Exoplanet Revolution Period About a Dwarf Star

Quote of the Day

If you want something you've never had, you must be willing to do something you've never done.

— Thomas Jefferson


Introduction

Figure 1: Example of the Habitable Zone for 40 Eridani.

Figure 1: Example of the Habitable Zone for 40
Eridani, a nearby star smaller than our Sun
(Source).

I have been reading about the possibility  exoplanets around stars that are relatively near our solar system. I usually think about exoplanets about stars similar to the Sun. While news articles in the popular scientific press often refer our Sun as an “ordinary star”, in fact it is somewhat large compared to the general star population – about 70% of stars in our galaxy are red dwarfs, which are stars that have masses between 7.5% and 50% that of our Sun. Because red dwarfs are so numerous, some planetary astronomers are asking if life could form on exoplanets that orbit red dwarfs – the habitable zone will be closer to the star and the exoplanet's orbital period will be relatively short (e.g. Figure 1).

In this post, I look at the habitable zone around stars smaller than our Sun and the orbital radii and periods of potentially habitable exoplanets (i.e. having temperatures near that of Earth). I became curious about this subject when I read the following statement in an astronomy magazine (Source):

A planet orbiting a star like the sun would have to complete an orbit approximately once a year to be far enough away to maintain water on its surface. "If you’re orbiting around a low mass or dwarf star, you have to orbit about once a month, once every two months to receive the same amount of sunlight that we receive from the sun," Cowan said.

My plan here is to

  • Confirm the statement that an exoplanet around a low-mass star will need to have an orbital period of one or two months in order to receive enough light to have an Earth-like temperature.
  • Determine the nominal orbital radius for an exoplanet in a red dwarf star's habitable zone.

Background

Definitions

Equilibrium Temperature (AKA Effective Temperature)
The planetary equilibrium temperature is a theoretical temperature of a planet if it is assumed to be a black body being heated only by its parent star. (Source)
Luminosity
Luminosity is the total amount of energy emitted by a star, galaxy, or other astronomical object per unit time. It is related to the brightness, which is the luminosity of an object in a given spectral region. (Source)
Exoplanet
An exoplanet or extra-solar planet is a planet that orbits a star other than the Sun. (Source)
Red Dwarf
A red dwarf is a small and relatively cool star on the main sequence, either late K or M spectral type. Red dwarfs range in mass from a low of 0.075 solar masses (MSun) to about 0.50 MSun and have a surface temperature of less than 4,000 K (Source).

Key Equations

Orbital Period Equation

Equation 1 allows us to compute the orbital period of an exoplanet given the mass of the star it orbits and orbital radius. Equation 1 can be derived using Newton's and Kepler's laws (Appendix A).

Eq. 1 $latex \displaystyle {{P}_{{\text{Planet}}}}=\frac{{2\cdot \pi \cdot R_{\text{Planet}}^{{\frac{3}{2}}}}}{{\sqrt{{G\cdot {{M}_{\text{Star}}}}}}}&s=2$

where

  • MStar is the mass of the star about which the planet is revolving.
  • RPlanet is the orbital radius of the exoplanet.
  • G is universal gravitational constant.
  • PPlanet is the orbital period of the exoplanet.

Planet Equilibrium Temperature Equation

An exoplanet's average temperature (TEffective) is temperature at which the amount of energy absorbed equals the amount of energy radiated away. Equation 2 provides an approximate value for this temperature. Appendix B contains a derivation of this equation.

Eq. 2 $latex \displaystyle {{T}_{\text{Effective}}}=\sqrt[4]{{\frac{{L_\text{Star}\cdot \left( {1-a} \right)}}{{16\cdot \pi \cdot \sigma \cdot \varepsilon \cdot {{R_\text{Planet}}^{2}}}}}}&s=2$

where

Many analyses omit the emissivity term because it usually is close to 1.

Star Luminosity Versus Mass

There is an empirical relationship between the mass of a star and its luminosity – it is very rough and has errors for low mass stars. Equation 3 shows curve fit for the data stellar luminosity versus mass data shown in Appendix C. For my work here, I will assume that this relationship is close enough.

