Dark Side of a Planet

This is a long one, folks, but hold in there. I think it's one of my best yet. 

Since I declared Astrophysics as my major a year and a half ago, my friends seem to have delegated it to one of my primary defining characteristics.  "Do you know Moiya?"  "Oh yeah, she studies astrophysics, right?"  I'm okay with this.  One, there are weeks when I spend more time at the Center for Astrophysics than I do in my own room, so I can't deny that my life pretty much revolves around the subject.  Two, I absolutely love talking to people about astronomy.

My friends know this, so they ask me a lot of questions about the universe.  Sometimes, their questions are very specific: "What do they mean when they say the universe is expanding?"  But most of the time, their questions aren't really questions at all, but requests to hear something, anything, about astronomy.  Because of this, I've developed a sort of astro fact kitty that I keep in my back pocket at all times.  One of my favorite facts to pull out is: The moon is tidally locked with the Earth, which means that the time it takes the moon to rotate is exactly the same as the time it takes to revolve around Earth, so that we always see the same side of it. 

"Tidal locking of the Moon with the Earth" by Stigmatella aurantiaca - Own work. Licensed under CC BY-SA 3.0 via Wikimedia Commons - https://commons.wikimedia.org/wiki/File:Tidal_locking_of_the_Moon_with_the_Earth.gif#/media/File:Tidal_locking_of_the_Moon_with_the_Earth.gif

I love this fact because it's quick, relatively simple, and I spent a week last year learning about the science and math that explains it.  But I never thought much about the broader applications of this knowledge.  In my mind, the moon was tidally locked with the Earth and that was it.  But my mind was wrong, because all sorts of things get tidally locked, including planets!

One question you might have at this point (especially if you didn't follow the link above) is:  How does tidal locking work? 

If you want a really in-depth answer, I'm here to tell you that Google is your friend.  If a more qualitative answer will suffice, Googling isn't necessary.

Say you have two massive bodies in space, A and B, where A is much more massive than B.  The gravitational force from A literally changes the shape of B, forcing it to elongate, or bulge, along the axis that points towards A.  So, instead of looking like a basketball, planet B now looks more like a rugby ball.

       

Before B is tidally locked to A (B's rotation speed does not yet match its orbital speed), that bulge travels around B.  Depending on the relationship between the rotational and orbital periods (which one is longer than the other), the bulge will lag behind the planet in its orbit or point in front of it.  This asymmetrical bulge creates all kinds of messy forces, which act on the system until the bulge faces planet A, thus tidally locking B to A.


Cool. Now you (hopefully) have a better understanding of how tidal locking works.  But that's not really the point of this blog post. The point is to explore what it would be like to live on a planet that was tidally locked to its star. (Just the scientific implications, because, as a student of mythology, I could go on an on about the mythological and cultural implications of having one side of a planet in perpetual light and the other in perpetual darkness.)

I'm going to try to do some math later, so let's make some assumptions about this tidally locked system.

1. The planet is about 1/10th the size of Earth:  \(R_P = 6.4\times10^6 m\)
2. The planet's atmosphere is made of mostly oxygen.
3. The star around which that planet orbits is roughly the same size and temperature as our Sun:  \(R_{sun} = 7\times10^8 m\) , \(T_{sun} = 5800K\)

The way I understand it, there are two possible outcomes.

In the first, the planet rests right on the edge of the space where oxygen freezes:

$T_P^4 = \frac{T_*^4R_*^2}{4a^2} \Rightarrow a = \sqrt{\frac{T_*^4R_*^2}{4T_P^4}}$

This was found by setting the flux received by the planet from the star equal to the flux emitted by the planet. 

A quick Google search told me that oxygen freezes at \(\approx 50K\), so that's the temperature we'll use to find the distance, a, from the planet to its star. 

\(a_P \approx 10^{12} m\)

For reference, this is about 10 times farther than the Earth is from the Sun.

 Because it's just on the edge of oxygen's freezing point, the side of the planet that faces the star is almost frozen, and the side of the planet that faces away from it is frozen solid. Eventually, the ice would encroach upon the "warm" side and the entire atmosphere would become ice.  That would be the end of any life that could have possibly lived under such extreme conditions.

The second outcome is, at least for me, more exciting.  In this case, I want the planet to be close enough to its star that the whole planet would be warm enough to sustain life.  

When I was originally thinking about this, I toggled back and forth between thinking such a thing was possible and thinking the dark side would freeze no matter what.  I had memories that backed up both theories. I remembered reading about a deep, deep crevice on earth that was colder than the coldest places on the moon because it never received any sunlight, and I remembered spending summer afternoons sitting in shady spots that never really felt cooler than the sunny spots a few feet away.  Eventually, the memories of lazy summer days won, but I needed to back that intuition up with science.

To do this, I needed to revisit my old friend (when I say "friend," I really mean "bane of my existence") from a class on partial differential equations, the Heat Equation.  This equation, intuitively enough, can be used to describe the distribution of heat in an area over time.  Sounds pretty perfect for what I'm doing.  I recognize that this might be giving some people flashbacks to traumatic physics class experiences, so I won't treat this as a rigorous physics problem, but will instead do some mostly qualitative assessments.

We can project the sphere of a planet onto 2 dimensions because we're concerned with the distribution of heat over its surface.

 

This projection becomes an oval, and it's fairly simple to solve the heat equation over a plane.  

$T_t = c^2\left ( \frac{\delta^2 T}{\delta x^2} + \frac{\delta^2 T}{\delta y^2}\right )$

where c is determined by the initial conditions, which are just how much flux the bright side of the planet is receiving.  Now that I have all of the equations, all I need to do is set my boundary conditions (an acceptable range of temperatures, say \(275K < T_P <320K\) that can sustain life) and I can solve for the right set of initial conditions.  Yay!  

Okay, now we know how the math behind creating a habitable, tidally locked planet would work.  But how would that manifest itself physically? In other words, what must the physical characteristics of this planet be in order to maintain a reasonably uniform temperature? 

• The planet has to have a lot of liquid (maybe water).  
• Those of you who use water to heat your home know that it's a really efficient way of transporting and re-radiating heat.
• There has to be some way to trap the heat in. 
• This could be like our Greenhouse Gas Effect, which uses Carbon Dioxide and other gases to trap the heat within our atmosphere. 
• If there's a Greenhouse Gas Effect, it means there has to be something producing that much greenhouse gas, which likely points to life existing on the planet! 
• There has to be some internal heat source.
• This could be in the form of volcanoes, which means there's plate tectonic activity on this planet, which some believe is essential for forming and sustaining life.  

I don't quite know what the implications are of all this yet; I just thought it was fun and cool to think about. Do with this what you will  :)