Full transcript
0:00Kepler's second law is a bit hard to
0:03picture in a way because what it says is
0:06let's say we have a planet orbiting the
0:09Sun and if we had an imaginary line from
0:12the Sun to the planet in a certain
0:16amount of time let's say a day it would
0:19sweep out a certain amount of area that
0:21big imaginary rope would sweep out a
0:23certain amount of area if we then took a
0:26day at any other point in the orbit the
0:29area swept out would be the same so this
0:34area here is the same as this area as
0:36long as we're measuring the same amount
0:38of time so it's a bit weird to picture
0:42again just for now think this triangle
0:44is the same as this triangle but I
0:47always like to think there's kind of
0:48this hidden meaning of Kepler's second
0:51law cuz think about this sure this is
0:54the same amount of time as this wait a
0:58minute look how much longer this is than
1:00that this path is so much longer than
1:04this even though it's the same amount of
1:05time so Kepler's second law you can
1:08really think of it it means when we're
1:10closer to the Sun we're moving faster
1:13the disk speed the planet orbits the Sun
1:16is not constant it's as we get closer it
1:19speeds up that's why this length is so
1:22much longer and again this is greatly
1:23exaggerated for the purposes of these
1:25figures just to make it clearer but the
1:27idea is there when we're closer to the
1:30Sun our planets actually orbiting faster
1:33and thus this area is the same as this
1:37but it means we're traveling faster when
1:40we get closer to the Sun finally
1:44Kepler's third law is the most mathy if
1:47you will it's relating to properties if
1:51we look at the period of a planets
1:53orbital motion and let's deconstruct
1:55this one for another time the period is
1:57the time for one remember period is the
2:02time for one occurrence okay and the
2:06planets orbital motion
2:08well how that means the period the time
2:10to do one or
2:13but for the time for the planet to orbit
2:15around the Sun once so in case of Earth
2:18that's one year so that number squared
2:23so that times itself represented by P
2:26period for piece in and we want to
2:29square that and that's proportional
2:31that's what this symbol here means
2:33proportional it's proportional to the
2:37semi-major axis cubed remember
2:40semi-major axis is basically the average
2:43distance from the planet to the Sun so
2:46the time for the planet to orbit the Sun
2:49once take that number and square it
2:51that's proportional to the semi-major
2:55axis the average distance from the Sun
2:57cubed and we're actually going to take a
3:00look at an example of this and do
3:01question because this is arguably the
3:05hardest formula you're gonna have in the
3:07book but I want to make sure you
3:09understand how to use it and what we are
3:11gonna take advantage of is the fact that
3:13if we put this in terms of units of
3:14Earth so how many times it takes to go
3:19around or how long it takes to go around
3:21a star in terms of Earth years and how
3:24far it is in terms of the astronomical
3:26unit which is the average distance
3:28between Earth and the Sun if we do that
3:30we can more or less put this as an
3:31equality put this equal to that we'll
3:34see you later there it's a bit of an
3:35oversimplification but we'll get there
3:38later all right
3:39so let's try an example we're told a
3:42planet orbits the Sun at a distance of 4
3:45au 4 astronomical units and we want to
3:48figure out the orbital period or in
3:50other words how long it would take to do
3:51a full orbit around the Sun and we're
3:54going to get that in terms of Earth
3:56years from our formula so remember we
3:59are given a our average distance from
4:02the Sun our semi-major axis it's 4 but
4:06the thing is we don't want a we actually
4:10want a cubed that's what we got to put
4:12into our formula a cubed so 4 times 4
4:15times 4 or 64 punch that in your
4:17calculator put 4 cubed or 4 times itself
4:20and then it times itself again you get
4:2264 great now we are
4:26use it we're again able to say that
4:28these are basically equal P squared
4:30equals a cubed thing is we want the
4:34orbital period this is the orbital
4:36period squared so somehow we have to get
4:39rid of this squared and I've
4:41accidentally clicked the slide so I've
4:42already given away the answer but in
4:44math when you want to get rid of
4:46something when you're solving an
4:47equation you do the opposite so if I
4:51have something multiplying another
4:53number I divide to get rid of it in this
4:55case if I am squaring I want to square
4:57root it to get rid of it but in math
5:00it's also important to always remember
5:02BFS both sides we have to do what we do
5:07to one side to the other you can't just
5:08take the square root of this without
5:11taking the square root of the other side
5:12as well and then the squared and square
5:14root cancels out so we'll be left with P
5:16we take a cubed which is 64 and take the
5:20square root of it we're gonna get 8
5:21again punch that in your calculator
5:24again this is a hard question but one
5:26that you will see on the midterm I'm
5:28telling you now know how to do this and
5:30don't just memorize this answer because
5:32I don't change the numbers so make sure
5:34you know how to do this if you had to
5:36take the square root you would just
5:37round off two if it was a couple
5:39decimals you got you just find the
5:42answer that was closest so ultimately we
5:45can say the orbital period would be
5:47equivalent to eight earth years orbiting
5:51four times the distance that Earth is
5:53from the Sun it's taking eight times the
5:57years to go one full orbit now
6:04Naza says Kepler's laws really let us
6:08determine the shape of our solar system
6:09help correct things got rid of those
6:11deference there was every cycles got rid
6:13of the idea this myth of the perfect
6:15circle no we know we're orbiting in
6:17ellipses and again you can see these
6:19images they almost look like circles but
6:22the problem was is that we got all these
6:25measurements he's like the orbital
6:27period and the semi-major axis but all
6:30we knew was relative to the distance
6:33Earth was from the Sun so we knew
6:35roughly that Venus was boat point
6:39an astronomical unit from the Sun and
6:41all of the other planets but we didn't
6:43know what an astronomical unit was we
6:46knew the relative ratios of the planets
6:50the distance from say Saturn to the Sun
6:52we knew it was nine and a half times
6:55that of Earth's distance but we didn't
6:58know exactly what Earth's distance was
6:59and we actually had to wait well many
7:03many years and what we ended up doing is
7:05bouncing a radar signal off of Venus
7:08when it was closest to us when we were
7:11lined up with it we knew that if we
7:14bounced the signal off of Venus and saw
7:17how long it took to get there bounce off
7:19enos and come back that would give us a
7:22time and we knew the distance from us to
7:25Venus would be point two reaction novel
7:27units because we're one astronomical
7:29unit away it's zero point seven so the
7:31distance between us must be point three
7:32and the only other thing we have to
7:34remember is that since the signal was
7:36hitting Venus and coming back we're
7:37actually gonna use half that time and
7:39using that and the fact we knew how much
7:43the velocity of the speed would be or
7:47how fast the speed of the radar would go
7:50we could actually determine the distance
7:54the true distance of what point three of
7:56an astronomical unit was and from that
7:58what an astronomical unit was and thus
8:00all of the sizes of our solar system but
8:03we had to wait until we could actually
8:06do that so we knew the relative shape
8:08and size but we didn't know the true
8:10dimensions until we could bounce the
8:11signal off Venus and from that we then
8:14knew that we could describe our solar
8:16system