CL - Distances (remote)[91] (1)
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California State University, Bakersfield *
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Course
1609
Subject
Astronomy
Date
Apr 30, 2024
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docx
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16
Uploaded by crystalized47
Computer Lab – Distances
(Virtual Lab Remote Edition)
Introduction
In this lab you will re-create experiments done by past astronomers to
determine distances within our solar system. The same ideas are still used today to
determine distances on Earth as well as in outer space.
Aristarchus Experiment
The Greek scientist Aristarchus (310–230 BC) realized that if we see the Moon
half lit, the Sun, Moon, and Earth must make a right-triangle. See the figure below.
The angle at the Moon’s vertex must be exactly 90° if we see the Moon exactly
half lit. But the angle a
next to Earth will be less than 90°, how much less depends on
how far away the Sun and Moon are from Earth.
The Sun is much further away from us than the Moon and the angle a
is very
close to 90°. Aristarchus did an experiment to measure a
, he physically measured the
angle between the Sun and Moon in our sky when the Moon appears half lit.
Aristarchus measured a value of 87° from which he (properly) calculated that the
Sun must be 20 times further away from us than the Moon.
Because the Moon and Sun appear equally large in our skies, Aristarchus
correctly reasoned that the Sun’s actual size must be 20 times larger than the Moon.
Because he discovered the Sun was so large, Aristarchus proceeded to create a
heliocentric model of the universe. The model was eventually revived by
Copernicus and validated by Kepler and Galileo almost 2000 years after Aristarchus
lived.
Distances – 1
Moon
The Sun is actually 375 times further away (and wider) than the Moon. The
experiment done by Aristarchus is challenging and difficult even with modern
equipment. For the time, Aristarchus’s results were excellent and were a milestone
in the development of the scientific method. Even using the Voyager 4 program, it is
not easy to get really good results for this experiment, we will do a different
experiment that uses the same idea.
Copernicus Experiment
We will do an experiment, first done by Copernicus, to measure the distance
between Venus and the Sun. Launch the Voyager 4 program (csub.apporto.com).
Select the Tools/Planet Report… menu and select Maximum Elongations from the
pop-up menu (which is usually hidden behind the Time Panel). Find the date of the
next maximum elongation for Venus and write the date and angle on the following
line.
Hmmm. The Venus values follow the Mercury values in that table but viewing
them is tricky, changing to Full Screen mode (
) worked for me. Here are the
Venus values so you don’t have to try to find them yourself. You can just note which
line gives the values for your “next” Venus maximum elongation.
Planet
Direction
Angle
Date
Venus
eastern
47.0°
Oct 29, 2021
Venus
western
46.6°
Mar 20, 2022
Venus
eastern
45.4°
Jun 4, 2023
Venus
western
46.4°
Oct 23, 2023
Venus
eastern
47.2°
Jan 10, 2025
Venus
western
45.9°
Jun 1, 2025
The elongation angle is the angular separation between the Sun and Venus in
the sky. Copernicus could measure this with simple. To get the maximum
elongation, Copernicus just had to measure elongations day after day and watch the
values for a maximum.
While still in the Planet Report window, select Angular Separations from the
pop-up menu, the white line shows how the elongation of Venus varies with time.
you picked one of the dates when the elongation angle reached a maximum.
Close the Planet Report window. Select the Chart/Set Time… menu, click on
the
Universal Time tab, and enter your date from above. Select the
Distances – 2
Center/Planets/Venus menu. Click the Physical tab in Venus' Info Panel, the
"Illumination" should be around 50%.
Why does Venus appear about half-lit (50% illumination) when at maximum
elongation? That is explained by the geometry shown in the figure below, if Venus
was anywhere else in its orbit the angle e
would be smaller. Copernicus lived before
the invention of the telescope, he could have predicted that Venus would appear
half-lit but had no way to check it.
The distance x
between the Sun and Venus is what we are trying to determine.
The distance between the Earth and Sun is always close to 1 AU. The Venus-Sun line
and Venus-Earth line are perpendicular because only that way would we on Earth
see Venus at maximum elongation, hence the 90° angle. The angle e
is the
elongation, the angle between Venus and the Sun as seen from Earth, the angle from
earlier.
