On
the page Can
You See Your Moon we found out how to tell if our
moons would be visible from the planet surface. We discovered
that the earth's moon, which is certainly noticeable in the
sky, has only half of a degree of visibility. What does this
mean?
The night sky arches over us like a great
upsidedown
bowl.
Scientists divide this semicircle
into 180 degrees.
In the picture, I have marked these off in groups of 10 degrees
each.
The moon takes up only half of one
degree in the sky. See the tiny yellow dot with its rays.
That represents our moon.
Sometimes one sees pictures that artists have
drawn which show huge moons.
Could planets really have moons
like these?
Let's find out.
On the page about The Roche Limit
we learned that the closest that our moon can be to earth without
breaking up is about 4 earth radii.
Earth's radius is 6378 km
Earth's radius * 4 = 6378 * 4 = 25,512 km.
Let's use the Can You See Your
Moon formula to find out what our moon would look like
at its closest to the earth.
0.068 is the Tangent of one half of Theta
To find the Tangent of Theta we multiply 0.068 by 2.
0.068 * 2 = 0.136 This number is the Tangent
of Theta
When we know Theta, we know how much of the sky is covered
by the moon.
Click here for
the Tangent Table.
0.136 is between 0.123 and 0.141
If the Tangent of Theta = 0.13
then the viewing angle of Theta
is nearly 8 degrees! Wow! Compare!
Scientists believe that
the moon was formed about 4.6 billion years ago when our solar
system was forming.
They theorize that a huge asteroid about the size of Mars smashed
into our world. Some of earth's material was torn away by the
impact. The rubble from the impact may have formed rings of dust
and rocks around the earth at first, but much of it was pulled together
by gravity to form our moon. At one time both the moon and the
earth were in a molten state.
After its formation, the moon
was very close to the earth, but outside the Roche Limit. At
that distance it would have orbited the earth in about 90 hours,
or about three and a half of our days. (Earth days were also
much shorter then.) Over billions of years the moon has slowly
moved farther and farther away. It is moving away at 4
centimeters a year right now. Right now it is about 60
earth radii away. When it is twice as far away, will we still
be able to see it?
Example:
The Moon is twice as far
away:
Radius of Earth's Moon = 1738 km
Radius of the Earth =
6378 km
Distance to Earth's Moon
(60 earth radii) = 382,680 km
Multiply 382,680
by 2 to make it twice as far
away.
Twice the distance
from earth to moon = (382,680 * 2) = 765,360 km
_ 1738_
= 0.00227
765,360
This is the
Tangent of Theta
2
Round to 0.002. This is the Tangent
of half of the angle Theta.
Multiply 0.002 by 2 to get the
Tangent of Theta.
0.002 * 2 = 0.004
Click here for
the Tangent Table.
Look for 0.004 in the yellow columns.
We see 0.003 with a viewing angle of 0.2
We see 0.005 with a viewing angle of 0.3
We don't see 0.004, but it should be between 0.003 and 0.005.
So we reason that the viewing angle is probably between 0.2 and
0.3
The viewing angle for the future far
away moon will be 0.25, which is a quarter of a degree.
The viewing angle for the moon right
now is 0.5 of a degree (half of a degree).
When it is twice as far away, the viewing angle will be 0.25
of a degree (a quarter of a degree).
We would still be able to see our moon
sailing among the stars in the sky.
Additional Question:
If the moon moves away from the earth at a rate of
4 cm a year, how long will it take for the moon to be twice as
far away from earth as it is now?
What we know 
Using the Math 
The moon is
about 382,680
kilometers
away.
1000 meters = 1 kilometer.
100 centimeters = 1 meter.

Step One: Calculate distance of moon in centimeters:
382,680
km * 1000 = 382,680,000 meters.
382,680,000
meters * 100 = 38,268,000,000 cm.

The moon
is moving away at a rate of 4 cm a year. 
Step Two: Divide distance by 4 cm a year:
38,268,000,000
/ 4 = 9,567,000,000 years.
This is more than 9.5 Billion Years!

What can we figure out from this?

Our solar system is
about 4.6 billion years
old. The lifespan of our sun is about 12 billion years. Even
if the earth and moon were to survive the death of the sun,
(which is very, very unlikely) there would be no one here to look at the moon when it is twice as far
away.

The moon has reached it's present distance
from the earth in 4.6 Billion years. It must have been moving
away from the earth more quickly earlier in its history.

If the moon's rate of moving away from
the earth is changing, will it eventually move away more
quickly or more slowly than it does now? Maybe we
need to change our calculations!
For more information, we
will need to study the rate of
change of the moon's drift away from earth.
© 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. 