What is Kepler's law of planetary motion?
German astronomer Johannes Kepler formulated three fundamental laws that describe planetary motion, and they are categorised into:
- Law of Orbits (Kepler’s First Law )
- Law of Areas (Kepler’s Second Law)
- Law of Periods (Kepler’s Third Law )
Now let us learn more about this:
1.Kepler’s first law - Law of Orbits
It states that all planets move around the Sun in an elliptical orbit, with the Sun at one of the two foci. This indicates that the Sun is located at one focus, not at the center and the distance between the planet and the Sun keeps changing
2.Kepler's second law- Law of Areas
This law states that the line joining the Sun to a planet (radial vector) sweeps equal areas in equal intervals of time. Indicates that when the planet is closer to the Sun, it moves faster, and when it is farther, it moves slower.
3.Kepler’s third law- Law of periods
The square of the time period of revolution of a planet is proportional to the cube of the semi major axis of the ellipse traced out by the planet.
T2 ∝ a3
This means Planets that stay away from the Sun take more time to complete one revolution; also, there exists a fixed relationship between distance and time period.
Universal Law of Gravitation
According to the Universal Law of Gravitation, every mass/body in the universe attracts every other point mass/body in the universe by a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
This law is applicable to all objects having weight, regardless of size; they are subject to this regulation. Two enormous objects may be referred to as point-like masses if there is a significant change between their sizes or if they are spherically symmetric. In these situations, a point mass at each object’s center of mass may be used to determine its mass.
F = force between the masses,
G = proportionality constant, also known as
The Gravitational constant: 6.67 × 10-11 Nm2 kg-2,
m1, m2 = mass of the objects,
r = distance between the objects.
Gravitational Constant
The universal gravitational constant is used in Newton’s Universal Law of Gravitation, Einstein’s General Theory of Relativity, and also Kepler’s Third Law of Planetary Motion to calculate the time period of a planet to complete one full revolution in its orbit.
The actual force exerted between two bodies can be given by the following equation,
Fg = G m1 m2 / r2
(G = 6.674 × 10-11 N m2 kg-2) where G is the universal gravitational constant
ACCELERATION DUE TO GRAVITY OF EARTH
The gravitational force acting on a body on the surface of Earth is:
F = GMm / R2 ……… (1)
Where:
- M – Mass of Earth
- m – Mass of the body
- R – Radius of Earth
The weight experienced by the body is:
F = mg ……… (2)
Comparing equations (1) and (2):
GMm / R2 = mg
Therefore,
g = GM / R2
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Get Expert GuidanceVARIATION OF ACCELERATION DUE TO GRAVITY
The value of acceleration due to gravity (g) varies as we go above or below the surface of the Earth. It also varies from place to place on the surface of the Earth.
The acceleration due to gravity at the surface of the Earth is given by:
For depth ‘d’, under the Earth of uniform density:
When d = R:
g = acceleration due to gravity at Earth’s surface
gd = acceleration due to gravity at depth d
R = Earth’s radius
d = depth below the surface
The acceleration due to gravity at a height ‘h’ above Earth’s surface is given by:
If h ≪ R, then:
Gravitational Potential (V)
The gravitational potential at any point is the work done per unit mass required to move an object from infinity to that point. The gravitational potential at any point is the gravitational potential energy per unit mass.
V = U/m
U is the gravitational potential energy.
m is the mass of the object.
The mathematical expression for the potential is given by
V= −GM /r
Gravitational Potential Energy
The energy possessed by an object because of its position in the gravitational field is called gravitational potential energy. The equation of gravitational potential energy can be calculated from the law of gravity, and it is equal to the amount of work done against gravity to bring the mass to the given point in space. The gravitational force is inversely proportional to the square of the distance; therefore, the force will be zero at a large distance.
The expression for gravitational potential energy associated with two particles separated by a distance r is given as:
This expression is most useful for calculating the escape velocity.
Escape velocity
The minimum initial velocity required for a particle to escape from the gravitational sphere of influence of a planet is known as escape velocity (ve).
