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 ellipses are the natural paths that any two orbiting masses will take, and that each mass orbits a common center of mass (CM) with a single orbital period. Demonstration of these concepts lies beyond the scope of this class; however, we can understand why the CM of 2 objects of equal mass lies exactly midway between them, and the CM always lies nearer to the more massive object.
If the Earth were the only planet to orbit the
then the Earth and Sun would each orbit the common CM, located at a
within a few hundred km of the Sun's center (the Sun is 333,000x more
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
Sun and Jupiter combine to have 99.95% of its mass (with the Sun having
and their gravitational force is much stronger than that of any other
pair. So to a good approximation the Sun and Jupiter orbit about their
CM, lying at a point
outside the surface of the Sun, with Jupiter having the much larger
and acceleration (the Sun is 1047x more massive than Jupiter). This
lies so close to the center of the Sun that we can still refer to the
"orbiting the Sun". The effects of the other planets are to cause the
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
0.1% from being at the Sun's precise center as Kepler had proposed.
Sun being stationary at the common foci of all of the planets'
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
incorrect for two orbiting bodies of comparable mass.
Kepler's 2nd 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 force on any mass: F = m x a; a is the acceleration of mass m (Newton's 2nd Law of Motion).
The force of gravity between 2 bodies and Newton's 2nd
Fgrav = M1 (G M2 / d2) = M1 a1 = M2 (G M1 / d2) = M2 a2
Since the distance between the Sun's center and the planet's center
changes, the gravitational force between them does too. Comparing the
for gravitational force with the general force law (Newton's 2nd
Law of Motion), one can see that acceleration of M1(let's
Sun) is G M2 / d2, and that of M2(the
is G M1 / d2. That is, the acceleration of a
in its orbit around the Sun depends upon the mass of the Sun and the
square of the planet's distance from the Sun. As the planet moves
away in its orbit around the Sun, the gravitational force exerted by
the Sun on the planet
If the force exerted on the planet decreases, the planet's acceleration, proportional
to Msun/d2, must also decrease, resulting in
lower orbital speed. We won't worry about how orbital acceleration is
into orbital speed in this class. Just think of accelerating a bowling
down the bowling alley - a ball undergoing a greater acceleration from
(in your hands) attains a higher speed moving down the lane. That's all
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)
kinetic + gravitational potential = constant) also explain Kepler's 2nd
Law. In the first case, as the planet's distance from the Sun (here
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).
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 3rd Law: p2(yrs) = a3(AU), meaning larger orbits (a) have longer orbital periods (p), and the average orbital speeds are slower for planets with larger orbits.
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: p2(yrs) = a3(AU)
/ (M1 + M2); where the masses are measured in
of our Sun's mass. Newton also showed that this equation may also be
in physical units as
p2 = 4pi2a3/ G(M1 + M2)
for p measured in seconds, a in meters, and M in kg (and pi = 3.141592653...).
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!Important: The force of gravity between the Sun and a planet can be larger than that between the Sun and Earth, yet that planet's acceleration can be either larger (if closer to the Sun) or smaller (if further from the Sun) than Earth's accleration in its orbit around the Sun. This is because the force depends upon the product of both masses and the square of the distance between them, whereas the planet's acceleration depends on just the mass of the Sun and the square of the distance between the Sun and planet. For example, the force between Jupiter and the Sun is 11.7x that between Earth and the Sun, and yet Jupiter's orbital acceleration is a factor of 27 smaller than Earth's (because Jupiter lies 5.2x further away from the Sun than Earth and 5.2 x 5.2 = 27). Review the discussion of Kepler's 2nd Law, above.
Back to the 3rd 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 3rd law (M1 + M2, 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 3rd 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 3rdlaw delves into physical concepts that are beyond this class.
Here are a few questions you had better think about.