Measuring Rotational Speed of Io w/Doppler & Relative Velocity to Jupiter

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I want to measure the rotational speed of Io using a spectrograph, using the doppler formula. Since it is rotating around Jupiter, i have to take the velocity relative to jupiter, but I wonder tough how this is possible when Io is not rotating at an right angle to my observation point. Is there some mathematical way to solve this?

 

[SpaceTiger]

I want to measure the rotational speed of Io using a spectrograph, using the doppler formula. Since it is rotating around Jupiter, i have to take the velocity relative to jupiter, but I wonder tough how this is possible when Io is not rotating at an right angle to my observation point. Is there some mathematical way to solve this?

You'll want to take the dot product of Io's velocity vector with your unit observation vector (a magnitude 1 vector pointing towards Io):

$$v_r = \vec{v}_{total} \cdot \hat{u}$$

This translates into red or blueshift in the usual way:

$$\frac{\Delta \lambda}{\lambda}=\frac{v_r}{c}$$

Note that you'll want it to be moving parallel to your observation vector if you want a large radial velocity.