Angular position (age of nebula)

[Kellytabares]

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.

 

* These are some relevant equations:

⋅Plate scale = 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

 

* Now, these are my calculations to my attempt to find the age:

-(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

Is there something I'm doing wrong in the calculations?

I hope my explanation makes sense.

 

Thank you in advance

[michael.h.f.wilkinson]

As your initial measurement shows a motion of just 0.5mm, measurement errors can have a profound impact on the result. Let us assume you can determine the position of pulsar and knot to within 0.2mm, this means the expected error (one standard deviation) is in the order of 0.28mm, assuming the errors are independent. Measuring multiple knots rather than a single one could improve matters, but obtaining a longer baseline in time would be even better. I also note that the calculation assumes a constant radial velocity. That might not be totally accurate (although it may well be a good first-order approximation)

[robin_astro]

The Crab nebula is a 3 dimensional object expanding in all directions. It is statistically unlikely that the knots you are measuring are all travelling outwards perpendicular to our line of sight and thus have maximum possible proper motion, which is what you are trying to estimate. (Consider for example a knot moving directly towards us. It will have no proper motion) This will inevitably lead to an overestimate of the  age of the nebula, though the size of the error you are finding is rather unexpectedly large, I suspect.

Robin 

[robin_astro]

A perhaps  easier way to estimate the age which should work for any feature  in the nebula and requires no scaling is to just measure the fractional change in distance of each of the knots from the origin, assumed to be the location of the pulsar. If the expansion is isotropic and the rate constant, the time between the two observations divided by the fractional change in distance from the origin should give you the age.  Even if the pulsar is no longer at the centre of the expansion, provided your measurement points are distributed around the nebula and average value should give you the answer.

Robin

[robin_astro]

Hmm....  using this technique for knot 1 gives

27/((105.9-105.4)/105.9) = 5720 yrs   

What values do you get for other knots ?

Robin

[robin_astro]

Are these the plates you are using?

https://dept.astro.lsa.umich.edu/ugactivities/Labs/crab/crab-full.html

If so I recommend rechecking your distances to the pulsar for your knots in the two images. I did a quick check of a couple and got a fractional expansion of ~0.03 which should give you ~3mm difference on a knot ~100mm from the pulsar (like knot 1)   This will give an approx age of 800 years 

Robin

[Tiki]

17 hours ago, Kellytabares said:

.....

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

If you rank these distances and plot the results sequentially you will be able to achieve a decent estimate of radial velocity. Plotting the smallest distance on the left all the way to the largest distance on the right and then draw a line of best fit which resembles a sine curve (0 to pi/2) .  ie. y-axis -> distance, x-axis ->  1 to 10 (or more). This curve should have an asymptote of the form x=constant. This constant is what you are after but you have to be careful with the units.

[Kellytabares]

Hi,

Yes, that is exactly the same plates I am using to try to calculate the age. And in fact, it is exactly the same lab work. - I agree that the distance of the knot's motion between the two images is considerably big. I am finding it a little difficult to follow your explanation above about rechecking my calculations of the distance of the knots to the pulsar. I forgot to mentioned that the plate scale for the reference stars A and B in the 1973 image is exactly the same in the 2000 image which is 1.74 "/mm as I mentioned above.

My distance for knot one, two, three and four in both plates, for instance is... (and recalculated the first knot and I got a lower decimal value for the 2000 image) :

*Knot1:

(1973) 105.4mm

(2000) 105.5mm

*Knot2:

(1973) 78.2mm

(2000) 78.5mm

*Knot3:

(1973) 87.6mm

(2000) 87.9mm

*Knot4:

(1973) 53.5mm

(2000) 53.7mm

My confusion comes from the distances of the knots and the conversions to arc seconds; so that the difference in arc seconds between the 2 images is then divided by the 27 years to obtain the arc seconds the knot is moving per year. So then, use the distance to the pulsar in 2000 and divide it by the arc seconds per year of the knot to find the age. As my assignment says "determine the total time from the given angular velocity and the angular distance to the pulsar". Am I correct with the calculation that I am using for the angular distance to the pulsar?

