Technically, we could go anywhere we wanted to. But not all planets are made equal, and while we love them all for their individuality, we can only pick one.
Our solar system consists of five rocky planets and three gas planets. Similar to Earth's system, our four innermost planets are rocky. Not so similar to Earth's system, these are followed by a gas planet, then another rocky planet, and then two more gas planets.
(For a long time we though that most solar systems must be similar to earths, with the rocky planets closest to the sun, and gas giants further out, but as we discovered more and more systems, we found that this does not seem to be the case!)
We rule out traveling to any gas giants early, not because they smell, but rather because these tend to be difficult to land on. They have a very high mass, and our spacecraft would be crushed before we hit the center, where we don't even know if there is anything solid to land on. They are definitely cool subjects of study, but for now, we want to land on something solid.
This leaves us with our rocky planets, none of which we know very much about other than where they lie compared to our star. However, this can already let us know quite a bit, such as an estimate of the temperature of the planets. The temperature is interesting in itself, as we know that between 0°C and 100°C, we find water in its liquid state. And where there is liquid water, there is potential for life.
This is why we tend to search for life in a zone around a star where the temperature allows for liquid water. You may have heard of it referred to as the habitable zone, or the goldilocks zone, and this is where we want to look for a suitable planet to land on.
So where is this zone?
The star our planets orbit releases a certain amount energy over a certain amount of time, referred to as its luminosity, and can be found from its flux and area:
\(L=FA\)
Keep in mind that the F stands for flux, not force, and is the energy passing from the star per area, over a small amount of time.
If we describe the star as a stable black body, an object that absorbs all radiation that hits it (it can still emit radiation of its own, so it does not have to look black), we can define flux by Stefan-Boltzmann's law, \(F=\sigma T^4\), where \(\sigma\) is known as the Stefan-Boltzmann constant and \(T\) is the temperature. The star sends out rays from all sides, so we can describe its luminosity from the flux of the star, \(F_*=\sigma T_*^4\), and the whole area, \(A=4\pi R_*\), where R is the radius of the star.
The planet, however, doesn't receive all of this, as the rays don't hit all of the planet, and it
can't absorb rays that aren't hitting it. Imagine you fall asleep sunbathing, while lying on your back. Your front will be burned, but your back will remain as pale as ever. Only about half of the planet will receive rays from the star!
There is also the issue that not all the rays hit the planet normal to it's surface, so the flux won't remain constant over the surface. Even if we assume that the rays have travelled over such a long distance that we say they are parallel to each other (and we do!), the surface of the planet is curved.
We instead decide to treat the flux as if it's hitting a flat surface, sort of the shadow of the planet, which we refer to as the effective absorption area, \(A_{shadow}=\pi R_p^2\).
We now treat our planet as a black body and assume that it absorbs all the rays that hits it (a bold assumption, but it simplifies things quite a bit). Because we don't want our planet heating up (no global warming here!!!), we then assume that the planet emits all absorbed rays in the form of thermal energy.
This energy will be released everywhere on the planet, not just the part hit by the rays from the star, meaning that the luminosity of the planet can be described as \(L_p=F_pA_{planet}\), where the area is that of the whole planet, not a 'shadow', and the flux can be described by Stefan-Boltzmann law.
Now we have the luminosity sent out from the star, \(L_*=F_*A_*=\sigma T_*^4 4\pi R_*^2\). This means that the flux the planet receives from the star, \(F_{*p}\), at a distance \(r_{*p}\) away from it, can be described as
\(F_{*p} = \frac{L_*}{A_{*p}}=\frac{\sigma T_*^4 4\pi R_*^2}{4\pi r^2_{*p}}=\sigma T_*^4\left(\frac{R_*}{r_{*p}}\right)^2\)
where \(r_*\) is the radius of the star, and \(T_*\) is the temperature of the star. We've simply rearranged the earlier expression for luminosity in order to define this flux.
We can now make two expressions, one for the total energy the planet receives from the star, and another for the thermal energy it releases:
\(L_{in}=F_{*p}A_{shadow}= \sigma T_*^4\left(\frac{R_*}{r_{*p}}\right)^2 A_{shadow}\)
\(L_{out} = F_pA_{planet}=\sigma T_p^4 A_{planet}\)<-- Note the temperature of the planet! Our unknown :0
We established that for the temperature of the planet to remain constant, we need the luminosity absorbed by the planet from the star to be the same as the luminosity emitted from the planet, \(L_{in}=L_{out}\)!
We rearrange this equation, as it is the temperature of the planet we are interested in:
\(T_p=\left(\frac{F_* A_{shadow}}{\sigma A{planet}}\right)^{1/4}=\left(\frac{\sigma T_*^4\left(\frac{R_*}{r_{*p}}\right)^2 \pi r_p^2}{\sigma 4\pi r^2_p}\right)^{1/4}=T_*\left(\frac{(R_*/r_{*p})^2}{4}\right)^{1/4}\)
We can use this expression to find the surface temperatures of the planets in our system if we know the temperature of the star, \(T_*\), the radius of the star \(R_*\), and the distance from the star to the planet, \(r_{*p}\).
