It is fun to design solar
systems, and to plan planets and moons. You have some information
on orbits now, and know about
The Roche Limit. However, there is still
a question to think about: when you are standing on your planet,
will you be able to see your moons? We can figure that out!
We will use trigonometry. If
you haven't taken any trigonometry yet, don't worry 
I will
lay this out so that you can do it. Trigonometry is about the
relationships of the sides and angles in triangles to each other.
It is useful for measuring things that we can't get to.
Here is a diagram of what we are going to do.
We decide on the radius (distance from the center to the outside)
of our moon.
We decide on the distance that our moon is from our planet.
We divide the distance into the radius and get a small number.
We divide that small number by 2.
Then we look up our answer in a Tangent Table to get how many degrees wide the moon is
in the sky.
Example:
A Theoretical System
The Radius of a moon is 1500 miles.
(This is bigger than our moon)
The Distance to this moon is 300,000
miles.
1500
= 0.005
This number is the (Tangent of
Theta)
300,000
2
Multiply 0.005 by 2
to get the Tangent of Theta.
The
Tangent of Theta = 0.010
Now we need to look up the Tangent of
Theta in a yellow Tangent
Table
here
The angle next to 0.010 is 0.6.
This is the
viewing angle, which is 6/10 of
one degree.
This moon looks a little larger than
our moon, which has a viewing
angle of 0.5, which is half of
a degree.
Tangent
of Theta 
Viewing
Angle in Degrees 

Tangent
of Theta 
Viewing
Angle in Degrees 

Tangent
of Theta 
Viewing
Angle in Degrees 

Tangent
of Theta 
Viewing
Angle in Degrees 
0.176 
10 
0.017 
1 
0.002 
0.1 
0.0002 
0.01 
0.194 
11 
0.035 
2 
0.003 
0.2 
0.0003 
0.02 
0.212 
12 
0.052 
3 
0.005 
0.3 
0.0005 
0.03 
0.231 
13 
0.070 
4 
0.007 
0.4 
0.0007 
0.04 
0.249 
14 
0.087 
5 
0.009 
0.5 
0.0009 
0.05 
0.268 
15 
0.105 
6 
0.010 
0.6 
0.0010 
0.06 
0.364 
20 
0.123 
7 
0.012 
0.7 
0.0012 
0.07 
0.466 
25 
0.141 
8 
0.014 
0.8 
0.0014 
0.08 
0.7 
30 
0.158 
9 
0.O16 
0.9 
0.0016 
0.09 
Example:
The Earth and the Moon
Diameter of Earth's Moon = 1738 km
Distance to Earth's Moon = 384,400 km
1738 = 0.00452
This is half of the Tangent
of Theta.
384,400
To find the Tangent of Theta, multiply
0.00452 by 2
Click here for the Tangent Table
Look for 0.009 in the yellow columns.
The angle next to 0.009 is 0.5 .
This is the
viewing angle, which
is half of one degree.
Example:
Jupiter and Europa
Radius of Europa =
.25 of Earth's radius
(See Page on Planets and
Moons)
Earth's radius = 6378 km
Europa's radius = (6378 * 0.25 ) =
1594.5 km
Round this to 1594 km
Distance from Jupiter to Europa = 670,900
km
1594 = 0.00238 This
is the Tangent of half of the angle Theta.
670,900
To find the Tangent of the angle Theta,
multiply 0.00238 by 2
which equals 0.00475
Round this to 0.005
Click here for the Tangent Table
Look for 0.005 in the yellow columns.
The angle next to 0.005 is 0.3.
This is the
viewing angle, which is three tenths
of one degree.
Our moon has a
viewing angle of half a degree (five
tenths), so Europa probably looks a little more than half as
wide as our moon from the surface of Jupiter. (You would need
to be in a space ship above the atmosphere to see the moon
though.)
Example:
Saturn and Tethys
Diameter of Tethys =
1060 km
(See Page on Planets and
Moons)
Earth's radius= 6378 km
Tethys radius = 530 km
(The radius is
half of the
diameter.)
Distance from Saturn to
Tethys = 295,000 km
530 = 0.0018
This
is the Tangent of half of the angle Theta.
295,000
Multiply by 2 to get
the angle of Theta.
Round this to 0.004
Click here for the Tangent Table
Look for 0.004 in the yellow columns.
The angle next to 0.004 is about 0.25.
This is
the viewing angle which is one
quarter of one degree.
If you were on the surface of Saturn
(which is gaseous) Tethys would appear to have a diameter about
one half of that of our moon.
© 2000,
2003. Elizabeth Anne Viau.
All rights reserved. This material may be used by individuals
for instructional purposes but not sold. Please inform the author
if you use it at
eviau@earthlink.net