I am so sorry if my questions seem silly, but I am very confused since m answer keeps being erroneous.

Thanks a lot.

[robin_astro]

I know it sounds crazy but are you sure you have images for the two dates and not duplicates of the same image?  The reason I ask is you are only seeing very small differences between measurements on the two images whereas I see about 3mm difference for a knot at ~100mm distance from the pulsar in the two image I linked to.

BTW The conversion to arcsec is a red herring which unnecessarily complicates things. The effects of any conversion should cancel out.  All you need is the fractional change in distance from the pulsar on two the plates over the 27 years  ie if say a feature moved 10% further away in 27 years then 10x27 years ago the distance from the pulsar must  have been zero ie when the explosion occured. (The nice thing about doing this is makes no difference where the feature is in the nebula, what angle we are viewing it or what velocity it has, provided the material moved outward from the centre of the explosion at a constant velocity.

Robin

[Kellytabares]

21 hours ago, robin_astro said:

I know it sounds crazy but are you sure you have images for the two dates and not duplicates of the same image?  The reason I ask is you are only seeing very small differences between measurements on the two images whereas I see about 3mm difference for a knot at ~100mm distance from the pulsar in the two image I linked to.

It does sound crazy, but they both do have the date on the lower right part of the picture to avoid confusion. However, I am growing very suspicious about my knot distances in mm. I did a lot of research today about my equations and absolutely everything seems to be followed step by step and I'm still getting the wrong answers. When you said to re-check my distances for the knots in mm, I did, but still getting the tiny decimal error between images. Or, I am measuring something totally completely different. --- Stars are the bold black dots whilst the knots are the very blurry, hard-to-see, opaque dots. The latter is the one I'm measuring, since stars are stationary and don't move but only slightly on the plane of the sky.

What I am not entirely sure is if my conversion of the distance in mm of the pulsar to arc seconds is correct. Which is, the scale calculated in the second step times the distance of the pulsar in mm?

What is one of your knots value that you measured against mine for example?

Quote

BTW The conversion to arcsec is a red herring which unnecessarily complicates things. The effects of any conversion should cancel out.  All you need is the fractional change in distance from the pulsar on two the plates over the 27 years.

I know though, but I have a structure to follow which asks me to find everything I've mentioned on my above question. So, I have to go all through it and try the confusion not to win :/

[robin_astro]

What are you choosing as knots and where are you measuring the knot distances from?  Measured  from the pulsar, they should be a few mm different from 1973 to 2000, not a few tenths. See attached example 

 

[Kellytabares]

Hi,

I am requested to estimate my measurement with a ruler to the nearest 10th of the mm. That's why you see the decimals in my calculations starting from the pulsar.

I am doing this the old school way with a ruler, paper and pen instead of calculating the distance with pixels as you've done it which seems well easier. Ignore stars A and B distance. All that is wrong there. Both images' stars should be 219.9mm (in case you ask).

And please, excuse my messy work... that is just for calculations, I am submitting a clean and tidy one

[robin_astro]

OK I see the problem. What you have identified as knots are actually faint stars (in the foreground or background) which explains why you are seeing effectively no change. The knots are features in the filaments in the nebula. (They look a bit like whisps in smoke) They are not well defined and isolated like stars and are harder to measure to but the change is quite significant so you should be able to measure them using your ruler ok. Use the one i found as an example and then try to find similar features.

Cheers

Robin

[Kellytabares]

On February 24, 2016 at 20:33, robin_astro said:

OK I see the problem. What you have identified as knots are actually faint stars (in the foreground or background) which explains why you are seeing effectively no change. The knots are features in the filaments in the nebula.

Hi Robin,

Oh yes, it does make much more sense now and I can see that the whisps you mentioned have actually some kind of outward movement.

Thank you again, so much!!