Before we begin analyzing our destination planet's surface, we want to stabilize our orbit. We are currently spinning around our destination planet in a very elliptical fashion, and that just won't do when we are trying to make precise measurements of the atmosphere. To achieve this, we perform an orbital injection maneuver. This means a boost that will put us into a circular orbit, with a stable velocity. We want to slow down to less than 10 km/s, and this boost should (ideally) slow us down enough to get there.
First, we need to find the stable velocity, or the velocity we want our rocket to have when in orbit:
\(v_{stable} = \sqrt{\frac{GM_p}{r}}\)
where \(G\) is the gravitational constant, \(M_p\) is the mass of our destination planet, and \(r\) is the distance to the planet. The orbital injection looks like the following:
where \(\mathbf{v} \pm = \pm \mathbf{e}_{\theta} v_{stable}\). The \(\pm\) means that we have to take into account whether we traverse the orbit clockwise or counterclockwise. \( \mathbf{e}_{\theta}\) is a unit vector, but this unit vector doesn't point along the position vector \(\vec{r}\) like we are used to; it is perpendicular to the position vector! And finally, we have \(\mathbf{v}_0\), which is just our velocity at the time of performing the orbital injection.
We choose to perform our orbital injection maneuver at perihelium. As we want a circular orbit, we check the eccentricity of our new orbit. An eccentricity of exactly 1 means a perfectly circular orbit. We are therefore very happy with an eccentricity of 0.9997. Next we check our speed, which should preferably be below 10 km/s. At 0.44 AU/years, or a little under 2.1 km/s, we are well within the speed limit!
Sadly, as we checked and double checked our trajectory, it turned out that we were unsuccessful after all. Luckily, we have NASA on hotkey, and with a little help from our more experienced friends, we are in a stable orbit. The speed was also off, as it was measured compared to the star, not our planet.
Now that we've stabilized our orbit, we can begin to descend on the planet. At this point, the gravitational effects from other astronomical bodies become minuscule in comparison to the planet, and so we can ignore the effects from all other planets and the star.
We know that the radius of a circular orbit is determined by the kinetic energy of the orbiting body,
\(K=\frac{1}{2}mv^2\)
where \(m\) is the mass of the spacecraft and \(v\) is the velocity of the spacecraft. One way to enter a lower orbit would be to lower our kinetic energy.
As we cannot change out mass (much), we want to slowly decrease our speed in order to decrease the average distance between our spacecraft and the center of the planet. To keep the stable, circular orbit as circular as possible, it is best to to the boosts at aphelium and perihelium. Even though these two points can be anywhere in a circular orbit, it is enough to decide on two points on opposite sides in the orbit where we slow down the rocket. (Or rather, this is what we would have done if we didn't get help from the Big Guys at NASA)
But watch out!
If we lower our orbit too much our spacecraft will enter the atmosphere. To be fair, this is in the plan, but not right away. We are therefore careful to check the distance between our spacecraft and the center of the planet to make sure we aren't too close. If the average of this distance remains constant over time, we can safely assume that we have not entered the atmosphere.
We do, however, want to enter the atmosphere at some point, and so we need to know its effects on the spacecraft during landing. For example, air resistance, which is dependent on the atmosphere's density profile, will have an effect when we try to land. This is why we set off some time to study the atmosphere!
In order to do this we will utilize the Doppler effect. Refresh the older posts on the Doppler effect, and on spectral lines, if you can't recall this! It's worth mentioning that in the post about spectral lines, we only mentioned the absorption spectrum, where we see black
Absorption spectrum vs emission spectrum.
lines where specific wavelengths have been absorbed. We could also look at an emission spectrum, where we look at the specific wavelengths that are ejected, when the electrons fall down from the excited state (they 'want' to have as low energy as possible which is a mood). This spectrum will be black where the absorption spectrum has colors, and where the absorption spectrum has black lines, it will have colored lines.
We want to use the Doppler effect to study the spectral lines in the atmosphere, as you know these are the lines that are absorbed by various gases in the atmosphere. Because each gas absorbs light of different wavelengths, they will leave different patterns, and from this we can decide which gases are present in our atmosphere! Our spacecraft has been equipped with a flux meter capable of measuring the flux of light with wavelengths between 600 nm and 3000 nm, which we will utilize in order to collect data. We will be searching for 'dips' in the flux curve around wavelengths we know are absorbed by different molecules.
