How Do You Calculate Observed Angular Velocity from a Reference Frame?

[Adorno]

Homework Statement

Determine the observed angular velocity from the origin of the reference frame for an object shown in the figure. The observed angular velocity is defined as the rate with which the observed direction on the object (measured in radians) changes in time.

Hint: The observed angular velocity is the ratio $\omega_0 = \Delta \varphi_0 / \Delta t$, where $\Delta t = t_2 - t_1$ and $\Delta \varphi_0 = \varphi_0(t_1^*) - \varphi_0(t_2^*)$ is the change of the angle $\varphi$, the polar angle which the star position had at times $t_1^*$ and $t_2^*$. These are the times when the light detected by the observer at $t_1$ and $t_2$ was emitted. The object is located at $(x_1, y_1)$ at time $t_1^*$ and it is located at $(x_2, y_2)$ at time $t_2^*$. However, $t_1$ and $t_2$,the observation times at the origin for these two events, are different from $t_1^*$ and $t_2^*$ because light takes a certain time to propagate from the object to the origin of the coordinate system.

Homework Equations

$\omega = \Delta \varphi_0 / \Delta t$
$\omega = \frac{|\vec{v}|sin(\theta)}{|\vec{r}|}$ (alternative formula for angluar velocity)

The Attempt at a Solution

Before I knew about the alternative formula for angular velocity, I tried solving this question like this:

$t_1 = d_1/c + t_1^*$, where $d_1 = \sqrt{x_1^2 + y_1^2}$ and
$t_2 = d_2/c + t_2^*$, where $d_2 = \sqrt{x_2^2 + y_2^2}$. Also $\varphi_0(t_1^*) = arctan(y_1/x_1)$ and $\varphi_0(t_2^*) = arctan(y_2/x_2)$, so that $\Delta \varphi_0 / \Delta t = \frac{arctan(y_1/x_1) - arctan(y_2/x_2)}{d_1/c + t_1^* - d_2/c - t_2^*}$.

I think that this is technically correct, but my lecturer said that this isn't what he wanted. He said that we should consider the angle $\Delta \varphi_0$ to be very small, and that this question could be solved using some basic geometry. I'm pretty bad at geometry so I can't see what I'm supposed to do, but I think it must involve the angle $\theta$ in the diagram and the alternative formula for angular velocity. Could anyone help?

 

[davidj89]

maybe expand out arctan (Taylor series) and see what happens? idk how that would make it prettier (only first approximation)

 

[Adorno]

Well, as I say, I don't think that what I did at first is what they wanted. I think I have to use the formula given in the question, $\omega_0 = \Delta \varphi_0 / \Delta t$, as well as the geometry of the diagram to get a new expression for the angular velocity. The question doesn't say what kind of expression I'm supposed to get. Now that I think about, it may be that they want me to derive the other formula for angular velocity, i.e. $\omega = \frac{|\vec{v}|sin(\theta)}{|\vec{r}|}$, using the geometry of the diagram. I'm not quite sure how to do this though.