Expression for the Escape Velocity from the Surface of the Earth
TE (Initial) = TE (Final) ………..(1)
TE (Initial) = KE (Initial) + PE (Initial)
⇒ ½mvi2 − GMm R ………..(2)
TE (Final) = KE (Final) + PE (Final)
⇒ ½mvf2 − GMm R∞ = ½mvf2 + 0 ………..(3)
½mvi2 − GMm R = ½mvf2 ………..(4)
vi = ve, vf = 0
½mve2 − GMm R = 0
⇒ ½mve2 = GMm R ………..(5)
⇒ ve2 = 2GM/R
⇒ ve = √(2GM/R)
ve = √(2gR) = 11.2 km/s
Where:
M = Mass of Earth
m = Mass of the projecting body
R = Radius of the Earth
Orbital velocity
The velocity with which the test mass orbits around a source mass is known as orbital velocity (v0). When the test mass is orbiting around the source mass in a circular path of radius ‘r’ having the centre of the source mass as the centre of the circular path, the centripetal force is provided by the gravitational force, as it is always an attractive force having its direction pointed towards the centre of the source mass.
mvo 2/r = GMm/ r2
v02/r = GM/r2
If the test mass is at small distances from the source mass
Then,
v0= √GM/r
The above formula suggests that the orbital velocity is independent of the test mass (the mass that is orbiting).
ullamcorper mattis, pulvinar dapibus leo.
What are Satellites?
A satellite is a body that revolves continuously around a comparatively large body. Satellites are categorized into two types:
Natural Satellite
A celestial body that revolves around a planet is called a natural satellite. For example, the Moon is a natural satellite of the Earth.
Artificial Satellite
The human-made bodies that are placed in specific orbits and are made to revolve around the Earth are called artificial satellites
Geostationary Satellite
A geostationary satellite revolves around the Earth with the same angular velocity and in the same direction as the Earth rotates on its axis.
Time Period of Satellite
The time period of the satellite is the time taken by the satellite to complete one revolution around the Earth. It depends only on the height from the surface of the Earth. A geostationary satellite therefore appears stationary relative to an observer on Earth.
Time Period of an Orbiting Satellite
Time period of a satellite is given by:
But orbital velocity,
Energy of an Orbiting Satellite
Kinetic Energy of a Satellite
Potential Energy of a Satellite
Total Energy of a Satellite
The total energy of the satellite is the sum of all energies possessed by the satellite in the orbit around the Earth. As only the mechanical motion of the satellite is considered, it has only kinetic and potential energies.
Total energy of the satellite = kinetic energy of the satellite + potential energy of the satellite
E = KE + U
E= – GMm/ 2r
E = -GMm/ [2(R+h)]
Weightlessness
Weightlessness refers to a situation in which the apparent weight of a body becomes zero because the normal reaction acting on it is zero.
Let m′ be the mass of the person inside the satellite.
Then gravitational force =
Reaction force F′ exerted by the satellite.
The net force ( GMm′ R02 − F′ ) is balanced by the centripetal force m′v2/R0.
From equations (1) and (2), we have:
or F′ = 0
Let’s summarise
- The law of gravitation is applicable to all bodies, irrespective of their size, shape, and position.
- The gravitational force between the Sun and the planets provides the necessary centripetal force for orbital motion.
- The acceleration due to gravity decreases as the height or the depth increases from the surface of the Earth.
- Kepler discovered the three laws of planetary motion.
- Kepler’s laws are applicable not only to planets but also to other satellites in orbit around any body.
- Escape velocity is the minimum velocity with which an object must be projected to escape the gravitational attraction of the Earth.
FAQ's
Kepler’s three laws are the Law of Orbits, Law of Areas, and Law of Periods.
It states that every mass in the universe attracts every other mass with a force proportional to their masses and inversely proportional to the square of the distance between them.
The acceleration due to gravity at Earth’s surface is given by g = GM/R².
The value of g decreases with height and is given by gh = GM/(R+h)².
Gravitational potential energy is the energy possessed by a body due to its position in a gravitational field, given by U = −Gm₁m₂/r.
Escape velocity is the minimum speed required for an object to escape the gravitational attraction of Earth, given by vₑ = √(2gR).
Orbital velocity is the velocity required for a satellite to remain in a circular orbit around a central body, given by v₀ = √(GM/r).
Weightlessness occurs when the apparent weight of a person becomes zero because the person and the satellite are in continuous free fall under gravity.
The time period is the time taken by a satellite to complete one revolution around Earth, given by T = 2π√[(R+h)³/GM].
Gravitational potential is the gravitational potential energy per unit mass, given by V = U/m.