How we see spectral lines in the flux we measure. If only it were this simple.
'Dips' such as these may point to the presence of a spectral line there, as this means that in this spot we absorb a smaller number of wavelengths.
However, when we measure these flux curves, our data will still have a lot of noise, described as mistakes in our data during the measurement process. This means that our curve will not be nice and smooth, with a few clear dips, but rather very noisy, jumping up and down.
MESSY STUFF
These are not the nice dips we wanted! How can we know what we are looking at here?
Let's take a moment to study what factors determine the shape of the dips:
The number of molecules present in the atmosphere determine the depth of the dip. The more molecules, the more they can absorb of the specific wavelengths they absorb, leading to a deeper dip! This flux has a minimum value between 0.7 and 1, found from earlier experiments.
The velocity of the spacecraft with regard to the atmosphere determine the position of (the center) of the dip.
vr is the velocity of the spacecraft in relation to the velocity, v, of the molecules in the atmosphere, F is the flux we measure, \(\lambda_0\) is the actual wavelength and \(\lambda\)is the wavelength we measure.
The velocity is measured from our moving spacecraft, meaning that we may not measure a dip exactly where we expect it. We've previously stated that when objects are moving, they create a Doppler effect where they either 'push' or 'pull' on the light waves, distorting them. Because of the Doppler effect, the dip may have been pushed a little off the wavelength we expect it to be for a certain gas. We control how wide this shift can be by setting our maximum speed (when we perform the spectral analysis of the atmosphere) in relation to the atmosphere as \(v_r = 10\) km/s, as Doppler shift depends on the velocity.
We can recall the relationship between velocity and Doppler shift from a previous post, and find that the Doppler shift that the molecules can have is defined as:
where \(\Delta\lambda_{max}\) is the maximum Doppler shift, \(v_{max}\) is the largest velocity we expect to find, \(\lambda_0\) is the original wavelength, and \(c\) is the speed of light. This is the maximum Doppler shift we observe!
The velocity of the molecules in the atmosphere determine the width of the dip. As the molecules in an atmosphere are not standing still, they too create a Doppler effect. The wavelength may have the correct wavelength in relation to the molecule even if it doesn't have the correct wavelength, thereby meaning that the molecules may absorb more wavelengths seen from the perspective of the atmosphere.
How the Doppler shift affects the spectrum. The colors represent red- and blueshift.
This shift is affected by the velocity of the molecules in the atmosphere. We know that temperature affects the speed of the molecules, but we do not know the exact temperature on the planet. We do know that it should lie somewhere between 150K and 450K, and so we assume that the maximum speed of the molecules is 10 km/s (in comparison to the spacecraft).
The parameters of the dip.
This means that we have three factors that determine the parameters of the dip, but we do not know their exact values. We do know that which spectral lines are absorbed depend on their velocity, and we know that the velocity for a gas follow a Gaussian curve! Previously we spoke about a method called the least square method, which we also used when we had a noisy plot, and a few parameters we knew the values of the curve we wanted should lie within. But when we used the least square method, the noise was constant.
However, looking at the picture of our data there is a lot (a LOT) of noise, and it varies just how noisy our noise is. Like when you were five, and turned the noise on the radio up and down and up and down til someone finally stopped you (or was that just me?). To handle all this noise, we use a method called \(\mathcal{X}^2\) minimization, as this method can handle data where the noise varies by downvoting noisy data points. If there is a lot of noise in one place, this method will make sure that those data do not influence our result as much as the less noisy data. This prevents inaccurate data from influencing our final result so much that it becomes completely inaccurate.
Let's explain this slightly more mathematically. Our final goal is the same as the least square method: Finding a smooth curve from the noise that is our data. if we let \(f_i\) denote the measured data points and \(f(t_i)\) denote the expected data points based on our mathematical model \(f(t)\), we can express the \(\mathcal{X}^2\) function as
where \(\sigma_i\) is the standard deviation in the noise of the measurements.
