Parallax |
Two widely separated observers could attempt this activity under the real sky, but usually is not practical with classes. So we will use the planetarium software to measure the distance to the Moon. The same method could be used with planets, but the smaller angles involved would make it much more difficult. And as the ancients learned, the parallax of stars just can't be measured without powerful telescopes.
We will assume two observers who have telescopes and can accurately pinpoint the Moon's position in the sky. One observer is in New York City, the other in San Francisco. We pick a night and a time in which the Moon will be visible to both observers at the same time. You could pick many other times, but for the sake of this activity, we will pick before dawn on December 22, 1999.
The observer in New York records the Moon's apparent coordinates at 4 a.m., EST. At the same time (although 1 a.m. PST), the observer in San Francisco does the same. By telephone or email they compare their findings and calculate the angular difference between them caused by parallax.
Then they figure the straight-line distance between New York and San Francisco (through the Earth, not over the surface) as their baseline. Effectively, the angular difference in their two readings can be viewed at the angle at the apex of a triangle with New York and San Francisco marking the other two apexes. It's simple math then to figure out the distance between the Earth and Moon (that is, the height of the triangle). To be more precise, an adjustment must be made for the fact that the observations were not from the center of the Earth, but we will ignore that.
1) Open your Planetarium software.
2) Across the top you should see the options:
File - Edit - Set - Field - Center - Animation - Miscellaneous -
Help .
From this bar click on "Field," then "Chart Mode"
and finally "Local Horizon."
3) Under "Set," change "Geographical Location" to New York, New York. Record the latitude and longitude of New York as shown in the box.
4) Again under "Set," set the date and time to December 22, 1999 at 4:00 a.m.
5) Under "Field," set "Field Size" to 180 degrees.
6) Under "Center," click "On point on horizon" and then "Zenith"
7) You should see the Moon near the right edge of the chart. If not, be sure "Solar System" is checked under the "Display" option in the Field menu.
8) Click on the Moon and a box should pop up with its coordinates (right ascension and declination). Record these.
9) Now, repeat this observation for the observer in San
Francisco, remembering to set the location to San Francisco and
the time to 1 a.m. (date the same).
To find the central angle (Ø)between San Francisco and New York, you will need to plug their latitude and longitude information into this formula:
cos Ø = sin Lat ny * sin Lat sf + cosLat ny *cosLat sf *cos(Long ny - Long sf )
the to find Ø in radians, take the Arccos of the answer. Lat ny is the latitude of New York in decimal degrees, and so on.
If you are not comfortable with the math, you can download a simple Excel spreadsheet for which all you need do is plug in the numbers. (Click here Parallax.xls . You must have a spreadsheet program that will accept this file.)
To find the straight line distance, D, between New York and San Francisco, use this equation:
D=R*cos(Ø/2) where R is the radius of the Earth (about 6400 km)
Now for the readings on the Moon's position, you can calculate the angular difference between them in the same way as you found the angular difference between San Francisco and New York. Call the angular difference ß. Be sure to change Declination and Right Ascension into decimal degrees, and substitute Declination for Latitude and Right Ascension for Longitude. (Or use the spreadsheet)
Finally, the parallactic distance (PD) between Earth and Moon (not adjusting for Earth's center), can be found with this equation:
PD = (D/ß)*57.3
Due to the inherent inaccuracies of this method, a ten percent error from the true value should be acceptable, although you may get within 5 percent. In fact, since presumably everyone will use exactly the same data, the results should be the same for everyone.
Now, try your hand at it. How far away will be (was) the Moon on December 22, 1999?
Click here here our answer: Answer.
