The Orbit of Venus
To the naked eye the planet Venus appears like a very bright point – the brightest “star.” Depending on whether it is east or west of the sun, it appears as the “evening star” or the “morning star.” Over the course of a year or so it moves back and forth between these positions, and sometimes it appears too close to the sun to be seen at all.
In this lab you will have copies of pictures of Venus taken with MHC’s telescope on three different dates:
|
Picture label |
MM/DD/YY |
HH:MM |
Elongation (degrees) |
|
Ven06bf |
11/20/00 |
15:19 |
40.5 |
|
Ven14bf |
01/01/01 |
14:03 |
46.4 |
|
Ven30 bf |
03/07/01 |
16:22 |
30.3 |
In the telescope Venus does not look like a point, but has a definite shape, rather like the moon, and shows phases like the moon that change with time. A natural explanation is that Venus is a sphere that we see illuminated by the sun at an angle V that changes with time, as shown in the diagram below:
If we could see the full circular outline of Venus, it would have diameter 2R, but because of the way it is illuminated, we see an outline that is not circular, and has a width only R(1+cos(V)). This gives a way to find V: we estimate both these quantities. Their ratio is (1+cos(V))/2, and we solve for V. The elongation angle E is just the angular separation of Venus and the Sun as seen from Earth, and was noted at the time the pictures were taken. (It is given in the table above.) Thus both angles E and V are known. The distance of the Earth from the Sun must be very nearly constant, since we see the Sun always with the same angular size. This constant distance is called the Astronomical Unit (AU). In Galileo’s day it was not known accurately at all, but for many purposes we don’t actually need to know its value.
On a sheet of paper draw a straight line representing 1 AU, with the Earth at one end and the Sun at the other. Then use your measured values of E and V, along with a protractor and straightedge, to construct the three positions of Venus relative to the Earth and Sun. Galileo was convinced by this construction that Venus moves in a circular orbit around the Sun, just as Copernicus had said, even in the absence of direct evidence.
Determine from your picture the Sun-Venus distance (in AU) for each contructed position. Also determine the Earth-Venus distance for each constructed position. You probably noticed that Venus sometimes appears large and sometimes appears small in the pictures. This is just what is to be expected if it is sometimes nearer and sometimes farther away. In fact, its apparent size should just be proportional to 1/D where D is its distance. Plot the apparent size 2R versus 1/DEV, where DEV is your constructed Earth-Venus distance, to see if they are really proportional. This is a kind of consistency check on the construction.
Since there are a lot of quantities to keep track of here, I suggest organizing them in a table with column headings like the ones below.
|
Picture |
E |
WD=2R |
BD=R(1+cos(V)) |
V |
E+V |
DSV |
DEV |
1/DEV |
Here I have used “WD” for “whole diameter” and “BD” for “bright
diameter.” A convenient formula for V
is
.