# What stellar conditions and orbital distance are needed to produce a year length of 515 Earth days and 9 Earth hours on an Earthlike planet?

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?

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