The Cosmological Distance Ladder

Relative Solar System Distances - Copernicus

As a consequence of the Copernican model in which the planets orbit the Sun rather than the Earth, it became possible to determine the relative distances between the Sun and the planets. How this can be done is described below.

In the three figures below, the Earth and Venus are shown orbiting the Sun, (they orbit counter-clockwise in the figures).  Because Venus is closer to the sun than the Earth, Venus orbits the sun faster than the Earth, so the three figures show Venus in three different positions relative to the Earth and Sun, (note that the Earth is moving too; I have simply drawn each figure with Earth at the bottom of the figure for simplicity).  Note that the angle between the Sun and Venus as viewed from the Earth (angle SEV which is called the elongation of Venus) increases until the line of site from Earth to Venus is tangent to Venus' orbit.  As Venus continues on its journey around the sun, the elongation again begins to decrease.

The key here is that at the maximum angle, since the line of site from Earth to Venus is tangent to the orbit of Venus, the angle SVE is a right angle. Then by definition of the sine of an angle within a right triangle,

and rearranging we can write,

Of course we don't know the distance SE, but if we let SE equal 1, (i.e. let distance measurements be in the units of AU), then we can easily determine the distance between the Sun and Venus (the distance SV) relative to the distance between the Earth and the Sun.  This is done simply by measuring the angle SEV at maximum elongation.

Step A

Start by measuring the angle between Venus and the Sun around the time of maximum elongation.  The angle will be seen to increase from night to night until it reaches a maximum, and then decrease on following nights.  The largest angle measured before it started decreasing is the angle used in the above equation.  When this is done, it is found that the maximum elongation of Venus is 46°18'.  Substituting into the above equation we get:

That's it.  That's all we need for step A.  We can measure the distance between the Earth and Venus with radar, and from there determine the distance between the Earth and the Sun, and from there, jump into stellar parallax.  And I promise to get to that very soon, but there is one other extremely important means by which Step A can be accomplished, so let's briefly consider Step A from one other point of view before moving on to Step B.