write Kepler's all 3 laws of planetary motion....
Answers
Answer:
Kepler First law – The Law of Orbits
According to Kepler’s first law,” All the planets revolve around the sun in elliptical orbits having the sun at one of the foci”. The point at which the planet is close to the sun is known as perihelion and the point at which the planet is farther from the sun is known as aphelion.
It is the characteristics of an ellipse that the sum of the distances of any planet from two foci is constant. The elliptical orbit of a planet is responsible for the occurrence of seasons.
Kepler's Laws of Planetary Motion
Kepler First Law – The Law of Orbits
Kepler’s Second Law – The Law of Equal Areas
Kepler’s second law states ” The radius vector drawn from the sun to the planet sweeps out equal areas in equal intervals of time”
As the orbit is not circular, the planet’s kinetic energy is not constant in its path. It has more kinetic energy near perihelion and less kinetic energy near aphelion implies more speed at perihelion and less speed (vmin) at aphelion. If r is the distance of planet from sun, at perihelion (rmin) and at aphelion (rmax), then,
rmin + rmax = 2a × (length of major axis of an ellipse) . . . . . . . (1)
Keplers Second Law
Kepler’s Second Law – The law of Equal Areas
Using the law of conservation of angular momentum the law can be verified. At any point of time, the angular momentum can be given as, L = mr2ω.
Now consider a small area ΔA described in a small time interval Δt and the covered angle is Δθ. Let the radius of curvature of the path be r, then the length of the arc covered = r Δθ.
ΔA = 1/2[r.(r.Δθ)]= 1/2r2Δθ
Therefore, ΔA/Δt = [ 1/2r2]Δθ/dt
\lim_{\Delta t\rightarrow 0}\frac{\Delta A}{\Delta t}=\frac{1}{2}r^{2}\frac{\Delta \theta }{\Delta t}lim
Δt→0
Δt
ΔA
=
2
1
r
2
Δt
Δθ
, taking limits both side as, Δt→0
⇒\frac{dA}{dt}=\frac{1}{2}r^{2}\omega
dt
dA
=
2
1
r
2
ω \frac{dA}{dt}=\frac{L}{2m}
dt
dA
=
2m
L
Now, by conservation of angular momentum, L is a constant
Thus, dA/dt = constant
The area swept in equal interval of time is a constant.
Kepler’s second law can also be stated as “The areal velocity of a planet revolving around the sun in elliptical orbit remains constant which implies the angular momentum of a planet remains constant”. As the angular momentum is constant all planetary motions are planar motions, which is a direct consequence of central force.
⇒ Check: Acceleration due to Gravity
Kepler’s Third Law – The Law of Periods
According to Kepler’s law of periods,” The square of the time period of revolution of a planet around the sun in an elliptical orbit is directly proportional to the cube of its semi-major axis”.
T2 ∝ a3
Shorter the orbit of the planet around the sun, shorter the time taken to complete one revolution. Using the equations of Newton’s law of gravitation and laws of motion, Kepler’s third law takes a more general form:
P2 = 4π2 /[G(M1+ M2)] × a3
where M1 and M2 are the masses of the two orbiting objects in solar masses.
Explanation:
Answer:
Kepler's three laws of planetary motion can be described as follows:
( 1 ). The path of the planets about the sun is elliptical in shape,
with the center of the sun being located at one focus. (The Law of Ellipses)
( 2 ). An imaginary line drawn from the center of the sun to the center of the planet
will sweep out equal areas in equal intervals of time. (The Law of Equal Areas)
( 3 ). The ratio of the squares of the periods of any two planets is equal to the ratio
of the cubes of their average distances from the sun. (The Law of Harmonies)
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