Relativistic Doppler Shift and a Star breaking up

TFM

Homework Statement

A distant star at rest with respect to an observer on Earth emits light of frequency 6.690 x 10^14 Hz. The star breaks up into two remnants of equal mass, which are observed to emit light of frequency 7.135 x 10^14 Hz and 4.282 x 10^14 Hz

Find the velocities of the remnant and the angle between line of sight and their direction of motion.

Homework Equations

$$v^{prime} = v \gamma [1 - \beta cos \theta]$$

$$v^{prime} = v \sqrt{\frac{1 + \beta}{1 - \beta}}$$

The Attempt at a Solution

I have solved the first part, finding the speed, using:

$$v^{prime} = v \sqrt{\frac{1 + \beta}{1 - \beta}}$$

assuming that since the star was initally at rest, teh centre of momentum will remain in the same polace, hence the two parts will fly away from each other.

I got $$\beta$$ to be 0.47, and the speed of the star remnants to be 0.235c

I am having a small problem with the second part, finding the angle. I am sure I need to use:

$$v^{prime} = v \gamma [1 - \beta cos \theta]$$

with the v primke being the orginal stars frequency. but I am nto sure which of the othe frequencies to use as v, and I am not sure how to get a value of beta to use in this part, since I am sure it will be different to the value in part A.

Any ideas will be greatly appreciated,

TFM

Answers and Replies

Mentor
I have solved the first part, finding the speed, using:

$$v^{prime} = v \sqrt{\frac{1 + \beta}{1 - \beta}}$$
That's the formula for the longitudinal Doppler shift when the source is approaching. It assumes an angle of 180 degrees, so it's not relevant here.

You can't solve the two parts separately. Hint: How are the speed and angle of the two remnants related?

TFM
Would you need to use the equation $$c = \lambda * frequency$$

so:

$$frequecy = c/ \lamda$$

so:

$$\frac{c}{\lambda}^{prime} = \frac{c}{\lambda} \gamma [1 - \beta cos \theta]$$

Or I am going in the worng direction?

TFM

Mentor
The equation to use is the first one you listed (I'll use f for frequency):

$$f' = f \gamma [1 - \beta cos \theta]$$

TFM
So:

$$f' = f \frac{[1 - \beta cos \theta]}{1 - \beta^2}$$

but we don't know beta or theta?

TFM

Mentor
but we don't know beta or theta?
Of course you don't--that's what you're trying to find! Set up two equations (using the two observed frequencies) and you can solve for the two unknowns.

Don't neglect to answer my question: How are the speed and angle of the two remnants related?

TFM
Would the relationship between speed and angle be:

$$v_{obseerved} = v*sin \theta$$

?

TFM

Mentor
How are the speed and angle of the two remnants related?
I meant this as two questions:
(1) How are the speeds of each remnant related?
(2) How are their angles related?

These should be easy to answer. Drawing a diagram will help.

TFM
Considereing coinservation of momentum, then the momentum before in the Earth Observer frame should be zero, since the star has zero speed. this means that the two stars remnants must have equal and opposite speeds, to keep the center of momentum frame the same. the angle between them must be 180 degrees (Pi)?

TFM

Mentor
Good! So how will the angles that you use in the Doppler formula relate to each other? How will their cosines relate to each other?

TFM
Would one of the remnants hav an angle 0f 0, thus the cos would be 1, whilst the other would have 180, with a cos of -1?

TFM

Mentor
Would one of the remnants hav an angle 0f 0, thus the cos would be 1, whilst the other would have 180, with a cos of -1?
No reason to think so. (But if one did have an angle of 0, then the other must certainly have an angle of 180.) Again, this is a more general case than the simple longitudinal Doppler shift.

Call the angle that one remnant makes $\theta$. (Draw that diagram!)

TFM
Is the daigram required?

TFM

Attachments

• Star Diagram.bmp
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Mentor
That's a good start. Now mark the angle that each remnant makes with the line of sight. (Assume for all practical purposes that both remnants are located just where the original star was.)

