A few notes on gravity and Kepler’s third law. First, a couple of things that I didn’t mention in class but that are useful to know: - Distance to use in gravitational force and orbit calculations is the distance from the center of a spherical body. One of the interesting things that Newton proved as he was figuring out the law of gravity is that spherical bodies behave as if all of their mass was concentrated at their center. For example, the force of gravity you feel at the surface of 퐺(푀푦표푢푀퐸푎푟푡ℎ) the Earth is 퐹푟푎푣 = 2 , i.e. it is based on your distance from the center of 푅퐸푎푟푡ℎ the Earth, not the surface of the Earth. And for orbits around the Sun, it is the distance to the center of the Sun that matters. So when the Sun starts to expand in size to become a red giant toward the end of its life, that won’t change Earth’s orbit even though the surface of the Sun will be closer to us, because our distance from the center of the Sun will still be the same. (Eventually the Sun might get big enough to engulf the Earth entirely, which is a different story. But until its surface actually reaches the Earth, the gravitational force stays the same.) - Kepler’s second law (equal areas in equal times) is another way of expressing conservation of angular momentum. We’ll talk more about angular momentum in a few weeks, but briefly (if you haven’t seen it before), it is another conserved quantity like energy, or momentum in a straight line, but it is associated with rotation. For a body on a closed path, like an orbiting planet, it is given by mvr, where m is the mass of the body, v is its speed in orbit at any moment, and r is its distance from the focus of its elliptical orbit. This is the same principle that causes a figure skater to spin faster by pulling in his or her arms; r decreases, so v increases in proportion in order to keep mvr constant. Second, I thought I’d spell out a little more systematically the different versions of Kepler’s third law that we talked about. It can be a little confusing at first to understand what applies when, so here’s a little guidance on that. 4휋2 푃2(푀 + 푀 ) = 푎3 1 2 퐺 The equation above is the full, complete, accurate version of Kepler’s third law. That applies to any orbit of any two bodies, and can be used in any units (but of course you have to keep track of your units on G and on the other variables to make sure that things cancel appropriately). If you’re ever in doubt, you can use that equation. That said, there are sometimes some simplifying circumstances that can make things a little easier to work with. A common one is when one of the two masses is a lot bigger than the other. For example, for the Earth orbiting the Sun, 푀퐸푎푟푡ℎ + 푀푆푢푛 = 1.000003 푀푆푢푛, so just using 푀푆푢푛 in a calculation is just fine. Even Jupiter is only about 0.001 푀푆푢푛. Second, working in other units is sometimes much more convenient. The constant in the 4휋2 equation above has a value of = 5.915 × 1011 kg s2 m−3 in the units we usually use for 퐺 physics problems (so-called mks units, for meters – kilograms – seconds). However, if we choose different units for mass, length, and time, then of course the constant has different units. In particular, if we choose the convenient units of solar masses, AU, and years, then it 4휋2 turns out that = 1 푀 yr2 AU−3 ! That’s right – the constant is one in those units. (Do 퐺 푆푢푛 the unit conversion yourself if you’re not convinced!) That’s not a coincidence; the equation has to be true for Earth’s orbit around the Sun, and those three units are defined by Earth’s orbit (where we have also accounted for the fact that 푀퐸푎푟푡ℎ + 푀푆푢푛 ≈ 푀푆푢푛 as discussed above). So we can write 4휋2 푃2(푀 + 푀 ) = 푎3 1 2 퐺 2 2 −3 3 푃 (푀1 + 푀2) = 1 푀푆푢푛 yr AU 푎 Or, if we just redistribute the units from the constant to other places in the equation, 2 3 푃 푀1 + 푀2 푎 ( ) ( ) = ( ) 1 yr 1 푀푆푢푛 1 AU That is also the full version of Kepler’s third law, that applies to any orbit of any two bodies. We’ve just written the constant in different units, and spread those units around in the equation. But often this version is much easier to use if we’re talking about planets, since we’ll often have the semimajor axis already in units of AU and/or the period already in units of years. It can also be applied to orbits not involving the Sun; as long as you plug in the masses in units of solar masses, you’re good to go. Finally, when we’re talking about orbits of small bodies around our own Sun, the sum of the masses is generally essentially equal to the mass of the Sun, so the mass term in the above equation ends up being one. So in that special case, we have 푃 2 푎 3 ( ) = ( ) 1 yr 1 AU Just be aware that this is a special case, and it’s only true if the total mass is 1 푀푆푢푛. .