Copyright © Michael Richmond.
This work is licensed under a Creative Commons License.
The goal of today's lecture is to show you how to calculate positions both differentially:
Equinox is concerned with the very slow but inevitable precession of the Earth's rotational axis. Over a period of about 26,000 years, the rotational axis of the Earth traces a circular path around the sky.
It's like a top spinning on a table: the top's spin axis precesses very rapidly due to the torque caused by gravity on its center of mass. The Earth's precession is also caused by a gravitational torque, due to the combined pulls of the Sun and Moon. So, although the Earth's rotational axis currently points close to Polaris (thus making it the "North Star"), it will eventually point elsewhere. Halfway through the 26,000-year cycle, it will point about 23.5 + 23.5 = 47 degrees away from Polaris, in the general vicinity of Vega.
Astronomers want to keep a coordinate system which is locked to a reference frame attached to the Earth. Therefore, they want "Dec = +90" always to mean "in the direction of the Earth's rotational axis." As the Earth precesses, the location of "Dec = +90" must move relative to the stars. In fact, the location of every (RA, Dec) coordinate will move very slightly. A star located at "RA = 0, Dec = +90" today won't be there next year (though the change is very small):
date RA Dec --------------------------------- 1950 23:58:43 89:43:18 2000 00:00:00 90:00:00 2050 00:01:17 89:43:18
So, when an astronomer provides a position, he needs to specify the coordinate system in which the measurements are made. Astronomers generally define systems at 50-year intervals:
You will sometimes see these abbreviated to "B1950" and "J2000". Why "B" and "J"? Go look it up!
There is another complication to consider when comparing positions today to those measured years ago. Proper motion is the shift of an individual star relative to its neighbors, due to its own intrinsic motion through space. A typical value for the speed of a star in its orbit around the Milky Way galaxy is 200 km/sec. Each individual star, however, has its own random velocity on top of this average motion. The magnitude of this individual component varies from around 10 km/s for young stars in the disk to around 40 km/s for older stars in the disk. Stars in the Milky Way's halo population may have very high speeds, over 100 km/s relative to the stars in the disk.
Of course, these stars are all very far away from us. How large will their apparent angular motion appear? Consider the following case: a star at a distance of 50 parsecs, which has tangential velocity 20 km/s.
Exercise:
- How large an angle does it move over a full century? Express your value in arcseconds.
- What is its proper motion? Express your value in arcsec/century.
Clearly, the position of a star with large proper motion depends on the date at which is was measured. The epoch of an observation is simply the date on which the data were acquired. It is usually expressed in fractional years: an image taken at 03:43 UT on Apr 28, 2002, has epoch 2002.321.
In order to explain the steps one must take to perform simple astrometric measurements, I will employ a concrete example: an image of asteroid 1107 Lictoria, taken at 21:48:30 EDT on Apr 11, 2002. The chart below shows a section of the image about 20 arcmin wide by about 14 arcmin high; North is down, and East to the right.
We will use stars "A" and "B" to determine the plate scale and orientation precisely (yes, in real life, one would use more than a single pair of stars), and then apply those values to find the position of asteroid Lictoria, marked with an "L". The steps will be
Using Aladin, this isn't hard at all. We simply acquire an image of the field (near RA = 07:17:21.45, Dec = +24 29 16, equinox 2000), ask for an overlay of stellar positions from the USNO-A2.0 catalog, and note the values for our stars. We can convert the USNO-A2.0 values from decimal degrees to sexigesimal notation if we wish.
star RA (J2000) Dec RA (J2000) Dec --------------------------------------------------------------------- A 109.413731 +24.452589 07:17:39.3 +24:27:09 B 109.309950 +24.474787 07:17:14.4 +24:28:29
Now, the USNO-A2.0 catalog is based upon photographs taken in the first Palomar Sky Survey. You can see from the detailed entry on each star that the epoch of the measurements is quite a while ago: 1954.972. That means that we must be careful: it's possible that one or both of these stars may have appreciable proper motion, in which case the positions in the catalog will NOT be the same as their current positions. But we'll ignore that point for now.
Again, this isn't very hard. We can simply display the images with XVista's tv command, move the cursor to each star, and then press the 'a' key.
distance from star B
star row col drow dcol
----------------------------------------------------------------
A 72.78 372.75 -30.77 121.42
B 103.55 251.33
L 120.05 283.88 16.50 32.55
The basic idea is to measure the distance between stars A and B in units of arcsec (via the catalog positions), and in units of pixels (via the image coordinates). Then we find the ratio of the two distances. It's simpler to break these distances down into components in each step.
So, first, the distance between A and B in arcseconds.
