Full transcript
0:00I do have a simulation this might be
0:01kind of tricky to picture especially
0:04just from a 2d image so let's take a
0:06quick look at a simulation so this is
0:08the same thing a Sun and a planet
0:10orbiting it and right now this is the
0:12direction of its velocity vector right
0:13here but this blue arrow is the force of
0:17gravity so you can see the force of
0:20gravity pulls it towards the Sun looping
0:24around the Sun and you can see it's just
0:27pulling closer as the they get closer
0:30the force gets much much larger that's
0:32the magnitude of the arrows showing how
0:34big the force is you can see it's always
0:36the same but in opposite directions
0:38because each one is pulling on the other
0:40and now if I wanted to I can actually
0:42increase the mass of the Sun in the
0:43simulation and what its gonna do is
0:45observe a bigger force and actually
0:47cause the plant to get closer and
0:51actually if I keep increasing the mass
0:53of the Sun then it's gonna hurt a bigger
0:57and bigger force and if I really go
0:59crazy
1:00well eventually it spirals in and
1:02crashes in this on thankfully not having
1:04to us because well the mass isn't really
1:06changing we're gonna see so hopefully
1:09that helps demonstrate it and again this
1:12explanation is why objects are orbiting
1:15Earth as well including things we put
1:17into space ourselves and you might have
1:20seen images of the astronauts in the
1:24space station and they're all floating
1:26and this tends to cause a lot of
1:27confusion to people they think well if
1:29there's a force the force of gravity
1:32from the earth acting on them why is it
1:35that they're floating around it's cuz
1:38the force of gravity is acting on
1:40everything the space station and the
1:43astronauts and as we just saw basically
1:46it causes them to be orbiting they're
1:48being pulled towards Earth the space
1:50station is constantly falling towards
1:52Earth and I usually describe it as it's
1:54falling towards Earth but constantly
1:55missing it's falling towards Earth
1:57because it had but since it has inertia
1:59it's also spiraling around the Earth
2:03constantly falling towards the Earth and
2:05the space station and astronauts are
2:07falling at the same speed and so it
2:10causes this idea of weightlessness in
2:12fact this is how they simulate
2:13weightlessness on Earth what they do is
2:15get a plane get it flying really high
2:17and then nosedives towards the earth and
2:20because everything's falling at the same
2:21speed everyone becomes weightless this
2:24is just in this case more of a permanent
2:26kind of thing because the space isn't
2:27completely looping around and around and
2:29around the planet it is worth noting
2:33that depending on how you put something
2:34into orbit it could eventually start
2:37falling towards the earth if the
2:39atmosphere acts on it and causes
2:42friction and makes them lose energy but
2:44otherwise these astronauts are falling
2:46around the earth and that's why they're
2:48weightless now all of these
2:51considerations that we've been talking
2:54about actually changes Kepler's law
2:57slightly those three laws we have to
3:00actually revisit two of them because we
3:02saw that the first law was all about the
3:06idea that we had planets orbiting the
3:08Sun in any ellipse and we said that the
3:10Sun was at one of the focal points
3:13technically that's not true because the
3:17Sun exerts a force on us we exert a
3:20force on it too and thus we end up
3:23actually having the center of mass of
3:26the planet Sun system at a focus okay
3:28well what does that mean let's take a
3:30look the center of mass is kind of what
3:32sounds like if you have two objects well
3:36if you had one big object the center of
3:38mass would be the very middle point
3:40where it's balanced the point where all
3:43the mass would be evenly distributed on
3:45all sides so if you found the center of
3:47mass you would actually be able to put
3:49your finger there and balance it in the
3:51case of two objects orbiting it's where
3:54you sum up the mass and where the middle
3:56between them would be in the case of two
3:58objects the same size it'd be the middle
4:00in the case of something like us and the
4:03Sun where most of the mass is the Sun
4:05you can see the center of mass is much
4:07closer to the Sun in fact it's within
4:11the Sun and this is true in our case so
4:14as much as this modifies Kepler's law
4:16instead of it being the Sun at the focus
4:19but the center of the mass well the
4:20center of mass is still within the Sun
4:22so it's this doesn't change it much it
4:26just puts
4:27lightly to decide a little bit more but
4:29it's still within the Sun itself the
4:32real modification is to Kepler's third
4:36law Kepler's third law
4:38remember we said it was proportional aq
4:41proportional P squared and that's
4:42because there was a missing value here
4:45we find that it actually depends on the
4:48masses of the two objects as well so the
4:52cube of the semi-major axis is actually
4:55equal to the orbital period squared but
4:58that's times the sum of the two masses
5:01now the reason this wasn't noticed at
5:03first is because when we're looking at
5:05the Sun the Sun has way more mass than
5:08any of the other objects and we're
5:11putting this number in terms of solar
5:13masses that's the important distinction
5:15so how many times the mass of our Sun
5:17well for the Sun it's 1 and for the
5:20planets it's a fraction a tiny fraction
5:22so this number is basically 1 and that's
5:24why this form at first wasn't noticed we
5:27didn't know this modification was
5:29required but it's incredibly important
5:33and incredibly useful because this is
5:36how we determine mass again we can't
5:38just go to these distant objects and
5:40pick them up and figure out their mass
5:42it wouldn't work that way for any number
5:45of reasons not the least of which is
5:47they're too far and would horribly kill
5:50us if we tried to pick up a star and I
5:51mean how would you anyway so we actually
5:54measure mass by determining the
5:59properties of something orbiting it so
6:01to determine say the mass of Jupiter we
6:05can look at some of the moons orbiting
6:06it look at their orbital the semi-major
6:10axis or the average distance that object
6:12is from the planet and how long it takes
6:15it to orbit once from that we can
6:19estimate this mass and what we're
6:21assuming always is that the 2nd mass is
6:23basically insignificant it's much much
6:26smaller compared to the object it's
6:29orbiting around so we can use this to
6:31estimate the mass of a planet or the
6:33mass of distant stars as well if we can
6:36figure out these traits of the
6:38characteristics of something orbiting it
6:40and so we're going to take a look at
6:42another example using this form of but
6:44now in its modified form is again this
6:46is incredibly important it's how we
6:48determine masses of distant objects