Unit 3 : Activity 3 : Inventory of the Solar System : Figuring Out Solar System Distances : Unit Exam









Figuring Out Solar System Distances

Now that we have given you the answer to the question of how big the objects in the solar system are and at what distances, we need to go back and look at how the astronomers figured this out over time. After all, when you look in the night sky with the naked eye, all you see are small dots of light.

The first step the astronomers of the past did was to figure out how far apart the various planets are from the Sun. In the case of planets, we have to wait for the Earth to move some significant distance along its orbit. At the same time, we have to watch one of the planets in its orbit. In this case, we will use Mars since the red planet is fairly easy to observe. From simply watching Mars in the sky, we know that it takes 686 days (1.88 Earth years) for Mars to complete one circuit around the Sun.

For simplicity sake, we call the distance from the Sun to the Earth 1 Astronomical Unit or "AU" for short. Using geometry, we can show that the distance from the Sun to Mars is 1.52 AU. In other words, by watching how the position of Mars changes with respect to Earth over the course of a year, we can determine how far Mars is from the Sun compared with how far Earth is.

We call the average distance from the Earth to the Sun the Astronomical Unit, or AU. It turns out to be easy to find the distances to the other planets in astronomical units but very hard to do the same in km or miles. Copernicus had the information needed to find the relative distances in AU, and Kepler used these distances to get his third law (period-squared vs. size-cubed).

Here is an example of how you can figure out how far Mars is from the Sun compared with Earth. Assume that both Earth and Mars have circular orbits. (That is a very good first approximation, even for Mars.) Keep track the dates when Mars is highest to the South at midnight, 6am, and 6pm. The following picture will explain your observations:

In this picture you can see that, moving at constant speed, Mars will take more time to go around the long arc (by "noon") than the short arc (by "midnight"). The farther away from Earth and the Sun that Mars is, the more nearly equal these two arcs will be. So by keeping track of these times, something astrologers had been doing for centuries, it is possible to figure out the sizes of the orbits.

For Venus, we can use a different trick to figure out its distance from the Sun in AU. The picture in this case shows that Venus can't get too far from the Sun. If we can find the farthest it gets from the Sun, we can draw (on paper) a triangle that is similar to the big triangle in space linking the Sun, Earth, and Venus:

Here Venus is as fas as it can get from the Sun in the Sky, as seen from earth. We can measure the angle a. Angle b is a right angle. This lets us draw a similar triangle on paper and find the distance from Venus to the Sun compared with the distance from the Sun to the Earth.

You may have noticed one problem with the above answers - the distances to the other planets are given in terms of the distance of the Earth to the Sun. So, how far is the Earth to the Sun? What exactly is an AU? How far is an AU? How many inches? Centimeters? Miles? Kilometers?

The answer to this problem was not so easy to solve by purely observational methods. Two things people tried were to observe a transit of Mercury or Venus and to measure the closest approach to Earth of an asteroid. Both gave good approximate numbers, and allowed us to estimate the mass of the Sun. Good values were not obtained until the 1960s when scientists bounced radar beams off the planets such as Mars, Mercury and Venus and were then able to calculate the average distance from the Earth to the Sun accurately.

For the record, the average distance from the Earth to the Sun, or 1 AU, is:

149,600,000 km
1.496 E13 cm
92,960,000 miles

 


1) How far is Mars away from the sun?

2.27 x 108km
1.496 x 108km
1.08 x 108km
0.72 AU
Note: Scientific Notation can be represented three ways. For example, 1.496 x 108km is the same as 1.496E8 km, and sometimes people will write 1.496 10^8. Calculators often use the E8 or e8 notation while books use the 108 style and some computer languages use 10^8.


So, why do we want to know how far the planets are from the Sun? Sure, it is interesting to know. But is there some practical reason? The answer is, yes. It is very useful to know the distances. Once we know the detailed distances of the orbits of the planets, we can then calculate the mass of the Sun. We can also find out the masses of planets and of distant stars using Kepler's laws, but that requires something more; we can not measure the mass of Earth by observing Earth's orbit around the Sun, and we can not measure the mass of Jupiter by observing Jupiter's orbit about the Sun.