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?

Question

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?

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.