3:42
When scientists look at a comet, they can measure its position and speed and tell whether we will see it again or not. For example, this is the 3I Atlas discovered in 2025. It's an interstellar comet, meaning it came from deep intergalactic space into our solar system, and we know that we will never see it again. It's going to go back into interstellar space. In contrast, this is the Halley's Comet, last seen in 1986, and we know we'll see it again sometime in 2061. So, the big question is, how can we just measure the position and the speed of a comet and tell whether we'll see it again or not? Well, it all has to do with the energy of orbiting systems. So, let's dig into it.
Let's consider a planet going around a star, like Earth going around the Sun, for example. You can also consider a satellite going around a planet, or you can also consider a comet orbiting the Sun, whatever you want, okay? Take any orbiting system. The question we're trying to answer is, what is the energy of that system? The total energy of this system. Well, let's think about it. First of all, because the planets and the planet and the star are interacting gravitationally, there must be some gravitational potential energy. What's the expression for that? Well, the gravitational potential energy between any two masses is given by this
expression, where M and M are the masses and R is the distance between them. But wait, why is the gravitational potential energy negative? Well, because when masses are infinitely far apart, the gravitational potential energy is maximum, right? Well, we choose that maximum value to be zero. That's our choice. That's our reference. And since the maximum gravitational potential energy itself is zero, for any other configuration, when the distance between them is finite, the potential energy ends up becoming less than zero. So, it becomes negative. Note that at this particular scale, we can pretty much treat both the celestial bodies as just point masses, so we don't have to worry
about the their radius and stuff, okay? And secondly, remember that this is the energy of the entire system. We don't have to consider the energy of the star separately and the potential energy of the Earth separately. We always consider potential energy for a system, so this is the potential energy of this entire system, okay? What else? Well, there's kinetic energy. Now, in this particular case, we're assuming that the star is at rest and it's the planet that's moving. So, all the kinetic energy is, you know, in the orbiting object. So, it'll be the kinetic energy of the planet or the kinetic energy of the satellite, whatever you want to call that. So, the total energy is going to be the sum of
these two, right? Okay, so what will that be? Well, if the speed of that satellite or the planet is V, then we can say the total energy becomes the potential energy plus the kinetic energy of that satellite or the planet, which is just 1/2 MV squared. Okay, so let's see what we can do with this. Well, first of all, let's assume that there are no other planets or stars in the neighborhood. Or let's assume they're very far away, so we can ignore their gravitational influence. Then, as the planet moves around, the total energy of the system must stay conserved because there is no external work being done on our system. So, the total energy must stay the same, right? However, the potential and the kinetic energies can
change. So, we can use this to predict some stuff. So, for example, a few months later when the Earth is somewhere over here, can we predict what's going to happen to its speed? Will it increase compared to this one or decrease or stay the same? Why don't you pause the video and think about this. All right. So, if you look at this expression, let's look at what happens to the potential energy. Well, the M and the small m and the G, that stays the same, but the R value has increased. So, the denominator increases, which means this number goes down, but there's a negative sign, which means the potential energy actually increases. Does that
make sense? I mean, a more intuitive way to think about it is you probably know when you throw a ball up, potential energy of the ball-Earth system increases, right? When masses go farther away, the gravitational potential energy increases. And the same thing's happening over here. Look, the Earth is going farther away from the Sun, so the gravitational potential energy increases. But the total energy must stay the same, which means the kinetic energy reduces. So, you'd expect the speed over here to be lower than over here. So, as the planet goes farther away, its speed reduces, and then as it comes closer, its speed increases, and so you should have the maximum speed when it's at, you
know, when it's closest to our star, and so on and so forth. The same thing would apply to an orbiting satellite as well, for example. But remember, Earth's orbit around the Sun is pretty circular. So, let's consider circular orbits now. Well, then the distance of the planet from the star stays exactly the same everywhere, right? Which means the potential energy is now a constant. Therefore, the kinetic energy of the orbiting planet or the satellite must also be a constant. In other words, the speed of the orbiting object should stay the same. So, this tells us that in circular orbits, objects' speed must stay a constant.
But wait, what should be the expression for that speed in circular orbits? I mean, we intuitively know if you are closer to a star, it should be orbiting faster. But if it's farther away from the star, the planet would be orbiting much slower. So, can we find an expression for it? Yeah, we can use Newton's second law, which says net force acting on an object equals mass times acceleration. So, over here, if we apply Newton's second law on the orbiting object, like the Earth over here, then the net force over here is just the gravitational force that the Sun is, you know, exerting on the planet. So, that's just FG. That equals the mass of our Earth, mass of the orbiting object times the acceleration. Now, which acceleration
are we dealing with over here? If the force is in this direction, acceleration is also in this direction. That is the centripetal acceleration. This is a very special case. The speed is a constant, and therefore the acceleration is always perpendicular to the direction of the velocity. That is the centripetal acceleration. It's directed towards the center, right? So, we can plug in. What is the expression for the gravitational force? Well, that can be found by Newton's universal law of gravitation. That is GMM / R squared. That equals the mass M times What's the expression for centripetal acceleration? Well, it's given by V squared over R. So, the M cancels out, the one of the R over here
cancels out, and if we rearrange, we get V squared equals GM / R. So, this shows that the orbital speed is independent of the mass of the orbiting satellite. But, it does depend upon the mass of the parent star. Again, that makes intuitive sense. If this was more massive, then you would need a higher speed to maintain your circular orbit. And notice how it's inversely related to the distance, which means the farther you go, the lower is the orbital speed to maintain a circular orbit. Okay. So, now that we know an expression for the orbital speed, we can plug this in over here and simplify the expression for our energy. Again, great idea to just pause the video and see if you can
simplify and gain some insights. All right. So, if we plug in for V squared GM over R, we end up with this expression. So, let's clean this place up a little bit and think about what this tells us. Well, first of all, I can simplify this now. I can write this as GMM divided by R times -1 + 1/2, which is -1/2. So, my total energy now becomes -1/2 GMM divided by R. Okay. So, the first thing I see is that my energy of this entire system is negative. Because, remember, these numbers are all positive, okay? So, that's the first thing. But, there's something else. Look at the expression for kinetic energy for
circular orbits. What do you notice? Well, it's similar to potential energy, except we don't have a negative sign and we have a half. Which means the kinetic energy is negative half of potential energy. And similarly, even the total energy can be written in terms of potential energy. We can say the total energy is half of potential energy. At this point, we might have even more questions. So, again, let's clean this up a little more. Okay, the first question we could be having is, how can total energy be half of potential energy? I mean, shouldn't it be more than potential energy because total energy is potential energy plus kinetic? So, how can it be half of potential energy?
