Co-ordinate systems?

[granthutchison]

Here are my raw data, if it's any assistance. My coords for J2000 agree with yours, Consty. I then precess to B1950, and derive my galactic coords from the B1950 coords.

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         alpha2000    delta2000  alpha1950    delta1950     l       b
Proxima 217.429167   -62.679444  216.466238  -62.456946  313.940  -1.927
Sirius  101.287083   -16.716111  100.728495  -16.662952  227.230  -8.890
UV Ceti  24.755529   -17.950278   24.152624  -18.203709  175.486 -75.699
Del Pav 302.181667   -66.181944  301.002060  -66.327793  329.767 -32.417


Grant

[Evil Dr Ganymede]

Well, that's all reassuring, kinda.

The RAs and Decs were from the RECONS website at http://www.chara.gsu.edu/RECONS/TOP100.htm , and that indicates that the epoch of the RA/Dec values is 2000 (in the labels at the top of the RA/Dec columns)

I found what I think are the RA/Dec co-ordinates of the NGP in epoch 2000.0 at http://www.projectrho.com/smap04.html

(and from http://www.geocities.com/alt_cosmos/galactic.pdf , I think he must have got the numbers from the same place)

Galactic center: RA = 17h45.6m; Dec = -28 56.3'
Galactic north pole: RA = 12h51.4m; Dec = +27 07.7'

So, the alpha and delta values for the NGP here are 192.85 and 27.12833... I replace the 192.25 and 27.4 values in the conversion equation with those, then I get the NGP being at b = 90 and l = 123 (which makes sense). For RA=0 and Dec=0, I get b = -60.192 and l = 96.43.

But for Sirius, I get b = =8.89 and l = 227.30 , which is closer but not quite the same as what Grant got (he got l = 227.23). Hang on. I'll post my versions of Grant's numbers in the next post.

[Evil Dr Ganymede]

So, Grant got these numbers:

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         alpha2000    delta2000  l       b
Proxima 217.429167   -62.679444  313.940  -1.927
Sirius  101.287083   -16.716111  227.230  -8.890
UV Ceti  24.755529   -17.950278   175.486 -75.699
Del Pav 302.181667   -66.181944  329.767 -32.417


If I put the 2000.0 NGP coordinates directly into the conversion equation, I now get:

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         alpha2000    delta2000        l             b
Proxima 217.4291667   -62.67944444   314.0120931 -1.928834853
Sirius  101.2870833   -16.71611111   227.3023151 -8.8821873
UV Ceti 24.75541667   -17.95027778   175.5772101 -75.69291785
Del Pav 302.1816667   -66.18194444   329.8351384 -32.42034188


Which is very close.

I'm still losing something though. Grant, how are you precessing the equations back to 1950.0 before you convert them? Are you doing it the same way I am?

[granthutchison]

But for Sirius, I get b = =8.89 and l = 227.30 , which is closer but not quite the same as what Grant got (he got l = 227.23).

The problem is in the assumption that the longitude of the ascending node of the galactic equator on the celestial equator remains unchanged at l=33 under precession - this could only happen by some wild coincidence. In fact, the J2000.0 coordinates of the north galactic pole and core that you posted (for which, thanks :)) indicate that this changes to l=32.93 for J2000.0 - a difference of 0.07 which exactly accounts for the residual error in your Sirius coords.

So that's nice ... now we can go straight from J2000.0 to galactic without the hassle of precessing back to B1950.

Grant, how are you precessing the equations back to 1950.0 before you convert them? Are you doing it the same way I am?

I confess I haven't quite gathered how you're doing it. I thought you were just trying to come up with enough data to make the shift straight from J2000.0 to galactic without having to pass through the hassle of doing the precession calcs on your star coords. I'm using another chunk of trig to make the conversion of all my J2000.0 coords to B1950.0 within the spreadsheet, and then taking galactic coords from there, using the equation I originally posted.
I'll happily post the precession equations if you like, but it looks like we've maybe solved your problem without needing that extra hassle :).

Grant

[Evil Dr Ganymede]

Well, what I'm did to correct my numbers was:

1) Used Spiff's ATAN2 method instead of th ATAN method I used initially.

2) Corrected the conversion from Dec to delta for -ve values

3) Took your equations:

b = asin[cos(delta)*cos(27.4)*cos(alpha-192.25)+sin(delta)*sin(27.4)]

l = 33 + atan{[sin(delta)-sin(b)*sin(27.4)]/[cos(delta)*sin(alpha-192.25)*cos(27.4)]}

And replaced 27.4 with 27.128333 and 192.25 with 192.85.

