Question

Calculate the size of the stellar aberration effect in Newtonian physics for
light coming straight down at speed c and a telescope moving straight
sideways at speed v. At what angle θ must we point the telescope to see
the star? Hint: In what direction is the light moving in the reference frame
of the telescope? Express θ in terms of v and c.

Answer 1

Let's choose x-axis along the telescope speed ${\vec v}$ and y-axis along the direction to the star (in accordance with the conditions of the problem angle between x-asix and y-axis is $\frac{\pi }{2}$). Then in the star's rest frame (we shall mark this values by apostrophe) the component of the velocity are $\vec u' = (0, - c)$ . Then in the rest frame of the telescope the velocity of the light will be

$$\vec u = \vec u' - \vec v = (-v, - c)$$

Thus the tangent of the angle between the y-axis and the direction to the star is

$$\tan (\frac{\pi }{2} - \theta ) = \frac{{{u_y}}}{{{u_x}}} = \frac{c}{v}$$

Thus

$$\tan (\theta ) = \frac{v}{c}$$