Orbits of the planets and centripetal acceleration

[sweetchoco1]

1. The table below lists the average distance from each planet to the Sun and the
time each planet takes to make a complete orbit around the Sun (“Orbital period”).
The orbits of all of the planets are roughly circular. Your job is to determine a
relationship between the radius r of a planet’s orbit (in meters) and the centripetal
acceleration arad of the planet (in m/s2). In particular, you are to see whether these
two quantities can be related by an equation of the form

a_rad = Crⁿ,

where C and n are constants. You are then to determine the values of C and
n. Based on your results, determine whether the acceleration of the planets increases,
decreases, or stays the same as the Sun-planet distance increases, and discuss
whether the size of the planet seems to affect its acceleration. Note that if you
take the natural log of the above equation, you get

ln a_rad = n ln r + lnC.

Thus to find the value of n, you will find it helpful to make a graph of ln arad versus
ln r. Can you see why?

Planet Average distance from Sun (106 km) Orbital period (years)
Mercury 57.9 0.241
Venus 108.2 0.615
Earth 149.6 1.000
Mars 227.9 1.88
Jupiter 778.3 11.86
Saturn 1429 29.46
Uranus 2871 84.10
Neptune 4498 164.86
Pluto 5915 248.60



2. a_rad=(4*pi²*R)/(T²)




3. I know how to solve for centripetal acceleration. I just don't get how to get C and n.

 

[rock.freak667]

From a_rad=(4*pi²*R)/(T²)

Find the various a_rad for each T.

Then you have a_rad = n ln r + lnC, which is in the form Y=MX+C, so what is Y and X?