Using 32.93 instead of 33 makes it a bit more accurate. The numbers still seem to be off by a few hundredth of a degree though, and I'm not sure why. Rounding errors, maybe?

I'd still be interested to see your precession equations though, if they're not too complicated...

[Spaceman Spiff]

I was doing some more investigation today, but the above two posts arrived before I finished composing this one. Still, here are the numbers I got with my knocked-up Excel spreadsheet are (with NGP 2000.0 as provided in Dr. G's post of Wed Mar 03, 2004 11:56 pm: R.A. = 12h51.4m, Dec. = +27° 07.7', and also l = 33°):

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Star           RA 2000.0      Dec 2000.0   Alpha 2000.0  Delta 2000.0  Lambda 2000.0  Beta 2000.0
Proxima:       14h 29m 43s    -62 40' 46"  217.4291667   -62.6794444   314.0120931     -1.92883518
Sirius:        06h 45m 08.9s  -16 42' 58"  101.2870833   -16.7161111   227.3023151     -8.88218738
UV Ceti:       01h 39m 01.3s  -17 57' 01"   24.7554167   -17.9502778   175.5772091    -75.69291764 
Delta Pavonis: 20h 08m 43.6s  -66 10' 55"  302.1811667   -66.1819444   329.8351385    -32.42034218


These numbers agree extremely well with Dr. Ganymede's, but not so well with Grant's. It means Dr. G's equations are now correct, but the input data may be inaccurate.

There's no need to bother with 1950.0 if the star and galactic data are for 2000.0. The only reason we got into precession from 1950.0 was because the equations in Grant's post of Mon Mar 01, 2004 7:10 pm were using 1950.0 galactic data.

By the way, note that Grant's R.A. for UV Ceti is 24.755529°, different from Dr. G's at 24.75541667°, so there's a duff input somewhere...

Otherwise, the funny thing about my differences from Dr. G's numbers is that I made the Excel spreadsheet again earlier today under Win NT 4.0 with Excel 97 and all the numbers were exactly the same as Dr. G's. Now I return to the Excel 2000 spreadsheet I made last night, and I get slight differences (non-identical numbers are Proxima Dec., UV Ceti R.A., UV Ceti Dec., and Delta Pav. Dec.). Which Excel do you use Dr. G? We may have a slight change in the ATAN2() algorithm, or treatment of rounding errors, between Excel's 97 and 2000.

However, I think Grant has the more accurate data. One source of lost accuracy is that that 33° (galactic longtitude of the ascending node of the galactic plane on the celestial equator) was too rounded for 1950.0, and must even change for 2000.0. Also, maybe even the NGP co-ords for 2000 at http://www.projectrho.com/smap04.html are still not as precise as what Grant is using. I tried tweaking NGP Alpha, Delta and that 33° around the final significant digits, and I find I can get close to Grant's numbers - but an iteration with three inputs is too complicated to succeed quickly. Whatever, those figures I gave in my post of Wed Mar 03, 2004 7:11 pm under "Near Star weirdness" (http://www.shatters.net/forum/viewtopic.php?t=4538) are definitely wrong, so please ignore them.

Note also those R.A. and Dec. co-ordinates of the NGP in epoch 2000.0 at http://www.projectrho.com/smap04.html give the NGP and the Galactic Centre, but you'll need that l = 33° figure instead, and it should be precisely computed. However, I used my spreadsheet to trim l until the 2000.0 R.A. and Dec. for the galactic centre was Lambda = 0° and Beta = 0° precisely: I found l = 32.931718°.

In summary:
1. Dr. G's equations appear to work correctly, just stick consistently to 2000.0 data.
2. There may be slight numerical errors in Excel 97 (?).
3. You need to ensure accuracy of NGP co-ords, GC co-ords and/or l. If you want arcmin accuracy for Lambda and Beta you'll need more than arcmin accuracy for the inputs.
4. Someone's UV Ceti R.A. is slightly wrong - I think it's Dr. G's.

That's it.

Spiff.

[Evil Dr Ganymede]

Thanks, Spiff. So basically, you reckon my numbers should be OK?

I'm using Excel 97 SR2 on Win2K.

What's Lambda and Beta, BTW? Is that what grant referred to as l and b - ie the galactic co-ordinates?

My UV Ceti RA = 01 39 01.3, Dec = -17 57 01. I got that from the RECONS list at http://www.chara.gsu.edu/RECONS/TOP100.htm

I found some 1950.0 epoch data for it on the Internet Stellar Database, which claims its RA = 1h36m25s and Dec = -18°12'42" - maybe that's the problem? Could Grant be using the old 1950.0 epoch data?

