Kepler's Laws of Planetary Motion: Law of Orbits, Areas and Periods
Johannes Kepler proposed three laws to explain the motion of planets around the Sun. These laws describe the shape of planetary orbits, the speed of planets during revolution, and the relationship between the orbital period and the size of the orbit.
Kepler's First Law (Law of Orbits)
Statement
Every planet moves around the Sun in an elliptical orbit with the Sun situated at one of the foci of the ellipse.
Important Terms
- Perihelion (AS): The closest distance of a planet from the Sun.
- Aphelion (BS): The farthest distance of a planet from the Sun.
Eccentricity of an Ellipse
The eccentricity of an ellipse is defined as:
e = SO / AO
Where:
- AO = OB = a (Semi-major axis)
- SO = ea
Therefore,
e = SO/a
Perihelion and Aphelion Distance
Perihelion Distance:
AS = AO − OS
AS = a − ea
AS = a(1 − e)
Aphelion Distance:
BS = BO + OS
BS = a + ea
BS = a(1 + e)
Important Facts
- For an ellipse, e < 1.
- For a circle, e = 0.
- For Earth satellites, perihelion is called Perigee and aphelion is called Apogee.
Kepler's Second Law (Law of Areas)
Statement
The line joining the Sun and a planet sweeps out equal areas in equal intervals of time.
Mathematical Form
dA/dt = Constant
where dA/dt is called the areal velocity.
Explanation
According to this law, a planet sweeps out equal areas in equal intervals of time.
- The planet moves faster when it is near the Sun (perihelion).
- The planet moves slower when it is far from the Sun (aphelion).
Proof of Law of Areas
The gravitational force between the Sun and a planet is a central force.
Torque acting on the planet is:
τ = r × F
τ = rF sin 0° = 0
Since torque is zero, angular momentum remains constant.
L = r × mv = Constant
Area swept in time dt is:
dA = ½ |r × dr|
Since dr = vdt,
dA = ½ |r × v| dt
Therefore,
dA/dt = ½ |r × v| = |L|/2m = Constant
Hence, the areal velocity of the planet remains constant.
Kepler's Third Law (Law of Periods)
Statement
The square of the time period of revolution of a planet is directly proportional to the cube of the semi-major axis of its orbit.
Mathematical Form
T² ∝ R³
or
T²/R³ = Constant
Where:
- T = Time period of revolution
- R = Semi-major axis of orbit
Proof of Third Law
For a circular orbit:
Centripetal Force = Gravitational Force
mv²/R = GMm/R²
v² = GM/R
Also,
v = 2πR/T
Substituting the value of velocity:
GM/R = 4π²R²/T²
T² = 4π²R³/GM
Since 4π²/GM is constant,
T² ∝ R³
or
T²/R³ = Constant
Central Force
A force that acts along the line joining a particle and a fixed point is called a central force.
The gravitational force between the Sun and planets is an example of a central force.
For a central force:
τ = r × F = 0
Therefore,
dL/dt = 0
L = Constant
Hence, angular momentum remains conserved and the motion remains confined to a plane.
Frequently Asked Questions (FAQs)
1. What are Kepler's three laws of planetary motion?
They are the Law of Orbits, Law of Areas, and Law of Periods.
2. What is Kepler's First Law?
Every planet moves around the Sun in an elliptical orbit with the Sun at one focus.
3. What is perihelion?
Perihelion is the closest point of a planet to the Sun.
4. What is aphelion?
Aphelion is the farthest point of a planet from the Sun.
5. What is eccentricity?
Eccentricity measures how much an ellipse differs from a circle.
6. What does Kepler's Second Law state?
The line joining the Sun and a planet sweeps out equal areas in equal intervals of time.
7. Why does a planet move faster near perihelion?
To sweep out equal areas in equal intervals of time.
8. What is areal velocity?
It is the rate at which area is swept out by the radius vector.
9. What is Kepler's Third Law?
The square of the orbital period is proportional to the cube of the semi-major axis.
10. Which Kepler's law is based on conservation of angular momentum?
Kepler's Second Law.
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