1. The Solar System Dimensions
Unit 2.3

2. Solar System Dimensions
Kepler’s laws allow us to construct a scale model of the solar system, with the correct shapes and relative sizes of all the planetary orbits, but they do not tell us the actual size of any orbit.

3. Solar System Dimensions
We can express the distance to each planet only in terms of the distance from Earth to the Sun because Kepler’s triangulation measurements used a portion of Earth’s orbit as a baseline.

4. Solar System Dimensions
Thus, our model of the solar system would be like a road map of the US showing the relative positions of cities and towns but lacking scale markers indicating distances in kilometers or miles.

5. Solar System Dimensions
If we could somehow determine the value of the astronomical unit – in kilometers, say – we would be able to add the vital scale marker to our map of the solar system and compute the precise distances between the Sun and each of the planets.

6. Solar System Dimensions
We might propose using triangulation to measure the distance from Earth to the Sun directly.

7. Solar System Dimensions
However, we would find it impossible to measure the Sun’s parallax using Earth’s diameter as a baseline.

8. Solar System Dimensions
The Sun is too bright, too big, and too fuzzy for us to distinguish any apparent displacement relative to a field of distant stars. To measure the Sun’s distance from Earth, we must resort to some other method.

9. Solar System Dimensions
Before the middle of the 20th century, the most accurate measurements of the astronomical unit were made by using on the triangulation on the planets Mercury and Venus during their rare transits of the Sun.

10. Solar System Dimensions
Because the time at which a transit occurs can be measured with great precision, astronomers can use this information to make accurate measurements of a planet’s position in the sky.

11. Solar System Dimensions
They can then employ simple geometry to compute the distance to the planet by combining observations made from different locations on Earth.

12. Solar System Dimensions
For example, the parallax of Venus at closest approach to Earth (13,000km), as seen from two diametrically opposite points on Earth is about one arc minute (1/60°).

13. Solar System Dimensions
We use the following formula to calculate the distance: 13,000km x 57.3° / (1/60°) 45,000,000km

14. Solar System Dimensions
Knowing the distance to Venus, we can compute the magnitude of the astronomical unit.

15. Figure above is an idealized diagram of the Sun-Earth-Venus orbital geometry.

16. Solar System Dimensions
The planetary orbits are drawn as circles here, but in reality they are slight ellipses. This is a subtle difference, and we can correct for it using detailed knowledge of orbital motions.

17. Solar System Dimensions
Assuming for the sake of simplicity that the orbits are perfect circles, we see from the figure that the distance from Earth to Venus at closest approach .3 AU.

18. Solar System Dimensions
Knowing that .3 AU is 45,000,000 km makes determining 1 AU straightforward – the answer is 45,000,00 / .3 or 150,000,000 km.

19. Solar System Dimensions
The modern method for deriving the absolute scale of the solar system uses radar rather than triangulation.

20. Solar System Dimensions
Radar is an acronym for radio detection and ranging. In this technique, radio waves are transmitted toward an astronomical body, such as a planet.

21. Solar System Dimensions
The returning echo indicates the body’s direction and range, or distance, in absolute terms – in km rather than AU.

22. Solar System Dimensions
Multiplying the 300 second round trip travel time of the radar signal by the speed of light, we obtain twice the distance to the target planet.

23. Solar System Dimensions
Venus, who orbit periodically brings it closest to Earth, is the most common target for radar ranging. The round-trip travel time can be measured with high precision: to an accuracy of about 1 km.

24. Solar System Dimensions
Having determined the value of the astronomical unit, we can re-express the sizes of the other planetary orbits in terms of more familiar units, such as miles or kilometers.

25. Solar System Dimensions
The entire scale of the solar system can then be calibrated to high precision.