The motion of celestial bodies such as the moon, the earth, other planets have been a subject of significant interest for a long time.
A famous Indian astronomer and mathematician, Aryabhata did the in-depth study of these motions.
After that, he proposed a theory of the elliptical path of planets where he stated that all the planets remain stable, and as they come closer to the sun because of attraction, their speed increases proportionately.
He also gave a conclusion in his book Aryabhatiya that the earth revolves around its axis and moves in a circular orbit about the sun and that the moon moves in a circular orbit around the earth.
A thousand years after Aryabhata, the brilliant combination of Tycho Brahe and Johannes Kepler studied planetary motion in significant detail.
Kepler formulated his important findings in his three laws of planetary motion. They are:
First Law: It states that all planets make an elliptical locus with the sun at a focus.
Second Law: The radius vector, r from the sun to the planet traces equal area in equal intervals of time.
Third Law: This law is also called the law of ellipses. This states that the square of the time period, T of the revolution of a planet is proportional to the cube power of the semi-major axis,r of the ellipse.
T ∝ r3
In the year 1665, an English mathematician, physicist, astronomer, and theologian named Isaac Newton studied the motion of the moon about the earth.
He stated that the laws of nature are the same for earthly and heavenly bodies.
This means all the objects in the universe fall freely under the influence of gravity such that the force acts towards the center of the earth.
The acceleration of a body falling near the earth’s surface = 9.8 ms-2.
He formulated an equation to showcase the force between the earth and the body, i.e.,
G = Universal gravitational constant whose value = 6.673 x 10-11Nm2/kg2
m = mass of the smaller body, and
M = Mass of a larger body, separated by the square of the distance ‘r.’
Newton further generalized the law by saying that not only the earth but all material bodies in the universe attract each other according to equation (1) with the same value of G.
All planets orbit in a counterclockwise direction. The inner planets orbit swiftly than the outer planets.
They all move in a path that obeys the laws of motion and the force that controls their motion is the gravity.
The earth is the third planet away from the sun, which takes 365 days to complete one orbit.
All the heavenly bodies like planets and satellites move in an elliptical orbit due to the attractive force of gravity, their centrifugal motion is balanced by the gravitational attraction.
The elliptical orbit is the elongated or skewed circle.
Instead of having a single center like a circle, ellipses have two centers called foci.
f1 and f2 (in Fig.1 (a)).
[Fig 1 & Fig 2- Image to be added Soon]
For planets in space, the center of the sun is always at the focus as shown in Fig.2.
So the larger is the distance between the two foci, the more elongated the ellipse is.
The amount of elongation of the orbit is given by the eccentricity of the orbit.
A planet like the earth has a low eccentricity where both the foci lie within the sun itself. So, we can say that Earth’s orbit is almost circular.
[Fig 2-Image to be added Soon]
Let’s understand the planetary motion by understanding Kepler's laws.
Where Kepler’s first law states that planets revolve around the sun in elliptical shape with the center of the sun being at one focus as you can in Fig.1 (b). This law is also called the Law of ellipses.
The second law states that an unreal line drawn from the center of a star to the center of a planet traces equal areas in equal time intervals.
This law is also called the law of equal areas.
Third law: It states that the ratio of the squares of the periods of revolution of any two planets is equivalent to the cubes of their mean distances from the sun.
[Image to be added Soon]
This is the law of Harmony’s for each point on an ellipse, the sum of the distances from each focus is a constant equal to the time the major-axis length.
In most planetary systems, the eccentricity is low enough that we can approximate the average distance between the star and the planet which is the major axial length of the orbit.
Q1: The Moon Completes Its Revolution Around the Earth in 27.3 days, and Its Distance from the Earth is 3.85 x 105 km. Find Its Acceleration.
Solution: We are given with T = 27.3 days, r = 3.85 x 105 km
Now using the formula for the angular acceleration:
α = ω2R, and
α=(2π/T)2 . R
= 4π2. R/T2
Putting the values:
= 4 x (3.14)2x (3.85 x 105)/(27.3)2
On solving, we get:
Q2: What are the Three Properties of Celestial Motion?
Ans: The celestial body like the earth makes three types of motion.
It rotates around the polar axis,
It revolves around the Sun, and
Swings smoothly as an unbalanced spinning top.
These three motions of the earth are unperceivable to us.
Q3: What is the Value of Kepler’s Constant?
Ans: Kepler’s third law is expressed as T = k r3
Where k is called the Kepler’s constant whose value = 4π2/GM.
Q4: What would Earth be like with Two Moons?
Ans: If Earth had two moons, then an extra moon would add an extra gravitational pull, leading to larger tides, and on colliding with each other, they will send their debris to the planets.