2. This problem relates to Kepler's Third Law. In this problem you will use the power law to determine the period of revolution about or distance from the sun for all planets given information about some of the planets. Let d be the mean distance (x106 km) from the sun and p be the period of revolution in days about the sun. You are given the following data [1] concerning four of the planets:

Planet

Distance d

Period p

Mercury

57.9

87.96

Earth

149.6

365.25

Mars

227.9

687.0

Jupiter

778.3

4337

a. The power law expression relating the period of revolution (p) to the distance from the sun (d) is given by
p = kd a,
where k and a are constants to be determined. Use the power law under Excel's Trendline to best fit the data above. Plot the data and the best power law fit, then have Excel write the formula on your graph. How well does the graph match the data?

b. In class, we will show that an allometric (power law) model has a straight line fitting the logarithms of data, giving

ln(p) = ln(k) + a ln(d)
for the formula above. In the table above, take the logarithm of the Distance (ln(d)) and the logarithm of the Period (ln(p)). Use Excel's scatter plot and linear fit under Trendline to see how this fits the data. Plot a graph of the logarithm of the data and the best straight line fit to these data. Show the formula for this straight line on your graph. Compare the coefficients obtained in this manner to the ones found in Part a. How well does the graph match the data?

c. Use the power law found in part a. to complete the table below:

Planet

Distance d

Period p

Venus

108.2


Saturn


10,760

Uranus

2871


Neptune

4497


Pluto


90,780

d. The Jet Propulsion Laboratory has an excellent website for astronomical data. Go to their website at
http://pds.jpl.nasa.gov/planets/welcome.htm
to obtain actual data on the distance and period on the planets listed in Part c., then find the percent error between your calculations in the table above and the actual values. (Note that 1 AU = 149.6x106 km.)

[1] Jay M. Pasachof, (1993) Astronomy: From the Earth to the Universe, Fourth Edition, Saunders College Publishing.