# Simple question, probably difficult answer: distance between Moon and object in sky?

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Hi all,

I'd like to use a spreadsheet to calculate the distance between a given DSO, and the Moon, in the sky. Sounds easy enough.

At first I thought this was simple: just do something like hyp squared=x squared+y squared, figure out what x is (whether RA1-RA2 or Az1-Az2), figure out what y is (whether Dec1-Dec2 or Alt1-Alt2), and you're done.

Which would work if this were a regular flat grid with cartesian coordinates. But it isn't, it involves polar coordinates for RA or azimuth. I think. I'm not a mathematician.

The more I think about this, the less I understand the problem or answer. I think that essentially the problem is calculating the direct line distance between two points on a globe. I've found some formulae that calculate an equivalent problem - between two points on earth, using lat and long - but they go around the surface of the globe, whereas I want to go direct. I think.

As for the answer, well I don't even really know what units this would be in! Degrees? Arcmin? It's sort of like asking how big is the sky.

So, are there any physics/maths geniuses who could offer some insight into this? Ideally with some sort of formulae that I could implement in Excel to help me with it? Or if someone could tell me this is a nonsensical notion to begin with, that would be fine, and it would help me to stop fretting about this.

Of course, I could just fire up Stellarium and look, but that would be too easy...

Thanks, Brendan

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After more thinking, this isn't about distances, it's about angles. The smaller the angle, the closer the objects. The largest angle would be 180 degrees, that is, opposite side in the sky.

So, if I know the difference between the alt for two objects, I have one angle. Same with the az, although I would need to start counting down when it goes past 180 degrees.

So given those two angles, would that be sufficient to derive the angle between the two objects? I have angle up, and angle across. Can I then get the 'diagonal' (for wanting of a better word, as I said, I'm no mathematician) angle?

I have a feeling this is either absurdly simple, or fiendishly difficult.

Stop me if I'm talking nonsense again.

Edited by BrendanC
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And this formula

cos(A) = sin(d1)sin(d2) + cos(d1)cos(d2)cos(ra1-ra2)

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Hey, thank you! It's all about the search terms I guess - didn't even know about angular distance. This could be exactly what I was looking for - and proves it wasn't quite as easy as I initially thought.

Ah, post-edit...

I don't understand how he's getting his sine cos etc values.

For example, he's saying that d1=-16.58.

Then he says sin (d1)=-0.285

If I do sin(-16.58) in Excel, I get 0.768.

This is the same across all the functions - very different results. There's clearly something very basic I'm doing wrong here. I've tried converting to radians etc, and cannot understand how he's obtaining those values for sins and cosines.

Any ideas?

Post-post edit...

The input to the sine etc functions needs to be in radians, so I just convert from degrees to radians. Then convert the answer back into degrees. All good.

Edited by BrendanC
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17 hours ago, BrendanC said:

I'd like to use a spreadsheet to calculate the distance between ...

3 hours ago, Gfamily said:

And this formula

cos(A) = sin(d1)sin(d2) + cos(d1)cos(d2)cos(ra1-ra2)

After more than a dozen years, an independent check to a formula I posted. Great!

The formula I posted and the formula at astronomycafe are equivalent, as shown by

cos(ra1 - ra2) = cos(ra1)cos(ra2) + sin(ra1)sin(ra2).

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Nice! I've managed to incorporate the formula successfully into my planning spreadsheet, so now I can quickly tell how far a proposed object is from the Moon on any given night.

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