RA: (A - B) = 109.413731 - 109.309950 = 0.103781 degrees (??)
Dec: (A - B) = 24.452589 - 24.474787 = -0.022198 degrees
But wait: this isn't quite correct. The separation in the RA
direction must be corrected for the change in scale of RA with Declination.
What's that?
Use an analogy with coordinates on the surface of the Earth.
Image a man flying a plane around the world, along a line of
constant latitude.
At the equator (latitude = 0), he must cover a circumference of
about 40,000 km.
But if he flies along the line delimiting latitude = 40 degrees North,
his path is much shorter, only about 30,700 km.
If he goes very close to the North Pole,
he can fly in a tiny circle of latitude = 89.9 degrees and
go "all the way around the Earth" in a mere 70 km.
In order to calculate distances properly, we must project the spherical sky onto a flat plane at the position of the image. As long as the distance between two stars is small -- just a fraction of a degree -- we can do so pretty easily:
separation in RA = delta_RA * cos(avg. Dec)
= 0.103781 * cos(24.46 degrees)
= 0.094467 degrees
= 340.1 arcsec
separation in Dec = -0.022198 degrees (no correction needed)
= -79.9 arcsec
The total distance between the stars is
sqrt [ (340.1*340.1) + (-79.9*-79.9) ] = 349.4 arcsec
Next, we must find the separation between the stars in pixels.
This is easy -- there are no corrections to make:
Therefore, the plate scale of the image is
In real life, one would use more than a single pair
of stars to determine this conversion factor.
Astronomers try to arrange their cameras so that the
rows and columns run along the cardinal directions,
but never get it exactly right.
One must determine the rotation of the camera's rows and cols
relative to North and East in order to transform
pixel offsets into proper astrometric values.
A simple way to find the rotation angle is to calculate
the angle between the same pair of stars in each coordinate
system, as shown.
Let's do it for our pair of stars.
The column labelled "angle" is simply the inverse tangent
of the two component distances.
It has not (necessarily) been placed into the proper quadrant.
Notice that I've figured out that "column" is the direction
corresponding to RA, and "row" the column corresponding to Declination;
just look at the image, and it's obvious.
The rotation angle between the two systems is exactly 1.00 degrees.
If we wish to convert a vector from (col,row) to (RA, Dec), we must
We already did that, back in step 2. See the table there.
We measure the angle 63.1 degrees as "degrees from positive row direction,
moving towards the positive col direction".
Or, in other words,
angle away from straight down on the image, measured towards the right.
Following the procedure listed above in step 4,
The result is a vector expressing the separation between star B
and the asteroid:
It is always a good idea at this point to go back to your picture
of the field and make sure that you have kept the rotation angles
correct.
Looking back at the finding chart,
we see that, yes, the asteroid is both North and East of star B.
Good.
It's even a greater distance East than North from star B,
which matches our offset.
Double good.
Finally, we need to convert this offset into an (RA, Dec) position.
Once again, we need to correct for the shrinking of the RA coordinate
with Declination.
= ( 44.1 arcsec N, 90.8 arcsec E )
It's even possible to convert this position to sexigesimal notation,
to make it look more official:
349.4 arcsec
plate scale = ----------------- = 2.79 arcsec/pixel
124.3 pixels
Step 4: determine image rotation angle
delta delta angle (*)
----------------------------------------------------------------
RA/Dec: 340.1 arcsec -79.9 arcsec 76.78 degrees
col/row: 121.42 pixels -30.77 pixels 75.78 degrees
Exercise:
Step 5: measure distance of asteroid from star "B", in pixels
vector V from B to L = ( +16.50 rows, +32.55 cols )
= 36.49 pixels @ 63.1 degrees
Step 6: convert distance to arcseconds North and East
V = 36.49 pixel * 2.79 arcsec/pixel = 101 arcseconds
V = 63.1 degrees + 1.0 degrees = 64.1 degrees
L - B = 101 arcseconds @ 64.1 degrees East of North
= ( 44.1 arcsec N, 90.8 arcsec E )
Step 7: calculate (RA, Dec) of the asteroid, using star "B" as a reference
RA (asteroid) = RA (star B) + (90.8 arcsec) * cos(24.46)
= 109.309950 deg + 82.7 arcsec
= 109.309950 deg + 0.02296 deg
= 109.3329 deg
Dec (asteroid) = Dec (star B) + 44.1 arcsec
= 24.474787 deg + 0.01225 deg
= 24.4870 deg
asteroid at RA = 07:17:19.9 Dec = +24:29:13
Copyright © Michael Richmond.
This work is licensed under a Creative Commons License.