# What apparent sizes of the sun an Earth like planet can have?

What is the smallest and largest sun an Earth like planet could have in its sky? What kinds of stars are those?

I saw this video and thought, yeah, some of those stars are really big in absolute terms, but Earth would have to be orbiting them proportionally further to keep the climate, making them smaller in the sky, so the video is not accurate.

• As a sidebar, the Alpha Centari one isn't accurate. They're about as far apart as the Sun and the outer planets. It's a pretty elliptical orbit, but even at it's closest, they wouldn't look like that at all from 1 AU. They'd be much further apart. – userLTK Apr 19 '17 at 20:52

I'll make a very simple dimensional calculation here.

Total power from of a star with radius $R$ and temperature $T$:

$$P_{tot} \ = \ R^2 \ T^4$$

Flux $I$ (power per unit area) at an orbital distance $r$:

$$I \ = \ \frac{R^2 \ T^4}{r^2} \ = \ T^4 \ \left( \frac{R}{r} \right)^2$$

Apparent size (radians, small angle approx):

$$\theta \ \approx \frac{R}{r}$$

So:

$$\theta \ \approx \ \frac{\sqrt{I}}{T^2}$$

Surprisingly, for a fixed incident flux of light (ignoring different planetary warming effects of different wavelengths) the apparent size of a sun viewed from a star varies roughly as the inverse square of its temperature, and does not depend on the actual size of the star.

Using Wein's dispacement law which relates the wavelength of maximum spectral radiance of a blackbody $\lambda_{max}$ with its temperature $T$ using Wien's displacement constant $b$:

$$\lambda_{max} \ = \ \frac{b}{T}$$

The apparent size scales with the square of the peak wavelength of a star's blackbody spectrum.

$$\theta \ \approx \ \lambda_{max}^2 \ \sqrt{I}$$

Yay! Reddish stars will appear much larger than bluish stars - the result we were all expecting.

And if you'd like do some landscaping, the 2008 Scientific American article The Color of Plants on Other Worlds is a good read in stellar spectra, atmospheres, and photosynthesis.

• So, largest apparent: red dwarf, smallest apparent: any young star? – Ansis Māliņš Apr 20 '17 at 11:42
• @AnsisMāliņš roughly speaking, the size doesn't matter, only the effective temperature. If a red star were twice as big, you'd be twice as far away, and it would appear the same size. It's up to you what the highest and lowest temperatures you't like to consider. I am not sure how cold something can be before it is no longer called a star. – uhoh Apr 20 '17 at 11:47
• @userLTK feel free to edit this answer and add more info like that if you like, or incorporate into a second answer. Goal in SE is to generate good answers and I think this insight will be helpful to future readers as well. – uhoh Apr 20 '17 at 18:06
• @uhoh OK. Answer coming. – userLTK Apr 20 '17 at 19:11