Cosmic Calculations
We now have a collection of tools adequate to explore a number of truly cosmic
phenomena.
We have tools with which we can
 weigh planets and stars
 comprehend the motions of planets in a solar system
 understand the shape of solar systems like ours
 explore how solar systems are formed
This is a pretty incredible outcome for a few laws of motion! Let’s look at a few
examples.
Although Kepler showed the orbits of the planets to be ellipses, most of the orbits in our
solar system are nearly circular. If we simplify the treatment to circular orbits, you have
the tools to weigh the Sun, weigh those planets with moons orbiting them, and understand
the relationship between orbital radii and orbital velocities.
Mass of the Sun:
Using the orbital radius for the Earth and its orbital period, we can calculate the mass of
the Sun.
r = 1AU = 1.5 x 1011 m
T = 1yr = 2.3 x 107 s
First, since the Earth’s orbit is approximately a circular path with constant velocity, we
can write an expression for the size of the force holding the Earth in orbit. This
expression for the net force is
We also know what the actual force responsible is and an expression for it. The
expression for the force of gravity, discovered by Newton, is
Setting the two expressions equal gives
Cancel common variables and represent the orbital speed in terms of the orbit
circumference and period to give an expression with
M(sun) = stuff ra/Tb where a and b are small integers and stuff consists of constants.
You have now constructed Kepler’s third law for planetary orbits, which relates the
period and radius of a planet’s orbit.
Using the data for Earth’s orbit above, calculate the Sun’s mass.
Now, using the radii for the orbits of the planets given below, calculate the orbital periods
for them. How do your calculated values compare with the measured data given in the
table? A handy value for the constants is GMsun/4π2 = 1 (AU)3/yr2.
Planet
Venus
Neptune
Jupiter
Pluto
Orbit radius
1.08 x 1011 m
4.5 x 1012 m
7.8 x 1011 m
5.9 x 1012 m
Period (Earth years)
0.612
163.7
11.86
248.0
The orbits of Venus and Neptune are nearly perfect circles, while that of Pluto is the most
eccentric of all the planets in the solar system.
You have already had a bash at calculating Jupiter’s mass from the orbits of its moons.
The large moons in our solar system have the following data:
Moon
Orbit radius (Gm)
Period (days)
Planet
Moon
Io
Europa
Ganymede
Callisto
Titan
Triton
0.384
0.422
0.671
1.07
1.88
1.22
0.354
27.3
1.77
3.55
7.15
16.7
16.0
5.88
Earth
Jupiter
Jupiter
Jupiter
Jupiter
Saturn
Neptune
Using the above and the relation
(G/4π2) M(planet) = r3/T2 ; r in AU,
T in days
with G/4π2 = 825 (Gm)3/dy2*MNeptune
You can calculate the masses of Jupiter, the Earth, Saturn and Neptune in units of
Neptune masses.
Formation of solar systems.
In space, there are many large clouds of gas and dust. Most are spinning about some
axis, as there are many more possible spinning configurations than there are non-spinning
ones. Since the cloud has mass, gravity pulls the material toward the center. The
material moving toward the center in a direction parallel to the axis of spin (think of
material falling toward the Earth from above the North and South poles) gathers in the
central plane of the sphere.
As the material in the center disk is pulled toward the center, what does conservation of
angular momentum say about the rate of spin (or orbital velocity) of the material?
Since this material is moving in a roughly circular motion about the axis, centripetal
acceleration and circular motion equations apply. As the material in the disk moves
inward, the question becomes when does it stop, or does it all collapse into a black hole
in the center? Use your answer above and the formulas for centripetal acceleration and
angular momentum to determine if the material reaches equilibrium before it collapses to
the center.
L = r x mv ac = v2/r , the velocity needed to stay in orbit at radius r.
Many moons and planets are scarred with impact craters from meteorites which fell to the
surface from space. The energy of impact is so great that the rock is melted as well as
blown into the sky. Using gravitational potential energy as PE = -GMm/r where r is the
distance from the planet, M the planet mass and m the rock mass, calculate the impact
velocity and kinetic energy of a 100 kg meteorite striking the surface of the Earth.
Assume it started from rest at infinite distance and apply energy conservation.