Eq. 3 $latex \displaystyle {{L}_{\text{Star}}}={{L}_{\text{Sun}}}\cdot {{\left( {\frac{{{{M}_{\text{Star}}}}}{{{{M}_{\text{Sun}}}}}} \right)}^{{3.5}}}&s=2$

where

  • LSuntar is the luminosity of the Sun.
  • LStar is the luminosity of the exoplanet's star.
  • MSun is mass of the Sun.

Analysis

Approach

My analysis approach is to:

  • Determine the luminosity of a typical red dwarf star using Equation 3.
  • I will then solve Equation 2 for the orbital radius of an exoplanet orbiting this red dwarf with the same effective temperature as the Earth.
  • I will then use Equation 1 to determine the orbital period of this planet.

Exoplanet Period

In Figure 2, I look at the orbital period for an exoplanet with the same equilibrium temperature as the Earth (-21 °C) about a red dwarf star with a mass of 0.3· MSun. I should note that the actual mean temperature of the Earth is ~16 °C. The temperature rise above the equilibrium temperature is caused by the greenhouse effect.

I wanted to test out Equation 3, so I used a nearby star (40 Eridani) as a test case. It was reasonably close (see green highlight).

Figure 2: Exoplanet Period Calculation.

Figure 2: Exoplanet Period Calculation.

My calculation shows that the planet would have an orbital period of about 2 months, which agrees with what I read in the magazine article. The orbital radius is only about 0.2 AU.

Using the framework established for Figure 2, I can plot the range of orbital radii and periods for exoplanets with Earth-like temperatures about a range of dwarf star masses (Figure 3).

Figure M: Orbital Radii and Periods for Exoplanets.

Figure 3: Orbital Radii and Periods for Dwarf-Star Exoplanets.

Conclusion

I have been able to confirm the statement that habitable exoplanets orbiting a red dwarf star will have orbital periods of ~2 months and radii of ~0.2 AU. Exoplanets with short orbital periods are easier to detect than long-period exoplanets, so we should have a good chance of finding these worlds.

The galaxy is full of red dwarf stars, with a number of these stars in our neighborhood (i.e. within 5 parsecs). I am sure that there  numerous challenges remain for life to form on these worlds, but there are lots of potential life-sustaining candidates. Some estimates claim as many as 60 billion habitable red dwarf planets are in the Milky Way.

The Wikipedia has a good article on this topic.

Appendix A: Formula for the Revolution Period of a Planet

Figure 4 shows how to derive Equation 2.  I also show all the various units and constants that I use in all of my other calculations for this post.

Figure M: Exoplanet Period Calculation.

Figure 4: Exoplanet Period Calculation.

Astronomical Unit Gravitational Constant Stefan-Boltzmann constant Solar Luminosity Earth Orbital Radius Solar Mass Bond Albedo Emissivity

Appendix B: Formula for the Effective Temperature of a Planet

Figure 5 shows how to derive Equation 2.

Figure M: Derviation of Formula for Effective Temperature of a Planet.

Figure 5: Derivation of Formula for Effective Temperature of a Planet.

Earth Equilibrium Temperature

Appendix C: Star Luminosity vs Mass.

Figure 6 shows a plot of stellar luminosity versus stellar mass. I will use the rough curve curve fit shown in Figure 5 as Equation 3.

Figure M: Star Mass-Luminosity Diagram.

Figure 6: Star Mass-Luminosity Diagram (Source).

 
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3 Responses to Exoplanet Revolution Period About a Dwarf Star

  1. Ronan Mandra says:

    Nice article. I looked for background on the formula for luminosity versus mass of a star and found this article in wiki: https://en.wikipedia.org/wiki/Mass%E2%80%93luminosity_relation

    This article states that the exponent in that luminosity equation can vary depending upon the mass of the star in question.

     
    • mathscinotes says:

      Excellent article. I see they broke the curve fit into 4 stellar mass regions. I can believe that approach to breaking up the problem based on the curve in Figure 6. Thanks for sharing.

      mathscinotes

       
  2. Pingback: Planets Around An Ultra-Cool Dwarf Star | Math Encounters Blog

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