# How would the characteristics of a habitable planet change with stars of different spectral types?

The tidal forces a habitable planet experiences increase with decreasing spectral type. So a habitable planet orbiting a smaller, less massive, and cooler star would experience much stronger tidal forces than one orbiting a bigger, more massive, and hotter star, and likely be tidally locked.

But what about other factors? How do other characteristics scale up with different stars?

For example would the orbital velocity of a planet in the habitable zone of a red dwarf be smaller than the velocity of a planet in the habitable zone of a more massive and hotter star?

The following table gives the mass, radius, temperature, and luminosity of an average star of several selected spectral types:

$$\begin{array} {d|c|c|} \text{Spectral Type} & \text{Mass} (\odot) & \text{Radius} (\odot) & \text{Temperature (K)} & \text{Luminosity} {(\odot)} \\ \hline \text{M8V} & \text{0.082} & \text{0.111} & \text {2500} & {0.00043}\\ \hline \text{M5V} & \text{0.16} & \text{0.199} & \text {3030} & {0.00299}\\ \hline \text{M2V} & \text{0.44} & \text{0.434} & \text {3550} & {0.0268}\\ \hline \text{K8V} & \text{0.59} & \text{0.587} & \text {4000} & {0.079}\\ \hline \text{K5V} & \text{0.68} & \text{0.698} & \text {4410} & {0.165}\\ \hline \text{K2V} & \text{0.78} & \text{0.763} & \text {5040} & {0.337}\\ \hline \text{G8V} & \text{0.94} & \text{0.909} & \text {5490} & {0.673}\\ \hline \text{G5V} & \text{0.98} & \text{0.982} & \text {5660} & {0.887}\\ \hline \text{G2V} & \text{1.02} & \text{1.01} & \text {5770} & {1.014}\\ \hline \text{F8V} & \text{1.18} & \text{1.25} & \text {6170} & {2.031}\\ \hline \text{F5V} & \text{1.33} & \text{1.46} & \text {6510} & {3.434}\\ \hline \text{F2V} & \text{1.44} & \text{1.61} & \text {6810} & {5.001}\\ \hline \text{A8V} & \text{1.67} & \text{1.81} & \text {7500} & {9.3}\\ \hline \text{A5V} & \text{1.85} & \text{1.94} & \text {8080} & {14.392}\\ \hline \text{A2V} & \text{2.05} & \text{1.97} & \text {8840} & {21.263}\\ \hline \end{array}$$

The following table gives the orbital distance, period, and velocity of an Earth-like planet receiving the same flux from its star as Earth does from the Sun, along with the radial velocity of the star caused by said planet and the tidal forces exerted on the planet relative to Earth:

$$\begin{array}{d|c|c|c|} \text{Spectral Type} & \text{Orbital Distance (AU)} & \text{Orbital Period (days)} & \text{Orbital Velocity (km/s)} & \text{Radial Velocity (m/s) } & \text{Tidal Forces} {(\oplus)} \\ \hline \text{M8V} & 0.0207 & 3.82 & 59.166 & 2.167 & 9166\\ \hline \text{M5V} & 0.0547 & 11.68 & 50.929 & 0.956 & 977.34\\ \hline \text{M2V} & 0.163 & 36.51 & 48.812 & 0.333 & 100.09\\ \hline \text{K8V} & 0.281 & 70.95 & 43.134 & 0.219 & 26.5 \\ \hline \text{K5V} & 0.406 & 114.84 & 38.518 & 0.17 & 10.11\\ \hline \text{K2V} & 0.58 & 182.93 & 34.525 & 0.133 & 3.98\\ \hline \text{G8V} & 0.82 & 280.06 & 31.877 & 0.101 & 1.7\\ \hline \text{G5V} & 0.942 & 337.48 & 30.375 & 0.093 & 1.17\\ \hline \text{G2V} & 1 & 365.56 & 29.973 & 0.088 & 1\\ \hline \text{F8V} & 1.425 & 572.18 & 27.102 & 0.068 & 0.407\\ \hline \text{F5V} & 1.853 & 799.11 & 25.231 & 0.057 & 0.209\\ \hline \text{F2V} & 2.236 & 1018.01 & 23.901 & 0.049 & 0.128\\ \hline \text {A8V} & 3.049 & 1505.21 & 22.041 & 0.039 & 0.058\\ \hline \text{A5V} & 3.793 & 1984.29 & 20.799 & 0.033 & 0.033\\ \hline \text{A2V} & 4.611 & 2526.01 & 19.859 & 0.029 & 0.02\\ \hline \end{array}$$