We write the model of the flux as \(F_{mod}\), our measured flux as \(F_{data}\), and the standard deviation as the usual \(\sigma\) (not to be confused with \(\sigma_i\)) . Then we can write the Gaussian line profile of the gases in the atmosphere as \(F_{mod}\):
where \(F_{min}\) is the smallest observed flux, \(\lambda\) is the observed wavelength, and \(\lambda_0\) is the actual spectral line that is being absorbed.
The model is generally dependent on a set of parameters - in our case, the parameters our dip is dependent of! We use the parameters we set for these to minimize \(\mathcal{X}^2\) or reduce it below a satisfactory error boundary.
To summarize them, these parameters are:
The depth, \(F_{min}\), which we measure between the interval \(F=[0.7,1]\), as previously stated. We've normalized the data so that \(F=1\) is the expected flux where no spectra lines are present.
The interval of the shift from the center of the dip, \(\Delta\lambda\), which is dependent on the Doppler effect, \(\Delta\lambda_{max}=\frac{v_{max}}{c}\lambda_0\), as previously stated.
The width, \(\sigma\), as a function of the temperature T, which we set between an interval \(T=[150K-450K]\).
This next part may be a little dry, but the math is not complicated, and we should understand why we can write \(\sigma\) as a function of the temperature!
We can relate the temperature to the spectral lines by deriving an expression for the standard deviation of the line profile, \(\sigma\), as a function of the gas temperature, \(T\), the mass of a particle in the gas, \(m\), and the central wavelength of the spectral line \(\lambda_0\).
Because of the Gaussian line, we can define width using \(\sigma\) or FWHM. The latter is given by:
where \(\sigma\) is the standard deviation of the line constant, \(\lambda_0\) is the central wavelength of the spectral line, \(c\) is the speed of light, \(k\) is a constant, \(T\) is the gas temperature, and \(m\) is the mass of a particle in the gas.
We write an algorithm that finds the best approximation of the flux for each of the gases. The three parameters we will vary are \(F_{min}\), \(\lambda_0\) and \(\sigma\). We test 20 different values for each, using the intervals defined above. As we are only interested in a handful of spectral lines, we approximate our flux-function over small intervals \([\lambda_0 - \Delta \lambda, \lambda_0 + \Delta \lambda]\). Every combination of those \(20 \cdot 20 \cdot 20 = 8000\) combinations (for every gas!) creates a smooth line, which we then subtract from the measured data. The line with the least difference from the actual data will be our Gaussian approximation.
One thing is still missing, though. The radial velocity of the molecules! (Could you tell?) Due to a misunderstanding between these basement dwelling rockets scientists, and the big guys at NASA, we didn't realize that radial velocity should have been a part of the \(\mathcal{X}^2\) minimization until it was too late. Running our approximation all over again would have been very time consuming, and we chose to press on. Our results will be inaccurate regardless, and we will have to make do with what we have for now.
We can now analyze this data in order to examine the presence or absence of common gases found in planetary atmospheres. In the table below are the gases we will be searching for, and we assume these are the only gases present.
Spectral lines of common gases found in planetary atmospheres:
Gas
Spectral lines
O2
632
690
760
H2O
720
820
940
CO2
1400
1600
-
CH4
1660
2200
-
CO
2340
-
-
N2O
2870
-
-
Each box does not necessarily have three spectral lines, therefore the blank boxes.
Stay tuned, next post we will get to the more exciting part of actually determining what our atmosphere consists of!
The \(\pm\) means that we have to take into account whether we traverse the orbit clockwise or counterclockwise. \( \mathbf{e}_{\theta}\) is a unit vector, but this unit vector doesn't point along the position vector \(\vec{r}\) like we are used to; it is perpendicular to the position vector! And finally, we have \(\mathbf{v}_0\), which is just our velocity at the time of performing the orbital injection.
One thing is still missing, though. The radial velocity of the molecules! (Could you tell?) Due to a misunderstanding between these basement dwelling rockets scientists, and the big guys at NASA, we didn't realize that radial velocity should have been a part of the \(\mathcal{X}^2\) minimization until it was too late. Running our approximation all over again would have been very time consuming, and we chose to press on. Our results will be inaccurate regardless, and we will have to make do with what we have for now.