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# Comparing a Galaxy to a Pencil

Astrometry Observing Outreach
9 replies to this topic

### #1 PollAirUs

PollAirUs

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Posted 21 October 2021 - 05:33 PM

Hello everyone!

Sometime ago I saw an astrophoto of a galaxy, and in the image description was written something like "this galaxy is X light years in diameter and it's Y light years from earth. Taking a picture of this galaxy is like taking a picture of a 10 centimeter pen Z miles away." And I'd like to know how to calculate this so I can do similar comparisons with other objects.

I've tried some formulas but nothing that I was sure about the result. I'm having trouble with the units of measure, like using kilometers for distance and the real size of the galaxy but baing forced to use arc minutes for the apparent size of it.

If anyone knows how to do the math, I'd be very greatful.

### #2 Asbytec

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Posted 21 October 2021 - 06:04 PM

http://www.ifa.hawai...parsec = 1 AU /

http://astro.physics...-angle-formula/

http://www.1728.org/angsize.htm

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### #3 jeffreym

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Posted 21 October 2021 - 06:15 PM

Maybe I'm thinking of this wrong, but isn't it just a simple ratio?

Needle Galaxy 191,000 LY across (or long due to edge-on view) at 39,000,000 LY away.

Pencil at 17.78cm long at some unknown distance away in cm.

(39,000,000 x 17.78) / 191,000 = 3,614cm

So, a pencil at 36 meters.

Also, you could convert the object into a unit of one and divide the distance by that.

So, 39,000,000/191,000= 204.19 units

Pencil of 17.78cm = 1 unit

17.78cm x 204.19 units away = 3,630 cm

So, a pencil at 36 meters.

Have fun!

Jeff

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Posted 21 October 2021 - 06:37 PM

As I understand your question, you're OK doing ratios of distances but don't know how to do it with arcminutes.

If the angular size is given in arcminutes, divide by 60 to convert to degrees, then divide the result by 57.3 to convert to radians. The tiny result that you get from that is -- to good approximation -- equal to the length of the pencil in cm, divided by the distance away in cm. (Use the same units top and bottom, in that fraction.) If the distance is an unknown, x, then you can calculate x in cm by simple algebra.

This is essentially the method given in the links in the first response; the only difference is that these links give the formula for arcseconds. (57.2958 * 60 * 60 = 206265)

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### #5 PollAirUs

PollAirUs

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Posted 21 October 2021 - 08:14 PM

Maybe I'm thinking of this wrong, but isn't it just a simple ratio?

Needle Galaxy 191,000 LY across (or long due to edge-on view) at 39,000,000 LY away.

Pencil at 17.78cm long at some unknown distance away in cm.

(39,000,000 x 17.78) / 191,000 = 3,614cm

So, a pencil at 36 meters.

Also, you could convert the object into a unit of one and divide the distance by that.

So, 39,000,000/191,000= 204.19 units

Pencil of 17.78cm = 1 unit

17.78cm x 204.19 units away = 3,630 cm

So, a pencil at 36 meters.

Have fun!

Jeff

If your math is right, then this is exaclty what i wanted! My only concern is that you're mixing light years with centimeters in the same expression.

### #6 Asbytec

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Posted 21 October 2021 - 11:41 PM

You could solve the small angle formula for arc seconds for both. The galaxy is given in arc seconds, so solve for a distance (not in light years) for the distance of the pencil where it subtends the given arc seconds.

It's really a similar triangle problem figuring the height with a set angle. The base defined by the galaxy has the same angle with a known height as the base of the pencil with an unknown height. Solve for the height of the triangle with the length of the base being the pencil. The angle will be the same as the galaxy.

### #7 jeffreym

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Posted 22 October 2021 - 07:58 AM

If your math is right, then this is exaclty what i wanted! My only concern is that you're mixing light years with centimeters in the same expression.

The math is correct.  That's why I did it two ways, to show the concept is correct.

The second method converted everything to a simple unit, which equaled one Needle Galaxy diameter.  Then calculated how many units away it is.   So, if unit equals PENCIL, PENCIL would be 204 "PENCILS" distant.

This is viewing things at a Galactic scale.  I think you could see a pencil at 36 meters.  However, at that scale your eye would have a lens many light-years in aperture. . .

### #8 jeffreym

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Posted 22 October 2021 - 09:22 AM

From our position in the universe, the Needle Galaxy has an apparent angular size of 16.8 x 2.9 arcmin.

Using simple trigonometry, you will arrive at the same answer.  A 17.78 cm pencil at 36 meters away will appear to have an angular size of 16.8 acrmin.

Jeff

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### #9 Delta608

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Posted 22 October 2021 - 11:48 AM

Wait a minute, with or without erasers..?

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### #10 jeffreym

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Posted 22 October 2021 - 03:55 PM

Wait a minute, with or without erasers..?

A new pencil out of the box is 7.5".  Sharpen it a couple times, with the full eraser, and I figured it would be down to about 7" (17.78 cm).  I had done my initial math in inches and feet and converted it.  I kept the two decimal places because in this example, each .01 cm equaled 107 LYs.  I thought that was a little much to just round it off.

Jeff

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