Planetary Motion: The history of an idea that led the revolution!!

MOTION OF PLANETS AND SATELLITES

Since the time of the scientist Copernicus, it has been known that various planets revolve around the sun. But in order to determine how the planets rotate and what their motions (planetary motion) are, it was necessary to know precisely the positions of the planets in the sky (at different times). Astronomer Tycho Brahe observed the positions of the planets for many years without a telescope (Galileo invented the telescope after Brahe’s death) and published a wealth of information. Then the scientist Kepler analyzed the data and discovered three laws related to the motion of the planets. These are known as Kepler’s laws of planetary motion.

KEPLER'S LAWS OF PLANETARY MOTION

Kepler’s laws are discussed below.

First Law: Each planet revolves around the Sun in an elliptical orbit and the Sun is at the focus of that ellipse.
Second Law: A straight line joining the Sun and a planet traverses equal areas in equal periods of time.
Third Law: The square of the period of revolution of a planet around the Sun is proportional to the cube of the semimajor axis of its orbit.

The motion and orbit of a planet are referred to as Keplerian motion and Keplerian orbit respectively.

Note that the motion of a natural or man-made object (such as a satellite) orbiting an object (e.g. a planet) is similar to the motion of the planets orbiting the Sun. That is, the motion of the satellites also obeys Kepler’s laws. However, it should be remembered that in cases where Kepler’s laws apply, the mass of orbiting objects is negligible compared to the object at focus, as in our solar system. Although the orbits of the planets of the solar system are elliptical, they are almost circular, that is, the eccentricity of the orbits is very low. Since the circle is a special region of the ellipse (eccentricity of the circle, e = 0), Kepler’s laws are equally applicable to circular orbits.

All of Kepler’s laws apply to circular and elliptical orbits (i.e. orbits where 0 ≤ e ≤ 1). Since hyperbolas and parabolas are open conic sections, Kepler’s first and third laws do not apply to such orbits. For example, a comet that makes a single trip past the Sun follows an elliptical or parabolic path.

ORBITAL VELOCITY OF A PLANET

Let us assume that a planet of mass m is orbiting the Sun of mass M₀ in an orbit of radius r.

The speed at which a planet travels in its orbit is called its orbital velocity (v). The value of this orbital velocity is a constant for circular orbits. Centripetal force required to move the planet in circular orbit = mv²/r. The mutual gravitational force between the Sun and the planets provides this centripetal force. This gravitational force is GM₀m/r².

So, mv²/r = GM₀m/r²

or, v² = GM₀/r

or, v = √GM₀/r ………(I)

Note that orbital velocity (v) is not constant in case of elliptical orbit. In this case, v ∝ 1/r if the distance r from the Sun to the planet at any instant.

SOME USEFUL DATA ON THE SUN THE EARTH AND THE MOON

MASS OF THE SUN

Let us assume that the mass of the Sun = M₀, the distance of a planet from the Sun = r, the orbital period of that planet around the Sun = T.

M₀ = 4π²r³/GT²

Considering Earth among the planets, mean distance of Earth from Sun is r = 1.5 X 10⁸ km = 1.5 X 10¹⁰ m, orbital period of Earth = 365 d (approx.) = 365 X 24 X 3600 s.

From above equation, putting all the value we got, M₀ = 2 X 10³⁰ kg (approx.)

MASS OF THE EARTH

The mass of the earth can be determined by analyzing the motion of the moon around the earth. Average distance of Moon from Earth r = 3.84 X 10⁸ m, orbital period of Moon T = 27.3 d = 27.3 X 24 X 3600 s.

From the equation putting all the value we got,

M = 6 X 10²⁴ kg (approx.)

If the value of the gravitational acceleration g at the surface is known, it is possible to determine the result of the earth in a much simpler way.

GRAVITATIONAL ATTRACTION BETWEEN THE SUN AND THE EARTH

Mass of Sun M₀ = 2 X 10³⁰ kg, Mass of Earth M = 6 X 10²⁴ kg, Average distance between Sun and Earth r = 15 X 10¹⁰ m. Hence, the mutual gravitational force between Sun and Earth is,

F = GM₀M/r²

Putting all the value in this equation we got,

F = 3.56 X 10²² N

GRAVITATIONAL ATTRACTION BETWEEN THE EARTH AND THE MOON

Mass of Earth M = 6 X 10²⁴ kg, Mass of Moon m = 7.33 X 10²² kg, Average distance between Earth and Moon r = 3.84 X 10⁸ m. Hence, the mutual gravitational force between Earth and Moon is,

F = GMm/r²

Putting all the value in this equation we got,

F = 1.99 X 10²⁰ N

Hence, gravitational force between sun and earth / gravitational force between earth and moon

= 3.56 X 10²² N / 1.99 X 10²⁰ N = 179 (approx.)

From this value we can say that the gravitational force between sun and earth is approximately 180 times more than the gravitational force between Earth and moon.

CONCLUSION

Planetary motion is the regular and predictable movement of planets around the Sun, which is governed by gravity and laws of physics. Johannes Kepler’s formulas of planetary motion, published in the 17th century, define these movements, demonstrating that planets travel in elliptical circles with the Sun at one point. Isaac Newton later stated that gravitational force keeps planets in orbit, with the force decreasing as the planet’s distance from the Sun rises. These concepts show that planetary motion is not random but follows natural rules, allowing astronomers to forecast positions and behaviours across time. This motion is critical for understanding the dynamics of our solar system, influencing everything from the length of each planet’s year to the occurrence of eclipses and other celestial occurrences. Overall, studying planetary motion is critical to understanding the physics of our solar system and universe.

Planetary Motion: The history of an idea that led the scientific revolution!!

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