# How do scientists measure the distance from earth to the sun?

Apr 3, 2016

Parallax method is used. I prefer the method detailed in my explanation below, for approximating 1 AU in a smaller unit of distance, with utmost precision.

#### Explanation:

Instant of time, for observation of the center C of the solar disc, is either equinox ( about Mar 21 for vernal equinox and Sep 23 for autumnal equinox).
At noon here, the Sun will be right over head. The line of centers of the Earth (E) and the Sun (S) passes through equinox.

Two locations A an B for observing parallax angle $\alpha$, as the angle between the directions, are on the equator, equidistant from the equinox, with longitudes differing by $2 \beta$.
The equinox longitude will be the average of these longitudes. Equinox is near Papa New Guinea Island. The spot on the east might be in the sea and has to be over a naval boat. The line of centers ES will bisect both the $\angle A C B = \alpha$ and $\angle A E B = 2 \beta . T h e \triangle A C B \mathmr{and} \triangle A E B$ are isosceles#.

The proof for the following formula is elementary.
The Earth-Sun distance
$E S = R + r \left(\cos \beta + \sin \beta \cot \left(\frac{\alpha}{2}\right)\right)$,
where R = radius of the Sun and r = equatorial radius of the Earth.

EA = EB = equatorial radius r of the Earth

1 AU = the average distance of the Earth from the Sun.
Mathematical average is semi-minor axis b and the instant at which SE = b is in the proximity of equinox instant. This is the reason for choosing time and locations in the above manner to get a good approximation, using the appropriate formula, in exactitude. Note the addition of R to make SE = SC + EC = precisely the distance between the centers..

ES = sum of the altitudes from S and E on the chord AB of the Earth
$E C = r \cos \beta + \left(r \sin \beta\right) \cot \left(\frac{\alpha}{2}\right)$.