Circumstellar Habitable Zone of an Orange Dwarf

I want to create a system orbiting an Orange Dwarf star, approximately 0.5 M☉.

I've a 5-planet system, one of which needs to be within the habitable zone of this star; this planet will be 2 M⊕

I know the formula for this is $\sqrt{\frac{1}{2}}$, but I'm unsure where exactly that would place my habitable planet and whether or not this means my habitable planet would be tidally locked.

So my question is thus; "Please calculate the habitable zone for my system above and let me know if the habitable planet would be tidally locked."

I would appreciate the answer to be in AU.

• You might also try asking on Astronomy. – JDługosz Oct 9 '16 at 12:56

While many people continue to assume tidal locking, I’d like to point out that a spin-orbit resonance that’s an odd half multiple is a more stable case. For example, our own planet Mercury.

To make a long story short, if there is another large planet farther out, you can expect this situation.

For your main question, I asked Google for habitable zone calculator and found this page and several others.

• I just did a search of my own and i got 2 sites with different answers in them, I'm not sure which one to believe. (Oh and both agree my planet would be tidally locked based on a one-planet system) – Raisus Oct 9 '16 at 12:43
• Are the answers far apart? Do they document the models and methods used? Does one have more inputs or more precision of input than the other? – JDługosz Oct 9 '16 at 12:49
• The one you suggested is asking for luminosity and effective temperature in Kelvin, whereas the other one just asks for Stellar Mass and Planet distance – Raisus Oct 9 '16 at 12:54
• and the second seems to make out that the planet would be tidally locked whereas the first doesn't seem to indicate one way or the other – Raisus Oct 9 '16 at 12:54
• So you had to convert your stated mass to temperature. Maybe different than what the other page had built in? – JDługosz Oct 9 '16 at 12:55

This formula for calculating the habitable zone can be found here

Calculating the HZ in the simplest case

If all of the complicating factors discussed above are ignored and the habitable zone is defined simply as the distance from a star where the effective temperature is in the range 0° to 100°C then it is straightforward to calculate the radii of the HZ's inner and outer bounds. The relevant formula is:

L = 4π r 2σT 4

where L is the star's luminosity, r is the distance from the center of the star, σ is the Stefan-Boltzmann constant (= 5.67 × 10-8 W m-2 K-1), and T is the effective temperature (in kelvin). For the Sun, this yields a range for the HZ of 0.7 to 1.5 AU. The HZ range for other stars can then be calculated easily since, from the above formula:

Lstar/Lsun = rstar2/rsun2

In the case of Vega, Lstar/Lsun = 53, which gives a range for HZ of 5.1 to 10.9 AU. In the case of Kapteyn's Star, Lstar/Lsun = 0.004 and the corresponding HZ range is 0.044 to 0.095 AU.

This is good for a rule of thumb way of calculating habitable zones. It can be used in conjugate with some of the more fancy habitable zone calculators on the internet.

• You still need to find the Luminosity for a particular mass main-sequence star. – JDługosz Oct 11 '16 at 20:41
• Of course. While this is reasonably easy for known stars, when it comes to worldbuilding there should be ball park figures for stars of a given Hertzsprung-Russell class and with a given mass. Our friends at Astronomy SE should be able to help too. – a4android Oct 12 '16 at 4:00