Measuring the Distance to an Exploded Star

[sapiental]

The gas cloud known as the Crab Nebula can be seen with
even a small telescope. It is the remnant of a supernova, a
cataclysmic explosion of a star. The explosion was seen
on the Earth on July 4, 1054 AD. The streamers in the
figure glow with the characteristic red color of heated
hydrogen gas. In the laboratory on earth, heated hydrogen
produces red light with frequency 4.568 × 1014 Hz; the red
light received from the streamers in the Crab Nebula
pointed at the Earth has a frequency of 4.586 × 1014 Hz.

a) Assume that the speed of the center of the nebula relative to the Earth is negligible.
Estimate the speed with which the outer edges of the Crab Nebula are expanding.
b) Assuming that the expansion has been constant since the supernova explosion, estimate the
average radius of the Crab Nebula in year 2006. (Immediately after the explosion, the size of the
Crab Nebula was neligible.) Give your answer in light years.
The angular size of the Crab Nebula, as seen from the earth, is about 4 arc minutes by 6 arc
minutes, for an average of 5 arc minutes. (1 arc minute = 1/60 of a degree)
c) Estimate the distance (in light years) to the Crab Nebula.
d) Estimate in what year BC the supernova explosion actually took place.
A light-year (ly) is the distance traveled by light in one year.
You may find it helpful to use the the first terms of the binomial expansion: (1+ε)^p ≅ 1 + pε

Hi, I'm not too sute how to approach this question but I think it has soething to do with the Doppler effect for light. If this is the case I would employ the following formula:

f_r = sqrt((c-v/c+v))(f_s)

where f_r is the frequency measured by the reciever and f_s the frequency of the source. What confuses me here is that both the frequencies are the same so I'm not sure how to set up the problem.

But, since they say to assume that the speed of the center of the nebula relative to the Earth is negligible, could I write the formula as follows?f_r = (1 + v/c)^(-1/2)

4.586 × 1014 Hz = (1 + v/(3.0 x 10^8))^-1/2

before I start solving for v, could you guys give me any hints on whether my setup is correct or not?

thank you !

 

[Astronuc]

In the laboratory on earth, heated hydrogen
produces red light with frequency 4.568 × 10¹⁴ Hz; the red
light received from the streamers in the Crab Nebula
pointed at the Earth has a frequency of 4.586 × 10¹⁴ Hz.

The frequencies are different.

One needs to know the distance as a function of reshift, or vice versa.

The angle subtended by an arc is given by $\theta$=s/d, where s is the arc length and d is the distance, and the angle is in radians. For small angles, the arc length is very nearly the linaer dimension (shortest distance between two points).

 

[kyle8921]

Hello,

Yes, your setup is correct. The formula you have used is the correct one for the Doppler effect in light. As you have mentioned, since the frequency of the light emitted by the streamers in the Crab Nebula is the same as the frequency of the light received on Earth, we can set the two frequencies equal to each other:

f_r = f_s

From the formula you have used, we can rearrange it to solve for v:

v = c * (1 - (f_r/f_s)^2)

Plugging in the values given in the problem, we get:

v = 3.0 x 10^8 m/s * (1 - (4.586 x 10^14/4.568 x 10^14)^2)

v = 3.0 x 10^8 m/s * (1 - 1.0000145)

v = 4.35 x 10^3 m/s

Now, to estimate the average radius of the Crab Nebula in 2006, we can use the formula for the expansion of a sphere:

r = v * t

Where r is the radius, v is the velocity of expansion, and t is the time. We can assume that the expansion has been constant since the supernova explosion in 1054 AD, so the time we are interested in is the time between 1054 AD and 2006 AD, which is 952 years.

Plugging in the values, we get:

r = 4.35 x 10^3 m/s * 952 years

r = 4.14 x 10^9 m

To convert this to light years, we divide by the speed of light:

r = 4.14 x 10^9 m / (3.0 x 10^8 m/s)

r = 13.8 light years

Therefore, the average radius of the Crab Nebula in 2006 was approximately 13.8 light years.

To estimate the distance to the Crab Nebula, we can use the formula for the distance traveled by light in a given time:

d = c * t

Where d is the distance, c is the speed of light, and t is the time. In this case, the time we are interested in is the time between the supernova explosion and the year 2006, which is 1054 + 2006 = 306