The distance of the planet from the
earth is 3.7 x 10^5km. If the angular
diameter of the planet is 1.5°, calculate
the diameter of the planet?
Answers
You can calculate the angular diameter of the Earth using the equation:
a=arctanDd
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 rE=6378.1km, the diameter is therefore D=2×rE=12756.2.
The mean distance to the Moon is d=384399km. This gives a (mean) angular diameter of a=1.90065°.
At Perigee the distance is d=362600km. This gives an angular diameter of a=2.01482°
At Apogee the distance is d=405400km. This gives an angular diameter of a=1.80226°
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 rM=1738.14km. This gives the following values:
The mean distance to the Moon is d=384399−1738=382661km. This gives a (mean) angular diameter of a=1.90928°.
At Perigee the distance is d=362600−1738=360862km. This gives an angular diameter of a=2.02452°
At Apogee the distance is d=405400−1738=403662km. This gives an angular diameter of a=1.81001°
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).
Or about 2 degrees.