We decided to use our planet's initial positions (their positions when we first begin the simulation) as the distance between the star and the planets. There are other methods, such as calculating the temperature when a planet is closest and furthest from the star, giving an idea of the highest and the lowest temperatures, or finding the temperature over an average distance. Calculating the temperature for every point in the orbit is also technically a possibility. But we've decided to save ourselves and our computers a bit of power, and we stick with the temperatures at the initial positions.
| Planet index | Distance from star | Temperature in Kelvin/Celsius |
|---|---|---|
| 0 | 748 973 288 km | 350K / 75.5°C |
| 1 | 1 250 716 210 km | 271K / -3.5°C |
| 2 | 8 900 994 850 km | 101K / -172.7°C |
| 3 | 254 439 360 km | 600K / 325.8°C |
| 4 | 4 749 904 990 km | 139K / -135.3°C |
| 5 | 6 762 109 750 km | 116K / -157.8°C |
| 6 | 3 070 321 510 km | 173K / -101.4°C |
| 7 | 438 653 792 km |
457K / 182.8°C |
Keep in mind that these results do not account for the fact that planets are not really blackbodies. They do reflect some of the rays that hit them. Some planets and moons also have a hot inner core that gives more thermal heat to the planet, and atmospheric density can make a difference.
Keeping this in mind, are the results somewhat probable? Our home planet (0), is about the same size as Earth, but five time as far away from its star, yet it's much warmer than Earth. However, this system's star is around two times as big, and twice as hot as the Sun! Our planet is still pretty hot, but not impossibly so. And as the planets often reflect some sunlight, it may be a little colder than we see here.
Now the time has come to choose a planet. We want to look for planets with a temperature within the range [260-390] Kelvin (0°C-100°C), as water is liquid for these temperatures, but we leave an error margin of \(\pm \approx 15 \) K, seeing as we've made quite a few simplifications.
Planet 0 and Planet 1 are the only two planets within this range, which made this decision a lot easier! Planet 1 may seem cold (the Earth has a surface temperature of around 14°C and sometimes it's very cold here), but it also has an atmospheric density almost twice that of the Earth. Hopefully it isn't as cold as it seems.
Planet 1 seems a good choice, but we should take into account that life needs a lot of time and stability to evolve. Sudden changes in the climate doesn't make for the best base for life, so we want to make sure that our planet doesn't move too much within the habitable zone.
We find this zone, the distance from which the temperature is within the interval for which we find liquid water, by rearranging the equation we found the planet surface temperature from:
\(T_p=T_*\left(\frac{(R_*/r_{*p})^2}{4}\right)^{1/4} \longleftrightarrow r_{*p} = R_*\sqrt{\frac{1}{4}\left(\frac{T_*}{T_p}\right)^4}\)
We use this for the minimum temperature and maximum temperature we want, and find the distances these two occur on a planet's surface.
Our habitable zone! It stretches from about four to ten AU, and this is where we can expect to find liquid water. Any closer to the star, and it will probably have evaporated, and any further, it will probably only exist in the form of ice. (Though there is ice on Mercury, and possible buried oceans as far out as Pluto, but this isn't really what we're looking for. We prefer proper oceans and sunbathing temperatures.)
Let's zoom in and have a proper look at the planets in the habitable zone.
As we predicted, Planet 0 and Planet 1 are both well within the habitable zone, and seeing as we're currently living on Planet 0 (we love the hot weather), we'd like to travel someplace else. Planet 7 (grey) isn't too far off, but with a temperature of over 180°C we decided against traveling closer to the sun. Traveling someplace colder than -10°C is out of the question, as one of us gets cold very very fast and even just looking at pictures of a cold planet sounds terrible. Of course, gas planets are always interesting, but in the end that means watching our dear rocket crash and burn, and we just don't like that thought. :(
We decide to travel to Planet 1, as this seems most promising considering what we know! But keep in mind that there are many factors that contribute to a planets climate (both Venus and Mars are technically in the habitable zone, but I don't want to live there). We just don't know enough to be sure, so we work with what we have.
While we're on the subject of luminosity and flux, let's finish with a final thought.
When we super-successfully arrive at our destination, we'll need to power our lander unit with some source of electricity. We've decided to use solar panels because there were no space-gas stations on our way out, plus it's, like, super good for the environment. Unfortunately, management thinks we are taking way long with all this simulation, and as punishment we only get solar panels with an efficiency of 12%. Why? No idea, take it up with the big guys. All we know is that this means that we can get 12% of the energy our solar panels receive.
We need to supply our lander unit with 40 W (Watt) of electric power. This means that we need 12% of the energy the panels receive to be 40 W:
\(\frac{40 W}{0.12}=333.3W\)
The total energy needed for our solar panels to be able to get 40 W, is 333.3 W!
We know that the energy received is given by the amount of flux from the star on the planet and the area absorbing it, in other words we have \(L_{panels} = F_{*p}A_{panels}\). We already know how much luminosity we need, and how we can find flux, which means that the area of the solar panels can be found:
\(A_{panels}=\frac{L_{panels}}{F_{*p}}=\frac{40W/0.12}{\sigma T_*^4\left(\frac{R_*}{r_{*p}}\right)^2} \)
Inserting our star's surface temperature and radius, \(T_*=10240.7\) K and \(R_*=1 \: 746 \: 710 \) km, as well as the distance from the star to our chosen planet at its furthest point from the star, \(r_{*p}=1 \: 250\: 716\: 210\) km, we get that our panels need an area of \(0.274\) m2. This is the smallest it can be and still function, but we can go bigger for even more energy! As we chose to measure the distance at the furthest point from the star, we have also secured enough energy everywhere in the orbit.
For simplicity’s sake, we have made the assumption that the flux on the planet is the same as on our lander, and that the 40 W is enough to keep the lander unit going through the night, when there is no light. We've also decided to ignore the atmosphere on the planet (but we will look further into the atmosphere at a later time).
Now that we know where we want to go, we can finally prepare to launch for real! Stay put, next time we figure out where to launch from!