The mathematical relationship between the distances and angles for this
triangle involve trigonometry, we need to use the trigonometric “sine” function. In
particular,
sine of angle e
= Distances – 3
or
x
= (sine of e
) (1 AU) = sin(
e
) AU
So, x
, the distance between the Sun and Venus in AU, is just the sine of the
elongation angle when Venus appears half lit. Use your calculator (make sure it is
set for degrees, not radians) to compute this value or type “sine of 45.8 degrees” into
Google – but use your
angle from the earlier table.
Venus to Sun Distance = 0.7169 AU
Look up the actual Venus-to-Sun distance to check your result. Do that by
selecting the Tools/Planet Report menu, then select Heliocentric Positions from the
pop-up menu, and then you'll find the desired value in the Distance column.
Actual Venus-to-Sun Distance = 0.72699 AU
How well did your result work out? Our biggest error was assuming the Earth-
Sun distance was exactly 1.000 AU. The same procedure works for determining the
distance between Mercury and the Sun. And a similar procedure can be used to find
distances to the outer planets.
Distance to Mars
In the Planet Report window, select Angular
Separations from the pop-up menu and look for the orange/red line of Mars. You are
going to determine the date, as accurately as you can, when Mars will have an
angular separation from the Sun of 90°.
Note that the line for Mars is actually a string of squares, each represents one
day. Also note that the grid lines on the graph do not match the beginning and
ending of most months as seen at the bottom of the
window. Try to determine a date when Mars had a
90° value (note: on the graph 90° is only half-way
up).
Date of 90° Separation: February 1st 2021
Distances – 4
Close the Planet Report window. Set the date to what you just determined.
Select the Center/Planets/Mars menu. Click on the Physical tab in Mars' Info Panel,
record the percent Illumination value of Mars on this day.
Illumination = 88.6%
If we were watching Mars from the Sun, we would see only the lit side, 100%
illumination. Because we are away from the Sun, we see part of the dark side of
Mars, less than 100% illumination. Based on the illumination seen, we can figure out
the angle we are away from the Sun relative to Mars. That is, we can figure out the
angle e
in the figure below from the illumination value.
You can use the percent Illumination value to determine the angle e
in the
figure by using the following table:
% Illum
75
76
77
78
79
80
81
82
83
84
Angle e
60.0°
58.7°
57.3°
55.9°
54.5°
53.1°
51.7°
50.2°
48.7°
47.2°
85
86
87
88
89
90
91
92
93
94
95
45.6°
43.9°
42.3°
40.5°
38.7°
36.9°
34.9°
32.9°
30.7°
28.4°
25.8°
Distances – 5
Or you can use the following exact formula to calculate e
, the formula
involves an arccosine, so don’t attempt the formula unless you are familiar with
those functions. If the fractional illumination of Mars is f
(75% would be f
= .75), the
formula for calculating the angle e
is:
e
= cos
-1 (2
f
– 1)
Record the angle e
you determined by formula or table. If you're using the
above table and your percentage was between two of the percentages, interpolate to
get a better value for the angle e
. For example, if your percentage was 94.5, you
might use an angle around halfway between the 94 and 95 values, like e
= 27.0, don't
worry about being exactly correct.
e
= 49.45°
From trigonometry we can calculate the Mars-Sun distance (
d
), the formula is
as follows. You can have Google do the calculation by searching for “one divided by
sine of 39.3 degrees” (use your
e
value instead of 39.3).
d
= = 1.326 AU
Compare your calculated value to the value given for Mars on the Heliocentric
table of the Planet Report.
Mars to Sun Distance = 1.54964 AU
Your result will likely be off, because we had to estimate the date of the 90°
separation and because we didn’t include the precise Earth-Sun distance for the time
of the experiment. Still, it is a very good method.
This method for Mars is really the same as the method we used for Venus just
turned around. In both cases we knew we had a right angle (90° angle) between the
Sun and planets, either because we saw a planet at maximum elongation or because
we measured a 90° elongation angle in the sky.
Copernicus and Outer Planets
Copernicus actually used a different method to determine the distance to the
outer planets (he couldn't use the above method because he didn't have a telescope
and so he couldn't know the percent illumination). It's a bit more complex than the
Distances – 6
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