How can we detect the potential for life outside our own system?
You probably already know that we can't simply point a telescope at any star and spot potential orbiting planets- if this was the case discovering planets outside our own system would be a lot simpler. This is partially because the light from the star outshines any light reflected by the planet, and partially because when two objects are far enough away form us, and close enough to each other, they sort of 'merge' when we see them through a telescope. We need a better method.
We hinted in our last post that we could look at a star's light to find orbiting planets. How you ask? The planets movement around the star causes it to move in a elliptical motion. When you're looking at an astrophysicist pacing around in circles, pondering over the mysteries of the world or maybe what they want to eat for lunch, you'll see them walk towards you, then sideways, then away from you, sideways again, and the motion repeats.
See where we're going with this?
When the star orbits around a center of mass it will sometimes be moving away from us, and sometimes towards us, and this affects the light we see!
The effect is known as the Doppler effect, and you've probably experienced it. When a car drives by you, the noise changes as it passes. This is because the car is sort of pushing on the sound waves as it heads towards you, making the wavelengths smaller. When moving away from you, the wavelengths get longer as the sound waves you hear are going the opposite direction of the car. This also applies to light!
Different colors of light have different wavelengths (some wavelengths we can't even see with our bare eyes). We see longer wavelengths as red and shorter as blue, which is why we say light is being red shifted as the star moves away from us, and blueshifted when it moves towards us. By observing this change in the light waves, we can find the radial velocity of the star:
\(\frac{\lambda-\lambda_0}{\lambda_0}=\frac{v_r}{c}\)
where \(\lambda\) is the wavelength we observe, \(\lambda_0\) is the wavelength observed from a still system, \(v_r\) is the radial velocity and \(c\) is the speed of light.
We can only observe this effect if the star has a radial velocity to us, in other words, it needs to be moving either away from, or towards us.
We can plot the radial distance as a function of time, as we can convert the wavelength to radial velocity. We can then find the stars rotational period, P, and the peculiar velocity \(v_s\)
\(v_{r}(t) = v_s\cos{\frac{2\pi}{P}t}\)
Where \(v_s\) is the velocity of the star with respect to the center of mass.
We use this formula to plot the radial velocity of our star over a time period of 70 years:
We chose to observe only the x-component, as we are only interested in looking at what is happening in our line of sight, which is one-dimensional (the line-part of 'line of sight' is a giveaway).
The problem with radial velocity curves is that they don't look like this in the real world, which is a hot mess compared to the idealized world physicists tend to live in. When observing a moving star, the actual information will look something like this:
What we are looking at here is Gaussian noise. We know that the actual information we want is (probably) somewhere in the middle. Unfortunately, saying that something is "somewhere in the middle" is not an approved scientific method. Interpriting this graph knowing as much about our own star as we do is no problem. But what about when we're looking at a distant star? After all, we wan't to know its period, as well as its radial velocity and the time of one period. If we don't know this we can't read the plot!
Once again, we will have to use statistics to get a more precise approximation of reality.
When we say that the noise is Gaussian, we mean that it's random, but that it creates a Gaussian distribution if you look at enough points. We can look at each point individually and compare it to the familiar bell curve. The curve stretches between the maximum fluctuations in the noise, and the curve can, in theory, be anywhere along the line created by the noise in each point.
We need an expression for what the radial velocity curve looks like without noise, and for that we need good estimates of the orbital speed and the period (see formula for radial velocity). We can find this using the the least squares method.
\(\Delta(v_*,P,t_0)=\sum\limits^N_{i=1}\left[v_r^{obs}(t_i)-v_r^{mod}(t_i,v_*,P,t_0)\right]^2\)
where \(v_r^{obs}\) is the observed radial velocity curve, \(v_r^{mod}\) is the modeled radial velocity, which is just a general velocity curve which takes the parameters \(v_*,P,t_0 \):
\(v_{r}^{mod} = v_*\cos \frac{2\pi}{P}(t_i-t_0)\)
(\(t_i\) is every point in time where we have measured the data, and is therefore known) Basically, we subtract the absolute values of the modeled curve from the noisy data at every point in time \(t_i\), and add all of those differences together to get the overall difference between the modeled curve and the noise. We want to minimize the difference \(\Delta\) as much as possible. When we get a \(\Delta\) that is small enough to be considered satisfactory, we know that we have found correct (enough) parameters, and therefore a correct (enough) expression for the actual curve underneath all the noise!
If we assume the noise is Gaussian (and it is), we can express the observed radial velocity as \(v_r^{obs}(t_i) = v_r^{real}(t_i)+\delta v_i\) , in words as the true radial velocity plus the Gaussian distributed noise.
