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3.9 The Speed of Light
Imagine that light did not have a constant speed, but behaved in the manner expected from experience. Namely, if the source of the light is rushing toward you, the light will approach you faster; if the source is rushing away from you, the light will approach you slower. This is incorrect, of course, but it’s worth investigating the consequences of a non-constant speed of light because the failure to observe those consequences is evidence that the speed of light is constant. With that backdrop, consider a binary star system situated a very large distance $L$ from Earth. Let the angular velocity of the smaller star be $\omega$, as it orbits the larger star in a circle of radius $r$. We want to find the value of $\omega$ for which the light emitted by the smaller star, when it’s traveling directly away from Earth, arrives at Earth at the same moment as light emitted a little later, when the star is traveling directly toward Earth. Let’s work this out in stages:
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Recall from basic physics that if the angular velocity of the smaller star is $\omega$ then its linear speed $v$ is given by $v = \omega r$. In terms of $v$, what is the time $t_1$ that it will take light emitted by the smaller star to reach Earth, if that light is emitted when the smaller star's orbit has it moving directly away from Earth? (Again, assume—incorrectly—that light behaves as you would expect from everyday experience, with its base speed $c$ being increased or decreased by the motion of the source.)CorrectIncorrect
In terms of $v$, what is the time $t_2$ that it will take light emitted by the smaller star to reach Earth if that light is emitted when the smaller star's orbit has it moving directly toward Earth? (Again, assume—incorrectly—that light behaves as you would expect from everyday experience, with its base speed $c$ being increased or decreased by the motion of the source.)CorrectIncorrect
In terms of $v$, what is the time $t_3$ that it takes the smaller star to move from the position relevant to part (A) to the position relevant to part (B)—that is, how long does it take the star to complete one half of a full orbit around the larger star?CorrectIncorrect
What value of $\omega = v/r$ will result in the light emitted when the smaller star is traveling directly away from Earth reaching us at the same moment as the light emitted later, when the smaller star's orbit has it moving directly toward earth?CorrectIncorrect