World Builders™                                                                       Session One   --   Astronomy
Can You See

Use these equations to find out if your moon will be visible when you are on your planet's surface.

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.

 0.005 * 0.01 = 0.010

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

 0.00452 * 2 = 0.00904

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

(See Page on Planets and Moons)

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

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)

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.

 0.0018 * 2 = 0.0036

Round this to 0.004

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.

 For a great Trigonometry Site, go to Ptolemy's Ptools http://library.thinkquest.org/19029/