Question

How do I do the calculations for this chart (Table #2)?

Constructing the Model

Table 1 gives current measurements for the actual sizes and orbital distances of the nine planets.

Table 1: Measured Astronomical Distances in Solar System (*Kuiper Belt Object radii are not well known)

Object	Radius (km)	semi-major axis (km)
Sun	6.96 x 105	--
Mercury	2.44 x 103	5.83 x 107
Venus	6.05 x 103	1.08 x 108
Earth	6.38 x 103	1.50 x 108
Moon	1.74 x 103	3.84 x 105 (avg. distance from Earth)
Mars	3.40 x 103	2.27 x 108
Ceres	4.73 X 102	4.14 X 108
Jupiter	7.14 x 104	7.78 x 108
Io	1.82 x 103	4.22 x 105 (avg. distance from Jupiter)
Ganymede	2.63 x 103	1.07 x 106 (avg. distance from Jupiter)
Saturn	6.03 x 104	1.43 x 109
Titan	2.58 x 103	1.22 x 106 (average distance from Saturn)
Uranus	2.56 x 104	2.87 x 109
Neptune	2.43 x 104	4.50 x 109
Pluto	1.19 x 103	5.91 x 109
Charon (moon of Pluto)	6.35 x102	1.96 x104 (avg. distance from Pluto)
Quaoar*	5.84 X 102	6.49 X 109
Eris*	1.16 X 103	1.02 X 1010
Sedna*	7.45 x 102	7.51 x 1010

As you can see, even when expressed in the one of the largest units (km) used to describe Earth-bound distances, the sizes of and distances to the planets require numbers raised to large powers of ten. In order to fully appreciate the relative sizes and distances within the solar system, it is necessary to scale these numbers down to values small enough so that we can "see" them in terms of more familiar distances. We can accomplish this by dividing every number in Table 1 by some constant scale value.

To determine the scale value you'll need to know how much space you have. Suppose the length of a hallway in the campus in meters is 10 meters. We can choose a scale factor, so that we can fit all the planets from the Sun to Uranus in this hallway. Then, the scale value can be obtained through the following procedure:

If 10 meters are assigned to 2.87 x 10⁹ Km

• Then the scale factor for distances from the Sun is: 1 meter / (2.87 x 10⁸ Km)

For the size of the planets, we can choose in our scaled model the radius of the Sun to be 10 centimeters. Then, the scale value can be obtained through the following procedure:

If 10 centimeters are assigned to 6.96 x 10⁵ Km

• Then the scale factor for radii is: 1 centimeter / (6.96 x 10⁴ Km)

Use the scale factors to calculate the size of your object and the distance of the object from the Sun (round two decimal digits). Fill in these values in Table 2. To make it easier to make the model, find the distance from the previous object to the current object. Again, record the distance in Table 2.

As an example, below you will find the calculations for the first three rows:

• Scaled radius of the Sun: 6.96 x 10⁵ Km * [1 cm / (6.96 x 10⁴ Km)] = 10 cm
• Scaled radius of Mercury: 2.44 x 10³ Km * [1 cm / (6.96 x 10⁴ Km)] = 0.04 cm
• Scaled Distance Mercury to Sun: 5.83 x 10⁷ Km * [1 meter / (2.87 x 10⁸ Km)] = 0.20 m
• Scaled radius of Venus: 6.05 x 10³ Km * [1 cm / (6.96 x 10⁴ Km)] = 0.09 cm
• Scaled Distance Venus to Sun: 1.08 x 10⁸ Km * [1 meter / (2.87 x 10⁸ Km)] = 0.38 m
• Scaled Distance Venus to Mercury:

[(distance Venus to Sun) – (distance Mercury to Sun)] * Distance Scale Factor =

[(1.08 x 10⁸ Km) – (5.83 x 10⁷ Km)] * [1 meter / (2.87 x 10⁸ Km)] = 0.17 m

OR

• From Scaled Distances table directly:

[Venus Distance from Sun – Mercury Distance from Sun = Distance of Venus from Previous Planet] = [0.38 m – 0.20 m = 0.17 m]

Table 2: Scaled Distances
Object	Radius (cm)	Distance from Sun (m)	Distance from Previous Planet (m)	Distance of Moon from Planet (m)
Sun	10.00	0.0	0.0	N/A
Mercury	0.04	0.20	0.20	N/A
Venus	0.09	0.38	0.17	N/A
Earth				N/A
Moon		N/A	N/A
Mars				N/A
Ceres				N/A
Jupiter				N/A
Io		N/A	N/A
Ganymede		N/A	N/A
Saturn				N/A
Titan		N/A	N/A
Uranus				N/A
Neptune				N/A
Pluto				N/A
Charon		N/A	N/A
Quaoar*				N/A
Eris*				N/A
Sedna*				N/A

Answer 1