Parallax in theory and practice

One of the few reliable methods astronomers may use to determine the distances to objects in space is called parallax. It is based on trigonometry, so let's start with that.

There are special mathematical tools one can use to determine the lengths of the sides of a right triangle. In the figure below, suppose that we want to know the distance L.

If we can measure the side B, and the angle alpha, then we can use the equation

                       B
             L  =  ----------
                   tan(alpha)
to calculate L.

We can use the same method for a situation in which there are two such triangles, back-to-back:

We can still figure out the distance L with trigonometry. Once again, we calculate

                       B
             L  =  ----------
                   tan(alpha)

but we could also describe this as

                        1/2 (2B)
             L  =  --------------------
                   tan [ 1/2 (2*alpha) ]

or

                    half of the total baseline
             L  =  ----------------------------
                   tan [ half the total angle ]

Using trigonometry to measure the distance in a hallway

One of the tasks before you today is to measure the distance to a camera tripod, set up in the middle of the third-floor hallway. You may not approach it, but must stay at one end of the hallway. You are given materials:

Your job is to devise some measurement that will permit you to calculate the distance to the tripod. Keep in mind the diagrams above.

Parallax and distance measurements

Measuring angles is difficult (as you may have discovered from the tripod exercise). It's not an easy thing to determine exactly in which direction you are looking at a particular moment. Remember that astronomers, sitting on the Earth's surface, are constantly being rotated around the Earth's axis (once every 24 hours), and carried along with the Earth around the Sun (once every 365 days). It's very hard for us to keep track of the absolute direction of our telescopes when we look at stars or planets.

There are times when it helps to use parallax to determine relative angles. Relative measurements, in astronomy and in all of physics, are often much easier to make than absolute ones. Parallax is simply the apparent shift of a nearby objects relative to objects behind it as one moves from one end of a baseline to another: to see it, hold your arm outstretched in front of you, and hold one finger straight up. Hold your arm steady. Close one eye, and look past your finger at some distant object. Now close the OTHER eye and look past your finger again. You should see that it appears to move, relative to the distant object.

Astronomers can do the same thing. If they can observe the same nearby object from two different places, they may detect a difference in its appearance relative to background stars.

Suppose that two astronomers separated by a distance D observe an asteroid simultaneously. One sees the asteroid next to star X, and the other sees it next to star Y. If the astronomers know the angle between those two stars, and they know the distance between their observatories, they can calculate the distance L to the asteroid:

                     half of total baseline
              L  =  ---------------------------
                    tan [ half of total angle ]

                          D/2 
                 =  ----------------
                    tan [ theta/2 ]

Sticklers for the truth will notice that this method only works when the background objects -- the stars -- are much, much, much farther away than the asteroid. Only then is the angle between the stars, as seen from the Earth, the same as the angle between the stars, as seen from the asteroid. Fortunately, in real life, the stars really are much, much farther away than any body in our solar system, so we can use this method.

Using parallax to determine the distance to Mars

Use the Sky Map Pro program to observe the planet Mars on April 24, 1999 from the following places:

Make your observations simultaneously at 6:42 AM, Eastern Daylight Time. Note that Sky Map Pro will try to change the time if you move to a different time zone.

Zoom in close to Mars, so that the field of view is about 10 or so arcminutes. Set the limiting magnitude of the chart so that it shows stars down to fifteenth magnitude. You should see a faint star very close to the planet. What star is that?

Use the program to measure the separation between Mars and the star, as seen from each location. Write down your measurements. They will be in polar form: each consists of a distance (in arcseconds), and an angle.

Convert these measurements into rectangular form: the distance in arcseconds to Mars' east (x-direction), and the distance in arcseconds to Mars' north (y-direction). Write these values down, too.

Use the difference in x- and y-directions to calculate the total angle, in arcseconds, by which the star moved relative to Mars. Write down this angle in arcseconds, and convert to degrees.

Make a picture on graph paper which shows the relative positions of Mars and the star, as seen from the different locations. Indicate the size of Mars' disk on your picture. What's really happening is that, seen from different locations on Earth, Mars' position is moving slightly relative to the star. But it is equivalent, and easier to draw, if you pretend that Mars is remaining in place and the star is moving.

Calculate the distance of the baseline between the two cities. Use the baseline, and the angle, to calculate the distance from Mars to the Earth on April 24, 1999.

How does your value compare to the true distance between Mars and the Earth on that date?


Last modified May 9, 1999, by MWR.