# What is the angular diameter of Earth as seen from the Moon?

...today I must have left my brain home. I know it's a simple calculation but I keep failing simplest calculations today.

What is the angular diameter of Earth as seen from the surface of the Moon?

• google.com/… – LocalFluff Oct 21 '14 at 14:56
• @LocalFluff: Gives me a thousand results of angular size of the Moon as seen from Earth. – SF. Oct 21 '14 at 14:59

You can calculate the angular diameter of the Earth using the equation: $$a = \arctan \frac{D}{d}$$ where $a$ is the angular diameter, $D$ is the physical diameter of the Earth, and $d$ is the distance from the Moon to the Earth.

The equatorial radius of the Earth is $r_E = 6378.1 \textrm{km}$, the diameter is therefore $D= 2 \times r_E = 12756.2$.

These Moon-Earth distances are as seen from the centre of the Moon. To calculate the diameter from the surface of the Moon, you'll have to subtract the position of the observer along the Moon-Earth axis.

If the observer is on the Moon's equator and the Earth is at zero hour angle (i.e. on the local meridian), the distance to the Earth needs to be subtracted by $r_M=1738.14\textrm{km}$. This gives the following values:

The angular diameter of the Earth from the surface of the Moon is, therefore, between $a=1.80226$° (at apogee and the Earth is near the horizon) and $a=2.02452$° (at perigee and for an observer at the equator and when the Earth is at maximum altitude on the meridian).

The Earth viewed from moon will appear larger, in proportion to how much larger the Earth's diameter is versus the moons diameter.

• Earth diameter 7900mi
• Moon diameter 2100mi

So the Earth-view from the moon would appear 3.75 times as large as the Moon appears in the sky on earth.

I looked up the moon's typical angular diameter, it is 0.5 degree. So the Earth's typical angular diam would be 1.9 degree.

The average angular diameter of the Moon, as seen from the Earth, is about 31 arcminutes.

The angular diameter depends on the distance between the two objects and the diameter of the object being viewed. Specifically, for small angles, it is the diameter divided by the distance. When the distance is the same, the angular size is proportional to the diameter.

The distance remains the same when viewing the Earth from the Moon, but the Earth is larger. According to NASA, the diameter of the Moon is 3,476 km, and the diameter of the Earth is 12,756 km.

So, because it's proportional, the angular diameter can be calculated as follows:

$a_{Earth} = a_{Moon} \times {d_{Earth}\over d_{Moon}}$

$= 31 arcminutes \times {12,756 km \over 3,476 km}$

$\approx 114 arcminutes$, or just under 2 degrees.

This is approximate, because not only is this valid only for small degrees, where the tangent of an angle can be approximated by the angle itself (in radians), the Earth-Moon distance varies because the Moon's orbit around the Earth is an ellipse.