RASC Calgary Centre - Double Stars
Double Stars
by Jason Nishyama
Page last updated November 5, 2018
Measuring Double Stars
Double stars are two stars that are very close
together in the sky, so close that they look like one star without a suitable telescope. Sometimes
these two stars are orbiting each other in which case they are a binary star system. Other times
the stars are not related to each other, but are simply close together in our line of sight in
which case they are called an optical double.
Double stars in the form of binary stars are quite important in astronomy. By observing these
stars over time, we can compute the orbit of one star about the other and from that work out the
masses of the two stars. This has helped astronomers to determine how stellar mass and brightness
are linked, with larger stars being brighter, providing a key piece of the stellar evolution
puzzle. Many of these stars also move on timescales of a year or so, so monitoring and measuring
them over time can show differences in separation and position angle.
In this observing project we'll be measuring the separation and position angle of one star to the
other. This can be done either visually or photographically. We'll look at both methods.
For working on this project visually, you'll need a telescope, a stopwatch and a graduated reticle
eyepiece. A graduated reticle eyepiece is one where there are tick marks along both sets of cross
hairs in the field of view. The aperture of your telescope will limit the separation of the double
stars you can see as separate stars (resolution), so more aperture is better. By using the
following formula, which computes Dawes' Limit, the minimum separation double stars a given
aperture can be worked out ahead of time:
R = 4.56 / D
Where R is the minimum separation in arc seconds and D is the aperture in inches. Changing the
constant 4.56 to 11.6 allows D to be in centimetres. So with this formula, the C14 at the WCO
(D=14") has a minimum separation limit of 0.32 arc seconds. In practice this is never achieved as
atmospheric seeing will also affect this resolution. In the Calgary area the turbulence off the
mountains often puts a limit to resolution in the 1 to 4 arc second range.
Once you've determined your resolution limit, pick a suitable double star to practice on. The
RASC Observer's Handbook lists several double stars so all you need to do is pick a pair that is bright
enough to be viewed through your telescope as well as having enough separation that you will see
them as separate stars.
For visual measuring an equatorially mounted and driven telescope is not needed. You will need to
determine the number of arc seconds between each of the ticks in your eyepiece and this can be
done by setting up the eyepiece so that the cross-hairs run north-south and east-west. Allow a
star of known declination (you can look this up ahead of time) to drift through the field of view
of the telescope and line up one of the cross-hairs with the path of the star. This line is now
the east-west line. Using the stopwatch you can now measure the time a star takes to drift from
one tick to the next along the east west line. Do this a couple of times with the same star.
Now you're ready to measure your double star. Ensure that your cross-hairs are still oriented
east-west by allowing one of the stars in the pair to drift along the east-west line, correct if
required. Note the direction that the dimmer of the two stars is from the brighter star, being
north or south and east or west. It may help to watch them drift for a moment or two as they will
drift west in the eyepiece. To work out north or south, move the telescope towards the southern
horizon and note which way the stars move and determine which direction the dimmer star is from
the brighter star from that.
Next use the ticks on the cross-hairs to determine how far east-west and north-south the dimmer
star is from the brighter star. Record the number of ticks north/south as well as the number of
ticks east/west the dimmer star is from the brighter one.
Now that you have some juicy data, time to head inside where it is warm and do some math! First
we're going to work out how far apart, in arc seconds the ticks in your reticle are. If you did
more than one timing, average them together and use the average as the time. Next use the formula
below to work out how many arc seconds are between each pair of ticks:
a = 60 x t x 0.2506845 x cos d
where a is the distance between ticks in arc seconds, t is the drift time in seconds and d is the
declination of the star used in degrees.
Now you can convert the number of ticks you recorded into arc seconds by simply multiplying the
number of ticks by the number of arc seconds per tick you just worked out.
Working out the separation in arc seconds can now be determined by the simple use of the
Pythagorean Theorem. Multiply the distance north-south by itself. Next multiply the east-west
distance by itself. Add these two numbers together then take the square root of the sum. This is
the separation of the two stars in arc seconds.
The position angle can be determined trigonometrically. If the dim star was south of the brighter
one, make the north-south distance negative (i.e. 4.5 becomes -4.5). If the dim star was east of
the bright star, make the east-west distance negative as well. If the east-west distance is not
zero, divide the north-south distance by the east-west distance. Then add the appropriate number
from below:
- if both numbers are positive, add nothing.
- if the n-s number is positive and the e-w number is negative, add 180
- if both numbers are negative, add 180
- if the n-s number is negative and the e-w number is positive, add 270.
This gives the position angle of the dimmer star in degrees.
Now if the east-west distance was zero, the position angle is 90 degrees if the n-s distance is
positive and 270 degrees if the n-s distance is negative.
So now you have the separation and the position angle of the double star. You can compare it to
the values published in the Observer's Handbook. How close you get depends on how close together
the ticks in your reticle eyepiece are and thus how accurately you can measure the various
distances.
To be more accurate you can do the measurement photographically. To do this measurement with this
method you will need an equatorially mounted and driven telescope, either a DSLR or CCD imager and
software such as Photoshop or GIMP.
The observations for this method are rather quick. You'll take two images. The first image is of
the double star and the second is to help calibrate the camera.
Aim and focus the telescope so that your camera has the double star in the centre of the field.
Try to have the long side of the frame point roughly east-west. This isn't essential, but makes
things easier later. Take several shots of the double star. Star with a 0.5 second exposure and
double the time until you get to 32 seconds. This will ensure a useful image, one where the stars
are not too dark and not too over exposed. Once you have done this do not move the camera, because
you'll need to take the calibration shot.
The calibration shot will provide us with a couple of important pieces of information. First it
will tell us which way the cardinal directions are on the frame. Second it can be used to
determine how many arc seconds there are per pixel on the frame. With the telescope tracking, set
the camera up to do a 15 second exposure. Leave the tracking on for 5 seconds then turn off the
tracking for the remainder of the exposure. This should give you a trail across the frame with a
bright point at one end. Since the stars track across the sky east to west, the bright point will
be on the eastern side of the frame. For greater accuracy, do a second frame by setting the camera
to do a 10 second exposure with the tracking off for the whole exposure.
Once you have your photos, time to do some data reduction at the computer. Load the 10 second
calibration shot (if you took one, else use the 15 second one) into your favourite photo editor
(Photoshop or GIMP for example). Use the measurement tool to determine how long one of the star
trails is, in pixels. Also record what angle the trail is to the horizontal axis. Work out how
many arc seconds long the trail is by using the formula used in the visual section above to work
out the distance between ticks, in this case the time is 10 seconds.
The number of arc seconds per pixel is found by dividing the length of the trail in arc seconds by
the length of the trail in pixels.
Now with this information you can work out the separation of the double star. Pick the best of
your double star photos, that is the one where both stars are clearly defined. Using the
measurement tool work out how many pixels are between the centre of both stars. Multiplying this
by the number of arc seconds per pixel will give the separation of the two stars in arc
seconds.
Also using the measurement tool work out the angle from the horizontal the dimmer star is from the
brighter star. The position angle needs to be calculated as an angle counter clockwise from the
west. You may need to take the angle calculated and re-work it into a counter clockwise angle. You
will then need to correct for the angle of the camera from the horizontal using the angle measured
from the star trail. Once this is done you also have the position angle of the dimmer star.
As with the visual method you can now compare your values with those published in the Observer's
Handbook. Generally any major difference in position angle may be due to the conversion to the
counter clockwise angle and re-converting may fix this. If the angle is 180 degrees off, then the
angle was measured to the brighter star and not the dimmer.
So go out and measure away and enjoy another aspect of double star observing!
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