Crab Nebula Age Estimation - Angular Position Problem

[JoAstro]

Homework Statement

Hi,

I am trying to use the angular velocity and angular position to find the age of the Crab Nebula but my calculations seem to be erroneous. I am supposed to get a number around 800 but I keep getting 5000+ years of age. I'm going to try to be very specific in what I'm doing to see if there is something in my procedure that is wrong and I'm not noticing.

1) I am given two images from different years with 27 years of difference - 1973 and 2000. I have two reference stars in each image (A and B) which both are 385 arc seconds apart. I have to calculate the 'plate scale' for these two stars by measuring the distance of these stars in mm. and multiplying that distance by the arc seconds given.

2) Measure the distance in mm. to the pulsar from each image.

3) Measure the distance in mm. of at least 10 knots from the pulsar in the both images (1973 and 2000)

4) Convert the distance of the knots in mm. to arc seconds. This is done by multiplying the knot's distance by the plate scale (for both images too).

5) Calculate difference of arc seconds each knot is moving between the two images to get the change in velocity between the 2 years. This is the proper motion.

6) Divide the value from 'step 5' by 27 years which is the time between the two images. This should give me the arc seconds per year in which each knot is moving which is called the angular velocity or "μ".

7) Now, I need the angular position from the image 2000 from the pulsar (this is what I am struggling with).

8) (extra)... I asked if the angular position was measured in mm. and I was told it should be in arc seconds. But trying to calculate the age with mm first and then with arc seconds, it still gives me a wrong result.

Homework Equations

⋅Plate scale[/B] = 385 arc seconds × Distance in mm.
⋅Arc seconds conversion = knot distance in mm. × plate scale
⋅Change in velocity or Proper motion = knot arc second from 1973 × knot arc second from 2000
⋅Angular velocity = μ=Δ_x / Δ_t
⋅Age of the Nebula = angular position to the pulsar in 2000 image / angular velocity

The Attempt at a Solution

(plate scale)
385 arc seconds / 220.3mm = 1.74 "/mm

(distance of one of the knots in mm and conversion to arc seconds)
Knot 1 (from 1973 image)
105.4mm × 1.74 = 183.39 arc seconds

Knot 1 (from 2000 image)
105.9mm × 1.74 = 184.26 arc seconds

(proper motion or the difference of arc seconds of the knot between 1973 and 2000)
183.39 arc seconds - 184.26 arc seconds = 0.87

(angular velocity "μ" for the knot)
0.87 / 27 = 0.032 arc seconds per year

-Here 27 years corresponds to Δ_t which is the time between the two images.-

(age of Nebula) - The mm. value is the distance to the pulsar.
1. Attempt with angular position in mm:
153.2mm / 0.032 = 4787.5 years

2. Attempt with angular position in arc seconds:
153.2mm × 1.74 = 266.56 arc seconds

266.56 / 0.032 = 8330 years

Both results are wrong but I can't spot where my mistake is. I can't understand in full the last step for the angular procedure but it might also be something else.

I hope my explanation makes sense.

Thank you in advance

 

[gneill]

Homework Statement

Hi,

I am trying to use the angular velocity and angular position to find the age of the Crab Nebula but my calculations seem to be erroneous. I am supposed to get a number around 800 but I keep getting 5000+ years of age. I'm going to try to be very specific in what I'm doing to see if there is something in my procedure that is wrong and I'm not noticing.

1) I am given two images from different years with 27 years of difference - 1973 and 2000. I have two reference stars in each image (A and B) which both are 385 arc seconds apart. I have to calculate the 'plate scale' for these two stars by measuring the distance of these stars in mm. and multiplying that distance by the arc seconds given.

I think you mean you need to calculate a plate scale for each image. You have two images taken years apart and probably with different equipment, so you'll want to calculate a plate scale for each one to relate linear measurements on the image to angular separations. You should be dividing the given angular distance by the measured linear distance, not multiplying.

2) Measure the distance in mm. to the pulsar from each image.

Measure the distance to the pulsar from what?

3) Measure the distance in mm. of at least 10 knots from the pulsar in the both images (1973 and 2000)

4) Convert the distance of the knots in mm. to arc seconds. This is done by multiplying the knot's distance by the plate scale (for both images too).

5) Calculate difference of arc seconds each knot is moving between the two images to get the change in velocity between the 2 years. This is the proper motion.

6) Divide the value from 'step 5' by 27 years which is the time between the two images. This should give me the arc seconds per year in which each knot is moving which is called the angular velocity or "μ".

7) Now, I need the angular position from the image 2000 from the pulsar (this is what I am struggling with).

Angular position of what?

8) (extra)... I asked if the angular position was measured in mm. and I was told it should be in arc seconds. But trying to calculate the age with mm first and then with arc seconds, it still gives me a wrong result.

If your velocity is an angular velocity then you should be working with angles. Otherwise the units won't make sense.

Did you use two plate scales, one for each image? How accurate were your linear measurements? It looks like you're working with pretty small differences: 105.4 mm on one plate and 105.9 mm on the other.

 

[JoAstro]

Hi gnaill,

Sorry for the slow answer.

First, I am sorry about my poor explanation above. It was very late that night and my frustration levels were (are) very high. To answer your questions about my confusion for this lab work... You are totally right, to calculate the plate scale I take the arc seconds between the two stars ( A and B ) divided by its distance in mm. For some reason I wrote the wrong thing even though I had already done it correctly. Also, the distance of these two reference stars is the same on both images, meaning that my plate scale is exactly the same for the two years.

Then, the pulsar is measured in mm from (literally) the lower left part of the picture. Now, what I said angular position, I meant to say angular distance to the pulsar. That should be the distance in mm of the pulsar form the lower left times the plate scale to make the conversion to arc seconds and have the angular value. Even though I have done this, I am not entirely sure if it is correctly done.

I have re-checked my procedure several times and everything seems to be followed step by step. What I come to conclude is my distance measurements to the knots in mm. Are they correct with only such a small error between the two years? I read a piece of work for knot measurements online with different scale dates, and their knots distance error were considerably large. For one knot they had (first year)= 223.60mm; and (other year)= 237.27mm.

Some of my knots for instance are:
(1973) 78.2mm
(2000) 78.5mm

(1973) 64.3mm
(2000) 64.8mm

(1973) 53.5mm
(2000) 53.7mm

By the way, I am asked to estimate them to the nearest 10th of mm and not 100th as in the online work I found.

Thanks a lot!

 

[gneill]

The online data that you found indicates about a 6% change in length over whatever time period for those plates. Yours represents only about 0.4% change for the timespan of your plates. What are the actual timespans for each? Could it be that the plates you are using are actually taken much closer together in time than you think? How are the dates identified?

 

[JoAstro]

Indeed my images have a shorter lifespan with only 27 years apart and this number is the one I'm told to use to find the proper motion of the knots. The other online source was using scales 52 years apart.

But since my plates are both dated (1973 and 2000) I have to stick to the 27 year timespan and work out the age with this value. But my printed plates keep giving me small measurement errors everywhere I go around the nebula. And since one is clearer and has more detail than the other one, I doubt they are replicated.

These are the plates I'm using (find them linked on the 'https://dept.astro.lsa.umich.edu/ugactivities/Labs/crab/crab1973.gif -https://dept.astro.lsa.umich.edu/ugactivities/Labs/crab/crab2000.gif')

 

[gneill]

There's certainly a contrast difference in the plates, perhaps related to differences in exposure times. I can see that the plate scales are indeed essentially identical.

How did you establish your reference point for taking measurements? When you measure the position of a knot, where's it measured from?