TFM
Would that be like this?

TFM

Attachments

• Star Diagram - Mk II.bmp
292.9 KB · Views: 467
Staff Emeritus
Science Advisor
Would that be like this?
Yes - so now find the velocity components of both stars with respect (parallel) to the line of sight, and relate them back to the velocities of the stars. Note that the star moving away is at the same angle $\theta$ as the star moving toward the observer.

TFM
This would be breaking down into components, so, for star Remnant A:

Horizontal speed would be:

$$v*cos(\pi - \theta)$$

Vertical speed would be:

$$v*sin(\pi - \theta)$$

and for Remnant B:

Horizontal Speed would be:

$$v*cos(\theta)$$

Vertical speed would be:

$$v*sin(\theta)$$

?

TFM

Mentor
Would that be like this?
Almost. The angle should be measured from the position vector that extends from Earth to the star.

In the general formula for Doppler shift you should recognize that $\theta = 0$ corresponds to the ordinary longitudinal formula when the source is receding.

There's no need to find parallel or perpendicular components of the velocity. The $\beta$ in the Doppler formula represents the full velocity, not a component. (Any needed component is taken care of by the $\cos\theta$ factor.)

Fix your diagram (marking the correct angle), then set up two equations, one for each remnant.

TFM
Should it be like this then, since I believe you measure angles clockwise

?

TFM

Attachments

• Star Diagram - Mk III.bmp
292.9 KB · Views: 459
TFM
opps, wrong diagram:

Sorry,

TFM

Attachments

• Star Diagram - Mk III.bmp
292.9 KB · Views: 485
Mentor
Should it be like this then, since I believe you measure angles clockwise
The issue is not clockwise versus counterclockwise, but that you should be measuring the angle from the forward direction (a vector pointing towards the star). An angle of zero means the source is moving directly away from earth.

TFM
That makes sense, so:

?

TFM

Attachments

• Star Diagram - Mk IV.bmp
292.9 KB · Views: 471
Mentor
Good. Now set up your two equations.

TFM
$$f' = f \gamma [1 - \beta cos \theta]$$

So, Equation for remnant A:

$$(6.690*10^{14}) = (4.282*10^{14}) \gamma [1 - \beta cos \theta]$$

For Remnant B:

$$(6.690*10^{14}) = (7.135*10^{14}) \gamma [1 - \beta cos (\pi - \theta)]$$

Are these the right equations?

TFM

Mentor
For some reason you have f and f' reversed. Other than that, those are the right equations. Simplify this expression: $cos (\pi - \theta)$.

TFM
Would that simplify down to $$sin(\theta)$$

TFM

Mentor
No. Review your trig identities. Even better, recognize vertical angles in your diagram.

TFM
Would it be $$sin{\pi + \theta}$$

TFM

Mentor
No. It's much simpler than that. Look up trig identities related to supplements.

TFM
Would it be:

$$-cos(\theta)$$

TFM

Mentor
Yes. That's the one you want.

TFM
So for Remnant A, the equation is:

$$(4.282*10^{14}) = (6.690*10^{14}) \gamma [1 - \beta cos \theta]$$

And for remnant B:

$$(7.135*10^{14}) = (6.690*10^{14}) \gamma [1 + \beta cos (\theta)]$$

TFM

Mentor
Good. Now combine and solve for $\cos\theta$ and $\beta$.

TFM
I've canceled them down to:

Remnant A:

$$\beta cos(\theta) = 1 - 0.648\gamma$$

Remnant B:

$$\beta cos (\theta) = 1.0665 \gamma - 1$$

Does this look right?

TFM

Mentor
There are many ways to combine these equations. (I didn't check your arithmetic.)

However, there seems to be a problem with the data. There doesn't seem to be a solution for the given numbers. Unless I'm making an error, I'd say there's a mistake in the supplied data.