Demonstrating Tilted, Straight Lines in Rotating Sphere Spectra

[mystery_witch]

Homework Statement

How do you show that in spectra taken along the equator of a rotating rigid sphere the spectral lines are tilted but straight?

Homework Equations

There were no equations given in the problem, so I guess you can use whatever equations you know.

The Attempt at a Solution

From the Doppler equation

$$\lambda$$=$$\lambda_{0}$$$$\frac{\sqrt{1+\frac{v}{c}}}{\sqrt{1-\frac{v}{c}}}$$

you can see that you get a different result depending on if the velocity is towards us or away from us. I think this is enough to show that the lines would be tilted, but I'm not sure.
This is as far as I've come. I don't have any idea how to do the rest.

 

[Jhero]

To show the spectral lines are tilted but straight, you can use the concept of rotational Doppler effect. The rotational Doppler effect states that the frequency of light emitted from a rotating object will be shifted depending on the direction of rotation relative to the observer. In this case, the rotating rigid sphere is the object and the observer is located along the equator.

First, consider the spectral lines emitted from the equator of the sphere as it rotates towards the observer. The velocity of the equator towards the observer will cause a blue shift in the spectral lines, resulting in a shorter wavelength. This will cause the spectral lines to appear tilted towards the shorter wavelength end.

Next, consider the spectral lines emitted from the equator of the sphere as it rotates away from the observer. The velocity of the equator away from the observer will cause a red shift in the spectral lines, resulting in a longer wavelength. This will cause the spectral lines to appear tilted towards the longer wavelength end.

Since the sphere is rotating at a constant rate, the velocity of the equator will change as it rotates towards and away from the observer. This will result in the spectral lines appearing tilted but straight, as they will be a combination of the blue and red shifts.

In summary, the rotational Doppler effect can be used to explain the tilt of the spectral lines observed along the equator of a rotating rigid sphere.