Has this ever happened to you?
A friend calls you up all excited, and tells you there’s a SUPERMOON, so you rush outside
and...meh?
MAYYYBE it’s a little bigger than usual?
But you’re not really sure?
Of course, the moon isn’t ACTUALLY getting any bigger.
But because it has an elliptical orbit, the moon isn’t always exactly the same distance
from the Earth.
One side of the Moon’s path is closer to Earth than the other, so its APPARENT SIZE
varies.
The truth is, a supermoon just APPEARS a LITTLE bit bigger, because it’s a LITTLE bit closer.
Supermoons get a lot of press, but, really, it’s just a small variation in apparent
size.
Astronomers often talk about the ANGULAR SIZE of objects as seen from the Earth, rather
than their ACTUAL sizes.
You’ll sometimes hear the phrase “to subtend an angle”—that means, to cover an angle.
So for instance, the Moon typically subtends, or covers, an angle of about one-half of a
degree.
The farther away it is, the smaller that angle.
We say the angular size decreases, the farther away an object is, and the smaller its apparent
size.
But remember, its ACTUAL size hasn’t changed.
In order to determine the angular size, we can use a little TRIGONOMETRY to describe
this TRIANGULAR arrangement: The diameter is Capital D.
Lowercase d is the distance to the object.
And theta is the angle.
For a right triangle, the tangent of an angle equals the opposite side divided by the adjacent
side, so Tangent of this angle theta is equal to Big
D over little d.
If theta is a small angle measured in RADIANS, we can use the Small Angle Approximation,
whereby tangent of theta is approximately theta.
That means theta approximately equals Big D over little d.
Now let’s look at the way astronomers typically write the formula to quantify this relationship:
The Small-Angle Formula.
This formula is the mathematical way to say: for a given diameter, the ANGULAR SIZE decreases
the farther away an object is.
Are you ready?
Here it is: alpha equals 206,265 big D over little d.
Again, Capital D is the diameter, or linear size.
Lowercase d is the distance.
And Alpha is the angular size, in arc-seconds.
You may be asking yourself: where does that 206,265 come from?
This is how we get the units of alpha to be in arc-seconds.
Arc-seconds are usually preferable to radians or degrees in astronomy, since we’re typically
dealing with very distant objects, and very small angles.
Remember there are 360 degrees in a circle (and the circumference is 2pi radians.
Each angle corresponds to 60 arc-minutes, and each one of those minutes corresponds
to 60 arc-seconds, for a total of 1, 296,000 arc-seconds.
Divide by 2pi to get 206,265.
Notice that we didn’t define any units for the distance.
With this formula, you can use any units you like—as long as they’re the same for big
D and little d.
So you can use kilometers, light years, parsecs...remind me to tell you about astronomers’ favourite
distance units another time.
Let’s plug in some values for that supermoon situation.
At perigee, the closest point, the full moon is about 360,000 kilometers from Earth.
This is when we call a full moon a supermoon.
By the way, there’s also a MICROMOON, which is what happens when the full moon is at apogee,
the farthest away from Earth (about 405,000 kilometers).
We know the linear size (or the diameter) of the moon.
It’s about 3,475 kilometers, if you average out all those craters.
Let’s calculate the apparent size, or angular size, of the supermoon.
Our formula is alpha equals 206,265 times big D, over little
d.
So at perigee, alpha equals (206,265 times 3,475 kilometers ) divided by (360,000 kilometers)
equals 1991 arc-seconds.
{not sure}..Is that a lot?
Let’s compare it with the micromoon seen at apogee.
In that case, alpha equals (206,265 times 3,475 kilometers ) divided by(405,000 kilometers)
= 1770 arc-seconds.
Now we can see the maximum difference in how big the moon appears.
1991 divided by 1770 = 1.12, or we can say there's about 12% difference between the biggest
apparent size and the smallest apparent size.
Here’s a challenge for you—we can do the same kind of calculation for the Sun.
Hey, hold on! Wait a minute.
—DON’T LOOK DIRECTLY AT THE SUN.
Let’s just...TALK about it, okay?
Post your answers on your favourite social media, and don’t forget to tag Socratica
so we can come say hi.
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