Well, remember, both these are negative. So, for example, if you potential energy was, you know, minus two units, for example, the total energy would be minus one unit. Minus one is bigger than minus two. So, yes, the total energy is always bigger than the potential energy. But, the second thing that we see is that the kinetic energy will always be negative half of the potential energy. Again, that makes sense because for circular orbits, V is fixed. The planet needs to orbit with a very specific speed, so it makes sense that its kinetic energy should also be very specific, negative half of the potential energy. But, the thing is, this only applies to circular orbits. For elliptical orbits,
remember, the values of the potential energy and the kinetic energy changes over the orbit. However, what does not change is this fact. The total energy, even for elliptical orbits, would be less than zero. So, whether you consider circular orbits or elliptical orbits, the total energy should be less than zero. In other words, if the planet is bound to the star, then the total energy will always be negative, less than zero. And so, this represents the bound state. Okay, so what do you think would happen if the total energy was more than zero? What would that represent? For that, let's do
a thought experiment. Suppose we give a kick to our Earth, increasing its speed such that it escapes the Sun's gravitational influence and reaches infinity and stops. What I mean is, let's say Earth has just enough speed to reach infinity. Then, what would be the total energy of the system? Well, when you're at infinity, we know that the potential energy of the system is zero because that's our definition of zero potential. When two masses are infinitely spaced out, that's what we consider as zero potential energy. That's our reference. So, the potential energy of the system is zero at infinity, and we are also saying that the Earth just reaches infinity with zero speed, meaning the kinetic energy of the Earth is also zero. So, in this
particular situation, the energy of the system at infinity is zero. So, what happened is that after giving the hypothetical kick, we did work on the system, and so the total energy of the system increased. From negative, it became zero. So, when the planet or satellite has just enough speed to escape to infinity, the total energy of the system is zero. And this minimum speed needed to just escape the gravitational clutches of a star or a planet or whatever, we call that the escape speed. Mathematically, you can think of it as the speed at which the total energy of the system goes to zero. So, how do we derive an expression for the escape speed? Well, we know the total energy of the system is zero. At any point, the
total energy must be zero. So, when the planet is over here, the total energy, which is zero, that should equal the potential energy - GMM by R plus the kinetic energy 1/2 mv^2. And so, if we rearrange this, we get this expression. And from here again, we can cancel m. Rearranging, we get this expression. Taking square root on both sides, we get the expression for the escape speed. Again, notice the escape speed is independent of the mass that is escaping, right? But, it does depend upon the mass of the parent body. So, in this particular case, the mass of the star. That makes sense. If the star was massive, it's harder to escape that because it has more gravity. But, notice that it is inversely related
to the distance from the star. That also makes sense. If you're farther away from the star to begin with, it takes less energy to escape it, so less speed to escape it. And normally, whenever you're talking about the escape speed of a planet or a star for that matter, what we usually mean is its surface escape speed. So, in that case, the small r just becomes the radius of the planet or the star from which you're trying to escape. So, if you plug the values in for Earth, you will find the surface escape speed to be 11.2 km/s. That's the minimum speed with which you need to kick an object from the Earth's surface so that it never returns back to Earth. Of course, all of that is assuming no atmosphere and drag and all
of that stuff. The surface escape speed from the Sun is about 618 km/s. Anyways, we now understand if an orbiting satellite or an orbiting planet has this particular speed at a given position, then the total energy is zero and the object would just escape the parent body, right? Which means what if its speed was more than the escape speed? Well, then it'll have more than enough speed to escape, which means it'll reach infinity and it won't stop. It'll have some residual speed left as well. So, now the total energy would be more than zero, right? And this represents an unbound state. So, the planet is no longer bound to the star or the satellite is no longer bound to the planet. And so, we now have the
full picture. If the speed of the orbiting object is less than the escape speed, then the total energy will be less than zero. We are in the bound state. In a special case, the orbit will be circular, but in general, the orbit would be elliptical. It'll be closed bound state. In contrast, if the speed of the orbiting satellite is more than the escape speed at that particular given position, then the total energy of the system would be more than zero. Now, it'll be unbound. Now, the path will be a hyperbola. It'll take a hyperbolic path and it'll never come back. And so, look, if you can measure the speed of an object and, you know, its position, you can tell whether it's in the bound state or the unbound
state. And from that, we can tell whether we'll see that object again or it will escape the gravitational influence of the parent body. So, if you go back to the 3 Atlas, it turns out that the total energy is more than zero. It's in the unbound state. And therefore, we are sure that it'll never turn back. It takes a hyperbolic trajectory and escapes our solar system and goes back into deep interstitial interstellar space. In contrast, if you calculate the energy of the Halley's Comet, you will find that it's less than zero. And that's how we know it's a bound system. It is a very elliptical orbit, and it takes about 76 years to complete that. And that's why you will see Halley's Comet about every 76 years.