[selden]

l2, b2 of (0,0) = RA,Dec of (17H 45m 37.20s, -28d 56m 10.22s) = (266.404996, -28.936172) decimal degrees in the J2000 Equatorial coordinate system.

If it isn't, then the Chandra X-Ray telescope has been pointing in the wrong directions.

http://cxc.harvard.edu/toolkit/precess.jsp

And according to Simbad, the ICRS 2000.0 coordinates of UV Ceti are
01 39 01.5 -17 57 04 D at Galactic coordinates 175.49 -75.70

And it has a rather large proper motion (~3.3 arc seconds/year), so you have to be careful about the date of the position value that you're using.

[granthutchison]

By the way, note that Grant's R.A. for UV Ceti is 24.755529°, different from Dr. G's at 24.75541667°, so there's a duff input somewhere...

My conversion was in error, interestingly enough, since exactly the same spreadsheet formulae were used for all the alphas I quoted. My RA agrees exactly with Dr. G's, as I said, but conversion to alpha was wrong for UV Ceti - somewhere in its travels through many version of Excel, the spreadsheet has suddenly started ignoring the decimal seconds in some instances, but not others. Gah.
UV Ceti should have read

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         alpha2000    delta2000  alpha1950    delta1950     l       b
UV Ceti  24.755417   -17.950278   24.152512  -18.203709  175.485 -75.700

The others are unchanged.

One source of lost accuracy is that that 33° (galactic longtitude of the ascending node of the galactic plane on the celestial equator) was too rounded for 1950.0...

No, these values were defined in B1950.0 coordinates, and are therefore precise - the problem is that the galactic pole data precessed to J2000.0 are quoted to only a few decimals, and so can't match the infinite precision of the nice round B1950.0 coordinates - longitudes to 0.1 minutes of right ascension are accurate to only 0.025 degrees, which I'm guessing accounts for Dr G's residual problems in the hundredths of a degree.

What's Lambda and Beta, BTW? Is that what grant referred to as l and b - ie the galactic co-ordinates?

Lambda and beta are the generally accepted symbols for ecliptic longitude and latitude. I fear Spiff may have mistyped, but I hesitate to disagree with a Calvin and Hobbes fan.

Grant

[Evil Dr Ganymede]

l2, b2 of (0,0) = RA,Dec of (17H 45m 37.20s, -28d 56m 10.22s) = (266.404996, -28.936172) decimal degrees in the J2000 Equatorial coordinate system.

Now why didn't I think of doing that...!

OK. Let's see how this goes: I convert l=0, b=90 (the NGP) from galactic to J2000.0 Equatorial in that precess program, and I get that to be:

RA: 12 51 26.28
DEC: +27 07 41.70

Alpha: 192.859481
Delta: 27.128251

So I plug those into the conversion equations instead of 192.85 and 27.128333 (and use the 32.93 angle for the ascending node).... and my results finally come out right! Woohoo! They're the same as what precess outputs and what Grant's values were. Plus putting in the NGP and 0,0 RA/Decs into the equations do give b,l that are close as dammit (ie to many decimal places) to 0,90 and 0,0.

So the only niggling thing that I wonder about now is that the ascending note angle might not be totally right - how do you figure that out?

Here's what I'm plugging into Excel, BTW - this should convert J2000.0 RA/DEC (or their Alpha/Delta values) directly into galactic latitude and longitude:

Galactic latitude in radians:

l = =ASIN(COS(RADIANS(F3))*COS(RADIANS(27.128251))*COS(RADIANS(E3)-RADIANS(192.859481))+SIN(RADIANS(F3))*SIN(RADIANS(27.128251)))


Galactic Longitude in radians:

b = RADIANS(32.93) + ATAN2((COS(RADIANS(F3))*SIN(RADIANS(E3)-RADIANS(192.859481))*COS(RADIANS(27.128251))),(SIN(RADIANS(F3))-SIN(I2)*SIN(RADIANS(27.128251))))

where F3 is Delta in degrees, E3 is Alpha in degrees, and I2 is the Galactic Latitude (in radians).

To convert those to degrees, I just wrap each of those equations in the DEGREES() function.

Oh, and of course, I am once again in debt and remain eternally grateful to the great minds of this board who keep helping me out with all my awkward questions. Thanks very much to everyone who assisted here!

[granthutchison]

So the only niggling thing that I wonder about now is that the ascending node angle might not be totally right - how do you figure that out?

