The Cosmological Distance Ladder

Determining the Astronomical Unit

Having determined the relative distances between planets and the Sun within the solar system, we now wish to convert those relative distances into an absolute distance such as miles or kilometers.  The critical one that we are seeking is the distance between the Earth and the Sun.. the astronomical unit, because it is key to stellar parallax measurements.  Since we now know solar system distances relative to the astronomical unit, all we need is to make one absolute measurement and all the rest falls into place.  An absolute solar system measurement was finally achieved in 1961 when a radar signal was bounced off the planet Venus from the Jet Propulsion laboratory in California.  By measuring the time between sending the radar signal and the time we received the reflection of the signal from Venus, we can easily calculate the and absolute distance between the Earth and Venus.

In step A, we found that the distance between the Sun and Venus is 0.723 AU.  As shown in the diagram below, when the Earth and Venus are directly in line with the Sun, and Venus lies between the Earth and the Sun, the distance between Earth and Venus is 0.277 AU, (i.e. 1 AU - 0.723 AU = 0.277 AU).  If we bounce a radar beam off Venus at this time, we can determine the distance to Venus.

The radar beam travels at the speed of light which is 300,000 km/sec.  Since the radar beam travels from the Earth to Venus and back to Earth again, the total distance traveled is twice the distance between the Earth and Venus which is 0.554 AU, (i.e. 2 X 0.277 AU = 0.554 AU).  To determine the distance between Earth and Venus, all we need to do is measure the time between sending the laser signal and receiving it back, and then apply the basic relationship: Distance = Velocity X Time.  In this case the velocity refers to the velocity of the radar signal which is equal to the speed of light, and the time for a radar signal to make the round trip between Earth and Venus is measured to be 276.2 seconds.

But since this is the round trip distance for the radar signal, the distance between the Earth and Venus will be half this distance:

Since we also know that the distance between the Earth and Venus at this time was 0.277 AU we a conversion factor between km and the astronomical unit, (i.e. 0.277 AU = 41,430,000 km).  We could use this conversion factor as it, but it is much more convenient to divide both sides by 0.277 so that the conversion factor is in terms of 1 AU.

And this has been the holy grail up to this point.  As you will see in the next section, distances determined from stellar parallax depend on knowing the distance between the Earth and the Sun.

Before rushing on to the next section though, I hope you will take few minutes to read about the transit of Venus.  This was another method by which the distance between the Earth and the Sun was sought before the advent of radar.  What makes this especially interesting is that the transit of Venus is very rare, but we are just fortunate enough that two will occur in most of our lifetimes. Venus Transit