A planet has a year length of 515 Earth days and 9 Earth hours. It is the same size and has the same climate as Earth. What stellar conditions would be needed to produce this and what orbital distance is needed?


1 Answer 1


We can estimate this using the inverse square law and the mass-luminosity relation, $$L=M^a$$ where $L$ is the luminosity of the star, and $M$ is its mass, both measured relative to the Sun. (That is, the mass and luminosity of the Sun = $1$). For a main sequence star of mass similar to the Sun $a\approx4$, for larger stars $a\approx3.5$. Please bear in mind that this is just an estimate, and that a star gets more luminous as it ages.

To keep things simple, let's assume that orbits are perfect circles. By the inverse square law, a planet with orbit radius $R$ orbiting a star of luminosity $L$ receives the same intensity of light and heat as the Earth if $$R^2=L$$ where $R$ is measured in astronomical units.

From Kepler's 3rd law, $$R^3=MT^2$$ where $T$ is the orbital period in sidereal years.

Combining these equations we get $$M=T^p$$ where $$p=\frac{4}{3a-2}$$

Here are the results for $T=515.375$ days.

a 3.5 4
Mass 1.175885 1.147652
Lumin 1.763102 1.734766
Radius 1.327818 1.317105

I calculated those results using this Sage / Python script.

A star in that mass range is a heavy G-type star, or perhaps a light F-type star.

  • 1
    $\begingroup$ Stars (solar type stars on the main sequence) get more luminous as they age, the temperature is fairly constant. $\endgroup$
    – ProfRob
    May 21 at 7:19
  • $\begingroup$ What orbital distance is needed? $\endgroup$
    – Galactic
    May 21 at 8:22
  • $\begingroup$ @Galactic Using a=4, the orbit radius is ~ 1.317 au ~= 197036000 km. $\endgroup$
    – PM 2Ring
    May 21 at 8:25
  • $\begingroup$ @PM2Ring, what about using a=3.5? $\endgroup$
    – Galactic
    May 21 at 8:31
  • 1
    $\begingroup$ @uhoh Yes, there probably should be a colour correction, since the peak frequency of a more luminous star is higher. But this is only a crude estimate which ignores star age & metallicity. We only have good mass-luminosity data for a few hundred stars (mostly from nearby binary systems, see DEBCat) to fit the mass-luminosity function to. $\endgroup$
    – PM 2Ring
    May 22 at 2:53

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