Looking at both the figure and the formula, notice that the further away the object, O, is, the smaller the parallax angle we measure will be. Also, for a particular object, O, having a shorter baseline will result in a smaller parallax angle.

Let us look at our closest astronomical body, the Moon. The background reference point will not line up exactly with the target object from viewpoint X. As a result, we must use a little geometry to find our angle, a, by measuring the angles between the background reference point and the target object at both locations X and Y. Measure bx, the angle between the target and the reference point at position X. Then go to viewpoint Y and measure the angle, by, between the target and the reference point at that position. Looking at Figure 2, and remembering that the angles in a triangle must add to 180°, we see that:

Figure 2.

Since g + h = 180° (because they make up a straight line), we can substitute

But then, looking at the figure, and using what we know about parallel lines, we know:

So then, substituting for h in the two equations above, and doing a little rearranging, gives:

So the parallax angle, a, is just the difference between the two angles, b, that you measure between the target and the very distant reference point.  The target object, O, is the Moon, and as the distant reference point, V, we will use a bright star near the Moon.  If our baseline was only the size of a typical building, then the parallax angle, a, would be too small for us to measure because the Moon is so far away.  Instead, we will use a sky-simulation program called HNSKY (you will need to download and install it if you have not done so already).  

In order to get a sufficiently long baseline, we will us Ann Arbor as viewpoint X, and viewpoint Y will be 1048 km to the South in Macon, Georgia (distance found using http://www.indo.com/distance/). You will need the following information:

• The coordinates of Ann Arbor, MI are about 42.283 deg North, 83.750 deg West

• The coordinates of Macon, GA are about 32.837 deg North, 83.627 deg West

Ready to go? Follow the instructions below:

1. Open the HNSky program (if you used the default installation settings, there will we an icon on the desktop). Press, and hold, "Crtl" and press "e" on your keyboard to open the "settings" dialog box. Set the viewpoint to X (Ann Arbor) by entering the coordinates above and find the Moon (if you have trouble finding it, click on the "Search" menu, click on "Planetary", scroll down to the Moon, click to highlight, and click "GoTo"). Left-clicking on the Moon (or any other object) will show the objects information in the upper left. Right-clicking will center on the Moon. With the Moon centered, click "IN" a few times to zoom in. You should see some stars near the Moon.  Choose a star that is located close to the southern tip of the Moon or in a north-south direction.  Left-click on the Moon and note its coordinates (RA and DEC; note that the RA is quoted in hours from 0 to 24; i.e., 24 hours = 360 degrees). Next, click on the star and notes its coordinates. You can treat the geometry as being Euclidian (flat) since the Moon and star are close together. Therefore, you can use a simple triangle to figure out the angular separation between the Moon and the star. Enter this as bx in Table 1. REMEMBER TO CONVERT FROM HOURS TO DEGREES!
2. Press, and hold, "Crtl" and press "e" on your keyboard to open the "settings" dialog box. Set the viewpoint to Y (Macon) by entering the coordinates above and find the Moon. Zoom in and find the SAME STAR that you used from viewpoint X.  It should be in a slightly different place relative to the Moon.  Again, find the angular separation and enter this and bx into your Table 1.  REMEMBER TO CONVERT FROM HOURS TO DEGREES! Find your parallax angle, a, for the Moon.
3. Use the distance given above for the distance from Macon to Ann Arbor as the baseline.  Calculate the distance to the Moon using the parallax you have just calculated.
4. Go online and choose your favorite web searching tool (like Google, or AltaVista, etc...) to find the true distance to the Moon, and compare.

Discussion Questions:

1. What do you think was the leading source of error in your measurements of the distances using parallax?
2. Why does a parallax measurement become less accurate (higher percent error) as the target object gets to be very distant?
3. If you wish to use parallax to measure the most distant possible objects, is it better to use a short or long baseline? Why? 

Part Two: Solar Neighborhood

When we look into the night sky, we see many stars of different brightness and color, and in different groupings. Most of the time, you'll see single stars, or even some double stars, or possibly whole groups of stars hanging out near each other. But how can you tell if they are really part of the same group or if they just happen to lie along your line of sight? You have to get the distance! Now we know how to do this using parallax.

First we must decide what to use as the baseline. We cannot just use our eyes anymore, since, when you look at the stars, you cannot judge how far they are. So, we need to come up with a longer baseline (recall Discussion Questions 2 and 3, above). Well, we could take an observation from one side of the Earth and then run around to the other side and take another observation. That would then give us a baseline equal to the diameter of the Earth. Or even better, we could take one observation in January and another in July, when the Earth has moved half way around in its orbit, and use the diameter of the orbit as a baseline. As shown in the figure below, the first observation, taken in January, is at location 1. Six months later, the second observation, taken in July, is at location 2. Here, D is the distance to the star, the parallax angle is called Theta, and the "stars" labeled A and B are the apparent locations of the real star compared to the background of very distant stars.

So, we can take images of the sky on one day and compare them to images taken 6 months later and see if there are any stars that appear to shift. This Figure shows that the parallax angle, Theta, is part of a right triangle in which the length of the side opposite, Theta, is the Earth-Sun distance of 1 AU (Astronomical Unit ~150 million km ~ 93 million miles) and the length of the hypotenuse is the distance, D.  Using the definition for the sine function, we find:

We generally use this formula to calculate D after we have measured q, so it is more useful to solve it for D:

By definition, 1 parsec is the distance to an object with a parallax angle of 1 arcsecond (1"), or 1/3,600 degree. (Recall that 1 degree = 60' and 1' (arcminute) = 60") Substituting these numbers into the parallax formula, and using the fact that the sine function for small angles is roughly equal to just the angle, we can write the equation as:

Next, let's consider some interesting stars in the Solar Neighborhood. Recently there was a satellite, called Hipparcos, put in space to measure the parallaxes, among other things, to over 100,000 stars. From the Hipparcos website: "Hipparcos was a pioneering space experiment dedicated to the precise measurement of the positions, parallaxes and proper motions of the stars. The intended goal was to measure the five astrometric parameters of some 120,000 primary stars to a precision of 2 to 4 milliarcsec, over a planned mission lifetime of 2.5 years, and the astrometric and two-colour photometric properties of some 400,000 additional stars (the Tycho experiment) to a somewhat lower astrometric precision."   You are encouraged to peruse this site for a wealth of information. Below is a table with the common name and the HD (from the Henry Draper catalog) name of some of the stars that have planets orbiting them. You will need the HD name to use in the search page on the Hipparcos website.

Discussion Questions:

Copyright 2002 Lynette Cook. Used with permission.

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