So what can we measure our velocity compared to in space? It should be something we can observe pretty much wherever we are. Get out of town and look up at the sky on a clear night. Or cheat and look at the picture provided above.
What did you see?
Stars! Everywhere stars!! So many stars!!!
Inside these stars there are plenty of fusion reactions where hydrogen turns to helium and such. This leads to plenty of hydrogenatoms at the surface of a star. These absorb specific wavelengths of light sent out from the star, and this will apprear as dark lines when we observe the wavelength spectrum from said star. We know which wavelengths of light hydrogen absorbe, as these are characteristic to each element, and therefore we know where these lines should be!
However, when we measure them from a moving star, these wavelengths will have been blue- or redshifted. We can measure this difference in the observed dark likes and where we know they should be for hydrogen. Then we know the Doppler shift of a star!
You should recall that we've previously spoken about the Doppler effect on light, and that we had a look at how to find the radial velocity of the star by observing this change in the light waves:
\(\frac{\Delta\lambda}{\lambda_0}=\frac{v_r}{c}\)
Recall that \(\Delta\lambda\) is the Doppler shift in a spectral line, \(\lambda_0\) is the wavelength of the spectral line we are observing, in our case this is the \(H_\alpha=656.3\) nm spectral line, as seen from rest frame, \(c\) is the speed of light, and \(v_r\) is the radial velocity of the star we are observing.
Last time we assumed we were standing still, and we could find the radial velocity of the star we were looking at. This time, we assume that a star far away is standing still, and that it is instead measuring our (stars) radial velocity. We use this to determine the radial velocity of our star with respect to a reference star!
That's nice, but not what we are looking for at the moment. We want to know the space craft's velocity, so that we don't zoom past our destination planet at an incredible velocity, unable to ever stop! That would be super bad. Luckily, our space craft is equipped with instruments that can measure the doppler shift from our rocket at any point in time. This means that we can find the difference in the Doppler shift at our star and our space craft
\(\Delta_{tot}=\Delta\lambda_*-\Delta\lambda_s\)
where \(\Delta\lambda_*\) is the Doppler shift measured from our star (assumed to not be moving), and \(\Delta\lambda_s\) is the Doppler shift measured by our rocket. Thus, we can find our radial velocity in relation to our own star:
\(v_{r,s}=\frac{\Delta\lambda_*-\Delta\lambda_s}{\lambda_0}c\)
We are pretty close to having everything we need!
If our software for determining our rotational orientation works as it should, our space craft is 
able to point its equipment towards any known star. From there, it can measure the Doppler shift \(\Delta\lambda\) in the \(H_\alpha\) spectral line. We've had an excellent team of astronomers to punch some numbers and scratch their heads for a bit, and they have determined the best reference stars for us. These reference stars are the ones we will use when we find the radial velocity of our space craft, and they are at spherical coordinates \(\phi_1\) and \(\phi_2\).
When we use these coordinates we get two vectors for the velocity in the \((\phi_1,\phi_2)\)-plane. We can represent each 'direction' using unit vectors \(\hat{x}\) and \(\hat{y}\), where \(\hat{x} = (1,0)\) and \(\hat{y} = (0,1)\). We can use these to write a vector, like this: \(\vec{d}=d_x\hat{x}+d_y\hat{y}\), where many write the vector \(\vec{d}\) simply as \(\vec{d}=(d_x,d_y)\). (If you do the vector sum, you will notice that they are the same!) Choosing x and y as directions is common, but not always necessary or preferable. So long as we have two unit vectors (two dimensions!) in the same plane we have what we need.
We want to find dx and dy, as these can be written as the velocity components of our space craft. We can do this by sort of going the opposite way as what we did in the photo above.
NOTE! This does involve matrices and inverse matrices, and is not a neccesary read for following what happens next. We're putting it here for fun, and you can have look at it if you want to, and if you follow it closely you may see that it isn't as difficult as it looks at first :)
First we write \(\vec{d}\) on matrix form:
We can then multiply the inverse A-matrix on both sides og the equation:
We have a rule for inverting 2x2-matrixes:
Meaning we can write our inverse matrix as
Giving
This is the expression we use in order to transform our velocities so they are given in the (x,y) coordinate system.
As we said we need two stars at different coordinates to know both the x and y components of our spacecraft's velocity with respect to our star. Therefore we find the radial velocity using the total Doppler shift measured from the star at \(\phi_1\) and at \(\phi_2\). Inserting this into the equation we found in order to find the x- and y-components, we get our space craft's velocity in the x-and y- directions!
At this point, our software lets us know which angle we are looking at through our camera, and we can therefore find our two reference stars and measure their Doppler shift. Further, it lets us use this in order to find our velocity! Very practical stuff.
Stay tuned, in the next post we will finalize our software so that we actually know where on Earth in space we are!