RASC Calgary Centre - Planet Orbit
Planet Orbit
by Jason Nishyama
Page last updated November 5, 2018
Measuring a planetary orbit with astrophotography.
Once you've been armed with the basics of astrophotography, it's time to
use that for some science! In this article we'll look at how you can measure a planetary
orbit using a camera and some math.
This project can be done with a simple tripod mounted camera. Exposure should be the shortest
possible to avoid trailing the stars as we'll need circular stars for measurement. The exposure
formulae in imaging 2 article can be used to determine this exposure time.
At the time of this writing, Mars is suitably located for taking pictures for this particular
project so locate Mars in the night sky. Set up your tripod and camera with a suitable lens. For
this project a telephoto lens doesn't necessarily help so a wider angle lens will do just fine.
Centre Mars in your camera viewfinder. Try to have the long axis of the viewfinder parallel to the
east-west line in the sky, this will make the math easier later.
Since you'll need to know what direction east-west is, first you'll take a longish exposure photo
of 2-3 minutes to create star trails. This trail will naturally run east-west and can act as a
reference for the science image. Once you have taken the star trail frame, take the science image
of Mars using the exposure calculated to avoid star trails. It is important that you don't move
the camera between the star trail image and the science image or you will lose your reference
line. Record the time that you took your Mars image.
Now that you have one image of Mars (and a line reference) wait 5-7 days and take another set of
reference image and science image. Wait the same amount of time (5-7 days) and take a third set.
With three more or less equally (in time) spaced positions of a planet you can determine it's
orbit, our ultimate destination.
For this project you may be asking yourself "what about dark and flat frames?". Given what we are
doing neither are strictly required so you can save yourself some time at the camera and at the
computer and ignore them for this project. If you want to make "pretty pictures" of your science
images, then by all means make your darks and flats and process as necessary, but if you just want
to do the science, don't worry about them.
Now that you have your science and reference images, time to do some math. First you need to work
out how many arc-seconds a pixel on your camera sensor is. If you know the physical size of each
pixel in the sensor (often available in the owner's manual or from the internet) this is actually
quite simple. Using this formula:
P = (206265 x u) / (1000 x f)
where u is the physical size of each pixel (in micrometres), f the effective focal length of the
optical system and P the number of arc-seconds per pixel. Note that any "multiplication factor"
for focal length is NOT applied in this case. So a camera with 5.5 micrometre pixels using a 50mm
lens would have an image scale of about 22.6 arc-seconds per pixel. If you multiply this number by
the number of pixels wide you can then work out the field of view. So if there are 4000 pixels on
the side, the field of view is 4000x22.6 or 90400 arc-seconds or about 25 degrees.
If you can't find the pixel size for your particular camera, you can work this out from one of
your star drift frames. You'll need to know how long the exposure was and conveniently most
cameras store this information in the EXIF data in the image itself so don't worry if you didn't
write this down at the time.
At this point you'll need some kind of image editing software such as Photoshop or if you're like
me and into free - as in no cost - software GIMP (the Gnu Image Manipulation Program) which is
available for Windows, Mac OS X and Linux. Use what you are comfortable with.
Load up the star trail image into your editing software. Find a star that you recognise and look
up it's declination in a catalog or on a star chart (the SIMBAD data server will have the most
accurate declination, it's at
http://simbad.u-strasbg.fr/simbad/
and is free). You'll need the
declination later.
Use your image editing software to measure the length of the line (in pixels). Don't worry if it
doesn't run along a row of pixels as the software will measure it's actual length regardless of
its angle. To figure out how long the line is in arc-seconds use the following formula:
L = t x (0.2506845 x cos(dec)) x 60
Where L is the length of the line in arc-seconds, t is the length of the exposure in seconds and
dec is the declination of the star used. At this point it is simple to work out the arc-seconds
per pixel by this formula:
P = L / n
where P is the arc-seconds per pixel, L is the length of the star trail in arc-seconds and n is
the length of the star trail in pixels as measured by your software.
Now you're set to work out the actual position of Mars. This process is known as astrometry. The
process is actually quite simple thanks to a web application located at NASA's Extra Galactic
Database:
http://ned.ipac.caltech.edu/forms/tab-conv.html
. This form will take the right
ascension and declination of a known object as well as an x-y offset from the reference object to
Mars (or any other object) and compute the right ascension and declination of the other object. So
you will need to find a star that is close to Mars on your image, determine its RA and dec (from
SIMBAD or other source) as well as determine its x-y position on the image, using your image
editing software.
You will also need to know which way is up and down on your image. East/west is simple as your
star trail will run east-west. Assuming you have used a standard DSLR with lens, the image you
have will have north at the top and east on the left side. For standard camera images, pixel 0,0
is in the top left corner. This means the x offset increases towards the WEST and the y offset
increased towards the SOUTH.
The angle the y axis is rotated can be worked out from your star trail image using your image
editing software. Most have a function that allows you to draw a line and while you draw the line
it will tell you the angle of that line compared to the x-axis. So if you draw a line over the
star trail, from east to west you will get an angle either above the x-axis or below the x axis.
The angle the y axis forms with north will be the same angle. Now if the angle is above the x axis
and greater than 180, than your computer is computing the angle in a clockwise way (the opposite
would be counter clockwise). This is important to remember as you will need to make the angle
negative when entering it into the NED form if it is clockwise.
Finally you'll need to enter the pixel scale that you computed earlier (arc-seconds per pixel) in
the NED form.
Now all you need are the x-y offset to Mars. This is simple to find as all you need to do is use
your image editing software again to determine the x-y coordinates of the centre of Mars' disk and
then subtract the x-y position of the reference star like this:
x_offset = x_mars - x_reference
y_offset = y_mars - y_reference
You can now enter the x and y offsets into the NED form and hit submit and voila, the RA and dec
for Mars.
Repeat the process with your other two images and star trail reference frames. You will then have
a set of three dates, times and RA and dec's for Mars. This is enough to roughly compute Mars'
orbit.
Now computing an orbit from three observations by hand is a long and tedious operation and I won't
get into it here. There is software available (for free) that will do this for you though.
Find_orb is available for the Windows and Mac OS X and Linux (Mac and Linux users will have to
compile it from source code) from here:
http://www.projectpluto.com/find_orb.htm
. Another
calculator is orb_fit (
http://adams.dm.unipi.it/~orbmaint/orbfit/
) but it requires Mac OS X or
Linux and a FORTRAN compiler. I've used orb_fit but I'm a glutton for punishment. Other software
may be available.
Once you've entered your data into whatever software package you end up using, you'll end up with
the orbital elements of the planet Mars, which will basically define Mars' orbit in terms of it's
inclination to the Earth's orbit, the semi-major axis of Mars' orbit, it's eccentricity and three
other terms which are used to position the orbit with respect to the Earth's.
Now this technique can be used to compute the orbit of any body orbiting the Sun. In fact this is
an area where amateur astronomers routinely make contributions to the field of astronomy. There
are many, many asteroids for which their orbits are not well known. Though there are major
professional surveys attempting to remedy this, amateurs can provide valuable information to help
refine these orbits. In fact all you need to do is pick a couple of asteroids to observe, observe
and calculate their position as above for Mars, then send the data in!
The place to check this out is the Minor Planet Centre at
http://minorplanetcenter.net/iau/mpc.html
. Here you will find what you need to do to send in
observations of asteroids to help refine their orbits. You may need to upgrade to an equatorial
mount however, most asteroids are quite faint and exposures of 30 seconds to several minutes may
be required to image them. That being said, I find it cool that as amateurs it is still possible
for us to do some serious science!
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