You should know by now that **the second part of this law cannot
be quite
correct**. Newton's Laws of Motion demand that the planets *and
the Sun*
must accelerate, *because the planets pull on the Sun via the force
of gravity
just as hard as the Sun pulls on them* - the Sun
cannot
remain stationary. Newton's laws of motion together with the Law of
Gravity
demonstrate that

**If** the Earth were the only planet to orbit the
Sun,
then the Earth and Sun would each orbit the common CM, located at a
point
within a few hundred km of the Sun's center (the Sun is 333,000x more
massive
than Earth, so the CM lies this factor closer to the Sun's center
than to the
Earth's center) with a period of 1 year. However, in our actual solar
system
the
Sun and Jupiter combine to have 99.95% of its mass (with the Sun having
99.85%),
and their gravitational force is much stronger than that of any other
planet-Sun
pair. So to a good approximation the Sun and Jupiter orbit about their
CM, lying at a point
just
outside the surface of the Sun, with Jupiter having the much larger
orbit
and acceleration (the Sun is 1047x more massive than Jupiter). This
point
lies so close to the center of the Sun that we can still refer to the
planets
"orbiting the Sun". The effects of the other planets are to cause the
Sun
to loop around a bit (i.e., yet other accelerations) about a grand
solar system CM in roughly 12 years (Jupiter's
orbital period being 11.86 years). (Those who of you who are more
curious may visit this
site for more information, but you needn't worry about
understanding at that level of detail.)

Note that this CM point along the Sun-Jupiter distance differs by
just
0.1% from being at the Sun's precise center as Kepler had proposed.
Kepler had
the
Sun being stationary at the common foci of all of the planets'
elliptical
orbits, but you should know why this is almost *but not quite right*
for our Solar System. You should also understand, however, that this is
grossly
incorrect for two orbiting bodies of comparable mass.

**Kepler's 2 ^{nd} Law:** A line connecting the Sun and
planet
sweeps out equal areas in equal times. That is, the orbital speed of
any
one planet varies inversely with its distance from the Sun (actually,
orbital
speed varies inversely with the square-root of the distance, but you
needn't
worry about that detail).

This one you can understand conceptually, once you understand Newton's Laws of Motion and Law of Gravity. If the planet changes its distance from the Sun as it orbits, then the force of gravity between them must change. If the force that the Sun exerts on the planet increases (as the planet moves closer), then the acceleration of the planet must increase, resulting in a higher orbital speed, and vice versa.

**See? You don't need a single equation!** Let's review where
this
comes from, referring back to a couple of equations only as
reminders of the relationships between physical concepts:

For *any* **f**orce on *any* **m**ass**: F = m
x a;
a** is the acceleration of mass

The force of gravity between 2 bodies and Newton's 2^{nd}
Law
of Motion:

**F _{grav} = M_{1 }(G M_{2}
/ d^{2}) = M_{1 }a_{1}
= M_{2} (G M_{1}
/ d^{2}) = M_{2 }a_{2}**

Since the distance between the Sun's center and the planet's center
(d)
changes, the gravitational force between them does too. Comparing the
expressions
for gravitational force with the general force law (Newton's 2^{nd}
Law of Motion), one can see that acceleration of M_{1}(let's
say the
Sun) is G M_{2 }/ d^{2}, and that of M_{2}(the
planet)
is G M_{1 }/ d^{2}. That is, __the acceleration of a
planet
in its orbit around the Sun depends upon the mass of the Sun and the
inverse
square of the planet's distance from the Sun__. As the planet moves
further
away in its orbit around the Sun, the gravitational force exerted by
the Sun on the planet
decreases.
If the force exerted on the planet decreases, the planet's acceleration,** proportional
to M _{sun}/d^{2}**, must also decrease, resulting in
a
lower orbital speed. We won't worry about how orbital acceleration is
converted
into orbital speed in this class. Just think of accelerating a bowling
ball
down the bowling alley - a ball undergoing a greater acceleration from
rest
(in your hands) attains a higher speed moving down the lane. That's all
you
need to understand, and you already knew that.