These calculations were done with the planet being at such a distance that it recieves exactly the same flux from its star as Earth does from the Sun. In reality a planet can be significantly farther away or closer and still remain in the habitable zone.

You can see a very clear correlation: As spectral type decreases (star becomes cooler, smaller, and less massive),

• Orbital distance decreases,
• Orbital period decreases,
• Orbital velocity increases,
• Radial velocity of the star increases,
• Tidal forces on the planet increase exponentially.

I'm sure there are other factors that I didn't take into account but these are the most obvious ones I could think of.

1. I got the data for the mass, radius, and temperature of each star from here. This is an average relationship and not meant to be exact.

2. The luminosity was calculated using the Stefan-Boltzmann law, assuming a perfect sphere and a blackbody radiator, which most stars approximately are.

The main effect would be the radiation environment. A planet in the habitable zone of an M-dwarf would likely be subject to far more ultra-violet and X-ray observation for longer than a planet orbiting a G-dwarf of similar overall age.

The reason for this lies in the physics of stellar dynamos that power the magnetism of cool stars. Fast rotating cool stars have stronger magnetic field and this leads to more heating in their chromospheres and coronae. This in turn leads to more UV and X-ray radiation arising from the hot plasma that is contained within these regions.

Stars are born as faster rotators and then they lose angular momentum by a coupling of their stellar winds to their large scale magnetic fields. Thus as they get older, they become slower rotating and the magnetic activity lessens. For example, a star like the Sun, but of age 100 million years, would be thousands of times more X-ray active than the Sun is now.

For reasons that are not yet fully understood, it is observed that lower mass main sequence stars have longer spindown timescales and remain magnetically active for much longer - billions of years in the case of mid M-dwarfs (e.g. Bouvier et al. 2014). This means that for a similar total flux of energy from the host star, a planet orbiting an M-dwarf would receive much higher doses of UV and X-ray radiation over the course of billions of years.

This extra high energy radiation is likely to have a dramatic effect on the atmospheres of close-in planets around M-dwarfs and may of course have a bearing on whether life (as we know it) could develop.

The (assumed circular) orbital velocity of a planet in the habitable zone of a cooler, lower mass star, would be larger than for a sun-like star (which just arises from Kepler's third law), but that will not have any implications for the atmosphere or habitability.

The orbital periods, and thus years, of planets in the habitable zones of stars of different types may vary a lot, depending on how wide or narrow a star's habitable zone is and which types of stars can have habitable planets. Thus it may be possible for some habitable planets to have years tens times as long as others, possibly even hundreds of times as long as others.

Meterological seasons do not correspond exactly with astronomical seasons, but are based on them. Astronomical seasons are each a quarter of a planet's year.

So planets with widely different year lengths can have widely different season length, which could possibly have major effects on the hypothetical life on those planets.

Planets in the habitable zones of dim reddish stars would be tidally locked to their stars, and thus would not have seasons unless their orbits were quite eccentric. Instead one side would have eternal day and eternal summer, and the other side would have eternal night and eternal winter. That might make the planet uninhabitable unless the atmospehere and hydrosphere kept both sides at similar temperatures.

If a giant planet orbited in the habitable zone of a red dwarf star, and had one or more planetary mass moons that might possibly have life, those moons would be tidally locked to the planet instead of to the star. Thus their orbital periods around the planet, their months, would equal their cycles of light and darkness, their days. That would have significant climatic effects.

So there are possibly strong climatic differences between planets and moons orbiting stars of different spectral types, even in cases where their distances are such that they receive exactly the same total amount of radiation from their stars as Earth does.