The longitude of the equatorial ascending node is equal to the longitude of the north pole + 90 degrees (that's a general rule ... if you draw a picture you should be able to convince yourself). Latitude of the ascending node is by definition zero (since it's situated on the equator of the relevant coordinate system). So from your figures, the coords of the galactic ascending node are alpha = 282.859481, delta = 0. We just need to calculate the angular distance between that point and the galactic core (alpha = 266.404996, delta = -28.936172; thanks, Selden), which marks the zero meridian of the galactic coordinate system. The answer is 32.931918 - which is the angular distance along the galactic equator between the galactic zero meridian and the ascending node of the galactic equator on the celestial equator (and a match to Spiff's iterated figure until the fourth decimal ).

Grant

[Evil Dr Ganymede]

Excellent. I'm now accurate to within a millionth of a degree - that should be more than enough

Thanks very much

[Guest]

Greetings peeps,

right, where were we?

Lambda and beta are the generally accepted symbols for ecliptic longitude and latitude. I fear Spiff may have mistyped, but I hesitate to disagree with a Calvin and Hobbes fan.

Aawwgghh! It was Hobbes! He made me do it!

Yes, you appear to be correct, Grant: turning the page in Duffett-Smith shows that Lambda and Beta are indeed used for the ecliptic longitude and latitude.

No, these values were defined in B1950.0 coordinates, and are therefore precise

I didn't know that the 1950.0 conversion actually defined the galactic co-ordinate system. Thanks for the tip, Grant! I wondered why Duffet-Smith had such a round set of figures for epoch 1950.0, especially "33°". Incidently, that implies that the true galactic centre Sgr A* isn't precisely at l = 0°, b = 0°, doesn't it?

I have to admit I must have become rusty to not have remembered Lambda and Beta versus l and b, but I also find trying to focus on details, explain everything as clearly and precisely as possible, get the phpBB formatting right, and keep up to date with ever updating and overlapping posts makes me very error prone - monitoring several windows and repeatedly previewing posts to find yet another mistake... sigh. I'll have to find a more efficient way to do this, and how you get quotes with credits, formatting and emoticons to work properly...

Still, I must be more diligent in future posts (sets jaw squarely and tries to look heroically resolute...).

This topic has been a fascinating exercise in numerical astronomy - both teaching and learning wise. Do you think after that convoluted discussion, someone should summarise the thread:
1. what equations to use to convert from ecliptic to galactic co-ordinates,
2. what the right parameters are for epoch 1950.0 and 2000.0,
3. that the galactic co-ordinates are not precisely aligned to Sgr A*, to stop someone posting: "Hey! Sgr A* isn't precisely in the centre of the galaxy!!!" later on.

Oh, did I just volunteer?

Spiff.

[Spaceman Spiff]

... oh yes, and all this editing and browsing makes me forget to log in too.

[granthutchison]

IN SUMMARY
To convert from equatorial to galactic coordinates:

Convert RA and declination to decimal degrees: RA becomes equatorial longitude alpha, declination becomes equatorial latitude delta.

For B1950.0 coordinates, the galactic latitude b and longitude l are given by:

b = asin[cos(delta)*cos(27.4)*cos(alpha-192.25)+sin(delta)*sin(27.4)]

l = 33 + atan{[sin(delta)-sin(b)*sin(27.4)]/[cos(delta)*sin(alpha-192.25)*cos(27.4)]}


For J2000.0 coordinates, the galactic latitude b and longitude l are given by:

b = asin[cos(delta)*cos(27.128251)*cos(alpha-192.859481)+sin(delta)*sin(27.128251)]

l = 32.931918 + atan{[sin(delta)-sin(b)*sin(27.128251)]/[cos(delta)*sin(alpha-192.859481)*cos(27.128251)]}


Grant

[granthutchison]

Incidently, that implies that the true galactic centre Sgr A* isn't precisely at l = 0°, b = 0°, doesn't it?

Sgr A* is said to be at J2000.0 17h45m40.045s -29?00'27.9", which coverts to l=359.944, b=-0.046.

Grant

[Evil Dr Ganymede]

Incidentally, I notice that the Hipparcos catalog uses RA/Dec from the J1991.25 Epoch. I don't think this is very much different from the J2000.00 epoch though, so I should be safe using the J2000 equations to convert from the Ra/Decs given there to galactic co-ordinates, right?

[selden]

We had a discussion about this a couple of months ago here on the Forum. As I recall, the conclusion was that the coordinates provided actually corespond to J2000 and no precession is needed.

I'm not sure if it was in this thread or a similar one where you and others mentioned the desirability of a Galactic coordinate grid in Celestia. I've created a CMOD sphere that attempts to do this. See my recent posting in the Addon Forum at http://216.231.48.101/forum/viewtopic.php?t=4570