The two very important conservation
laws in nature: that of angular momentum (r x m x v = constant)
and
energy (here:
kinetic + gravitational potential = constant) also explain Kepler's 2^{nd}
Law. In the first case, as the planet's distance from the Sun (here
called 'r')
decreases, it's orbital speed must increase in order that its orbital
angular momentum (r x m x v) remains constant. Just the opposite occurs
as the planet's distance increases from the Sun. In the second case, as
the planet's distance from the Sun decreases, so does its gravitational
potential energy. Thus the planet's kinetic energy (proportional to the
square of its orbital speed) must increase to compensate to keep the
total energy =
sum of kinetic + potential energies constant (conservation of energy).
And of
course, just the opposite applies as the planet's distance increases
from the Sun - its potential energy increases, thus its kinetic energy
(and so orbital speed) must decrease.

**Kepler's 3 ^{rd} Law:** p

After deriving gravitational orbital motion from the laws of motion,
law of gravity and quite a bit of calculus, Newton found that Kepler's
3rd Law should actually be this: p^{2}(yrs) = a^{3}(AU)
/ (M_{1} + M_{2}); where the masses are measured in
units
of our Sun's mass. Newton also showed that this equation may also be
expressed
in physical units as**
p ^{2} = 4pi^{2}a^{3}/ G(M_{1} + M_{2})**

for

How can we understand this relation
between
the period of the orbit, mean distance, and sum of masses?* Just as
we did
above*. Recalling the underlined statement above, the acceleration
of
a more distant planet due to the force of gravity between it and the
Sun must
be smaller than the acceleration of a planet near to the Sun.
Result:
a more distant planet must orbit the Sun at a lower average orbital
speed.
Slower orbital speed and a larger orbit mean that its orbital period
must
be longer! **No equations!**

Back to the 3^{rd} law. Note that the relationship between
the
orbital period and the average separation between the masses actually
depends
upon the sum of the two masses orbiting the common center of mass. That
this relationship should involve mass
should
make some sense given what you know about the relationship between
force,
mass, and acceleration, and that the force of gravity itself depends
upon
masses of the bodies involved. *Surely, you can imagine that if the
star
that Earth orbited were more massive than our Sun, Earth's acceleration
would
be greater, it's orbital speed faster, and so its orbital period shorter*.

Note that for the case of our Solar System's planets orbiting the
Sun,
the sum of the masses correction to Kepler's 3^{rd} law (M_{1}
+ M_{2}, measured in units of our Sun's mass) amounts to a
factor
of at most 1.000955 (for the Sun + Jupiter = 1 + 1/1047). Thus Kepler
had it
nearly
right in his description of the orbits of the objects around our Sun.
On
the other hand, **this mass correction term is important**, i.e.,
significantly
different from a factor of 1, for all other cases involving the orbits
of
two masses when one of them is not our Sun (e.g., Earth-Moon,
star-star, or
even planet-satellite).

So does the __mass of the planet__ have a significant impact
upon its
orbital period (or orbital speed) about some star? Given that planets
are
by definition almost always much less massive than the stars they
orbit,
the practical answer is "NO." Is there *any* effect? Well, yes of
course
there is, since the force of gravity does consider the masses of pairs
of
objects. Look at Newton's corrected version of Kepler's 3^{rd}
law again. It does indeed include the masses of __both__ objects.
What
if Jupiter with its mass of 318 Earths orbited the Sun at Earth's
location
rather
than Earth (ignoring the presence of nearby Mars and Venus)? Well, then
the mass correction factor would be 1.000955
instead
of that of 1.000003 for Earth. In that case the orbital period of
Jupiter in Earth's orbit would be a tiny, tiny amount shorter (1.000476
times
shorter). How the sum of the two masses
comes into Kepler's 3^{rd}law delves into physical concepts
that are beyond this class.

**Here are a few questions you had better think about.**

- What would happen to the Earth's orbital period or orbital speed, if Earth had its present orbit around a star with 1/2 the Sun's mass? or twice the Sun's mass? A qualitiative answer is all that I require. i.e., does the period become longer or shorter? does the orbital speed decrease or increase?
- How can we determine the actual masses (in kg) of the Sun,
planets,
and moons? i.e., what information is needed in order to make these
measurements?

Two final notes to the more astute student of physics: (1) The conservation of angular momentum is somewhat more complicated than presented here (containing a vector cross product between the radius and momentum vectors, and some other details). Kepler's 3