# This question investigates the most extreme differences in the sizes of stars. Compute the ratio of the radii of a M supergiant star to that of a hot white dwarf star. Assume the M supergiant star has a luminosity of 2x10^5 solar luminosities and a surface temperature of 3000 K. Assume the white dwarf star has a luminosity of 10^3 solar luminosities and a surface temperature of 15,000 K.

The temperature, radius, and luminosity of all stars (not just main sequence stars) are related by this equation:

`L/L_0 = (R/R_0)^2 (T/T_0)^4`

Where L_0, R_0, and T_0 are based on the Sun, which has a radius of 6.96*10^8 meters and a temperature of 5780 K. From there, we can solve...

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The temperature, radius, and luminosity of all stars (not just main sequence stars) are related by this equation:

`L/L_0 = (R/R_0)^2 (T/T_0)^4`

Where L_0, R_0, and T_0 are based on the Sun, which has a radius of 6.96*10^8 meters and a temperature of 5780 K.

From there, we can solve for R:

`R = R_0 sqrt{L/L_0 * (T_0/T)^4}`

Plug in the numbers for the supergiant:

`R = 6.96*10^8 sqrt{2*10^5 * (5780/3000)^4}`

`R = 1.2*10^12 m`

Similarly for the dwarf:
`R = 6.96*10^8 sqrt{10^3 * (15000/3000)^4}`
`R = 5.5*10^11 m`

Despite one being a "supergiant" and the other being a "dwarf", they are not that different in size, because these terms are actually about luminosity and temperature, not about size per se. (Also, they're both huge; the distance from the Sun to the Earth is about 1.5*10^11 meters, so both of these stars would engulf us if they were swapped out for the Sun.)