deg) = r_j*/D_ej, solve for r_j*.
r_j* = D_ej*tan(0.267) = 2930942 km, or about 41 times the original size.

/r/askscience Thread