# Can Kepler's 2nd Law be applied to more than one planet?

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In summary, Kepler's Law states that the planets each sweep out the same amount of area in the same amount of time, but the outer planets have a longer year because their orbits are larger.
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I saw an explanation for why Jupiter has a slower tangential velocity in its orbit compared to inner planets and it stated:

"Remember that by Kepler’s second law, the planets each sweep-out the same area in the same amount of time. The outer planets’ elliptical orbits are considerably larger than those of the inner planets so, so over any given time period, they only need to complete a much smaller part of their orbit than do the inner planets and thus they have a longer year."

I've always heard and thought of the 2nd law with respect to a given planet, not as an explanation for the velocities of multiple planets in a system. To me it doesn't make sense to do it this way because of the differing initial conditions in each planets' formations.

As long as the central body is much heavier than the orbiting one, the second law has the same proportionality constant, which then only depends on Newton's gravitational constant and the mass of the central body.

Edit: I should add that this is true for circular motion. Naturally, you may have a situation with zero angular momentum regardless of the radius.

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The way the planets form isn't important, this is just the mechanics of a 2 body problem. Kepler's Law is a consequence of the much broader principle of conservation of angular momentum, which you most probably have already heard of.

Just to be clear, if we let a stopwatch run for 10 seconds and attach a line connecting the sun to Earth and a line connecting the sun to Jupiter then they will both trace the same area as each other.

Sorry, I do not know what I was thinking with ... The proportionality constant does depend on the radius. In fact, it has to depend on the radius just from dimensional analysis.

The argument for why outer planets have a lower velocity is to be found in the relation between the required centripetal force to the gravitational force:
$$m \frac{v^2}r = m \frac{GM}{r^2} \quad \Longrightarrow \quad v \propto \sqrt{1/r}.$$

Indeed, the areal speed is constant for a given planet but varies from planet to planet.
The relation between the distance and velocity follows from Newton's second law, not from Kepler's second law.
Their "explanation" seem to imply that all the planets have the same areal speed which is not true.

You can see a simple little proof as such :
The area swept out of a segment is
$$A = \int \frac{1}{2} r dr \delta \theta \ = \frac{1}{2} r^2 \delta \theta = \frac{1}{2} r^2 \frac{\delta \theta}{\delta t} \delta t \\$$ Where the RHS expression is the fraction swept out in a time dt. Using angular momentum,
$$= \frac{1}{2} r^2 \omega \delta t \ = \frac{L}{2m} \delta t$$
Because their is no torque acting on the planet the angular momentum is constant over time. Hence the area swept out in every time element, dt is constant.

This holds for individual planets. The angular momentum of Jupiter is not equal that of earth. Nor is the area swept out the same.

:::: Let me add. You should google the radius and orbital period and check whether I am indeed correct for yourself !

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## 1. What is Kepler's 2nd Law?

Kepler's 2nd Law, also known as the Law of Equal Areas, states that the line connecting a planet to the sun will sweep out equal areas in equal times. This means that a planet will move faster when it is closer to the sun and slower when it is farther away.

## 2. Can Kepler's 2nd Law be applied to all planets in our solar system?

Yes, Kepler's 2nd Law can be applied to all planets in our solar system. It is a fundamental law of planetary motion and applies to all objects orbiting a central mass.

## 3. Can Kepler's 2nd Law be applied to more than one planet in the same system?

Yes, Kepler's 2nd Law can be applied to multiple planets in the same system. Each planet will follow its own elliptical orbit around the central mass, and the law will still hold true for each individual planet.

## 4. How is Kepler's 2nd Law related to the speed of a planet?

Kepler's 2nd Law is directly related to the speed of a planet. As a planet moves closer to the sun, it will increase in speed in order to maintain the law of equal areas. Conversely, as a planet moves farther away from the sun, it will slow down in order to maintain this law.

## 5. Can Kepler's 2nd Law be used to predict the motion of planets?

Yes, Kepler's 2nd Law can be used to predict the motion of planets. By analyzing the size and shape of a planet's orbit, as well as its distance from the sun, we can use this law to accurately predict the speed at which a planet will move at any given point in its orbit.

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