\(P(\delta v_i) = \frac{1}{\sqrt{2\pi}\sigma_i}e^{-\frac{1}{2}\frac{\delta v_i^2}{\sigma_i^2}}\)
For several velocity values, we can multiply the probability in every point, and set \(\delta v_i = v_r^{obs}(t_i) - v_r^{mod}(t_i)\), where the real radial velocity curve has been replaced by our modeled velocity curve, which contains our unknown parameters, \((v_*, P,t_0)\). We want to find for which values of \((v_*, P,t_0)\) the probability is maximized and the difference is minimized, and this requires some trial and error.
As we have yet to hear from any extraterrestrial neighbors, we have to create our own curve for now. By analyzing the noisy curve we have already introduced, we can make an educated guess about the period and velocity like this:
We can see that the period is roughly 50 years, and the radial velocity is somewhere around 0.0004 AU per year at \(t_0\)=12. We plot the curve over 50 years to check:
And this is our final result. It's far from perfect, but when you have to do this manually, there are only so many values you have got time to check. We still have a pretty good idea about how the sun moves!
A solar system rarely contains nothing but a single planet. How will several planets affect the radial velocity curve? First things first. We now have several planets affecting the star's movement, and the center of mass will be in a different place. In the calculations we have done so far, the center of mass was lying along a line between the star and the plan, now it is somewhere between all of the planets. That means our new coordinate system has moved in both x- and y-direction! In order to find the initial position of the center of mass, we use this expression:
\(\vec{R} = \frac{1}{M} \sum_{i} m_i \vec{r}_i\)
We take the vector sum of all of the position vectors of the planets times their mass, and divide by the total mass in the system, star included. Once again, we move in the negative direction of \(\vec{R}\) and then along the vector from the star to the planet to get the position of the planet in the new coordinate system.
After moving three planets, we do do the whole integration over again, and use the same technique to plot the radial velocity curve as before.
We observe that the sun's velocity no longer oscillates around 0, it increases as it oscillates! It might slow down over a greater amount of time, but if so, we can't see it as we have only plotted over 50 years. (It may be a mistake on the part of the more-or-less qualified computer scientists in our team, but we are fairly certain of our results!)
What else can we find out about our (possible) neighbors? They most likely live on a planet, not a star. That means we have to a) find a planet and b) find out whether it is somewhat inhabitable.This time we will be utilizing another type of curve: a light curve. When a planet passes in front of its star, we will observe an eclipse, as long as it happens within our line of sight. The eclipse causes a change in the light emitted from the star, also called flux, and that dip in the flux curve yields a lot of information.
What is the radius of the planet? What is the velocity of the planet with respect to the mass? And most importantly, what is the density of the planet? The density of a planet may not sound like the coolest thing to find, but this is actually vital if we want to see if the planet is inhabitable!
Let's analyze the above curve, step by step. Everything that happens before A doesn't tell us much, as the flux is constant. Things are getting interesting when the planet starts eclipsing the sun (B), as we will see a drop in the flux. This lasts as long as it takes the planet to move the distance equal to its diameter (or two times the radius). After that, the whole planet will be in front of the sun (C), and the flux is once again constant, but lower than before. The planet keeps on moving, and when the planet starts to leave (D), we can once again observe a change in the flux. When the eclipse is over, the flux is once again stable at 1. (E)
First, let's find the radius. We know the time it takes the planet to get all the way in front of the sun (\(t_1 - t_0\)). The velocity of the sun \(v_*\) is also known, as we can use our radial velocity curve to calculate it. We can use the following equation to get the radius of the planet:
\(2R_p = (v_* + v_p)(t_1 -t_0)\)
The only thing missing is \(v_p\), or the velocity of the planet with respect to the center of mass. In order to find this, we have this equation:
\(v_p = v_* \frac{m_*}{m_p}\)
And we get yet another missing piece of information! Two, actually, as we don't know the mass of the star, nor the planet. As for the latter, we can't know the exact mass, we can only decide the minimal mass.That is because the mass is dependent on the angle between the stars orbit and our line of sight:
\(m_p = \left(\frac{P}{2\pi G}\right)^{1/3} m_*^{2/3}\frac{v_{*,r}}{\sin{i}}\)
We get the smallest possible mass when \(i = 90^{\circ}\), which means the denominator will be 1. \(v_{*,r}\) is the radial component of the suns velocity, or \(v_* \sin i\).
The mass of the star is measured by methods that are beyond the scope of this blog, but we can consider it to be known.
Now that we know the minimum mass of the planet, we can calculate the density of the planet:
\(\rho_p = \frac{m_p}{4/3 \pi R_p^3}\)
With this information, we can determine if the planet is a gas planet or rock planet. If the planet is 4-5 times denser than water, it is a terrestrial planet. That means there is a slight possibility of finding life there!
As you can see, we can tell a lot about a system even if we can't picture it directly! Knowing that there are planets is a good start when searching for life. But not all planets have life on them (really, we just know of one so far). How can we tell which planets are more likely to have life on them? Stay with us, and all your questions will be answered! In the meantime, take solace in the fact that while we don't know weather we are alone in the universe, at least we have each other <3