Motion of Celestial Objects in Space           

The night sky is filled with stars, planets, comets, asteroids, and satellites, all moving in predictable paths governed by physical laws. The motion of celestial objects in space is studied under celestial mechanics, a branch of physics that applies the laws of gravitation and motion to astronomical bodies.

For JEE aspirants, understanding this topic is crucial because it combines concepts from Newtonian mechanics, gravitation, energy conservation, and orbital dynamics. Questions related to orbital motion, satellite trajectories, and planetary dynamics often appear in exams, making this a scoring topic when studied in depth.

1.0Celestial Mechanics 

Celestial mechanics is built upon a few fundamental ideas. The most important is that celestial bodies interact primarily through the force of gravitation, which is a central force obeying the inverse-square law. This makes orbits mathematically elegant and predictable.

Another essential concept is the two-body approximation, where the motion of a smaller body (satellite, planet, or comet) is considered with respect to a much larger central body (Earth, Sun, or Jupiter). In this setup, the smaller body’s motion can be derived using conservation of energy and angular momentum.

Additionally, the principle of angular momentum conservation ensures that as a planet comes closer to the Sun, its velocity increases, while it slows down when moving farther away. This leads directly to the famous laws of Kepler.

2.0Motion of Celestial Objects

The motion of celestial objects is governed primarily by the gravitational pull between massive bodies in space. Stars, planets, moons, asteroids, and satellites all move along paths called orbits, which are determined by their velocity and the central gravitational force acting upon them.

In the simplest case, when a smaller body revolves around a much larger one, the motion can be approximated as a two-body problem. For example, Earth’s revolution around the Sun and the Moon’s revolution around Earth are classic cases. The shape of these orbits can be circular, elliptical, parabolic, or hyperbolic depending on the total energy of the system.

Key principles that describe this motion include:

• Newton’s Law of Gravitation, which provides the central force pulling objects together.
• Kepler’s Laws of Planetary Motion, which explain the elliptical nature of orbits, varying orbital speeds, and the relation between period and orbit size.
• Conservation of Angular Momentum, which ensures that objects move faster when closer to the central body and slower when farther away.

The motion of celestial objects is not always perfectly regular; external factors such as the gravitational influence of neighboring planets, the non-spherical shape of celestial bodies, and relativistic corrections can cause small deviations called perturbations.

3.0Kepler’s Laws of Planetary Motion

The understanding of planetary motion began with Johannes Kepler, who derived three empirical laws by observing Mars and other planets. These laws are still part of every JEE syllabus.

First Law (Law of Ellipses):Each planet moves in an elliptical orbit, with the sun at one focus of the ellipse.

Ellipse : Ellipse is the set of all points such that the sum of the distance from the foci (plural of focus) and any point on the ellipse is constant.

General equation of ellipse

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Second Law (Law of Equal Areas): A line joining any planet to the Sun sweeps out equal areas in equal intervals of time, i.e. the areal velocity of the planet remains constant.

At minimum distance from the Sun, velocity is maximum.

At maximum distance from the Sun, velocity is minimum.

Third Law (Law of Periods): The square of the period of revolution of any planet around the Sun is directly proportional to the cube of the semi-major axis of its elliptical orbit.

$$\begin{gathered} T^2\propto a^3 \\ \text{a : average distance of the planet from Sun} \\ a=\frac{r_{\max }+r_{\min }}{2} \end{gathered}$$

This law establishes a clear relation between the size of an orbit and the time taken to complete it.

4.0Newton’s Law of Gravitation and Celestial Motion

Newton explained Kepler’s observations through his law of universal gravitation, which states:

$F=G\frac{m_1{m}_2}{r^2}$

where $m_1$ and $m_2$ are the masses of two bodies, r is the distance between them, and G is the gravitational constant.

When applied to celestial objects, this force provides the necessary centripetal acceleration for orbital motion. For a body of mass m orbiting a central mass M at distance r,

$\frac{m{v}^2}{r}=G\frac{mM}{r^2}$

which gives the orbital velocity,

$v=\sqrt{\frac{GM}{r}}$

This equation is central to understanding the motion of satellites and planets.

5.0 Orbital Velocity and Energy of Celestial Bodies

Orbital velocity determines how fast a celestial body must move to remain in orbit. For circular orbits, the orbital speed depends only on the mass of the central body and the radius of orbit, not on the satellite’s mass.

In addition, the mechanical energy of an orbiting body is the sum of its kinetic and potential energy:

$E=\frac{1}{2}m{v}^2-\frac{GMm}{r}$

For bound elliptical orbits, this total energy is negative, indicating that the body is gravitationally bound. For parabolic trajectories, total energy is zero, while for hyperbolic paths it is positive.

For circular orbits, total energy simplifies to:

$E=-\frac{GMm}{2r}$

This relationship between energy and orbital radius is frequently tested in JEE numericals.

6.0Perturbations and Real-Life Deviations in Orbits

While the ideal orbits described above are neat and mathematically precise, real celestial motion is often influenced by additional effects known as perturbations.

These include gravitational pulls from other planets, the non-uniform distribution of mass in a central body, and even relativistic corrections in the case of Mercury’s orbit around the Sun. Satellites around Earth also face atmospheric drag at low altitudes, which gradually changes their orbits.

Another fascinating concept in celestial mechanics is that of Lagrange points. These are special positions in a two-body system where gravitational and centrifugal forces balance, allowing smaller objects to stay in equilibrium. These points are used to station space observatories such as the James Webb Space Telescope.

7.0Applications of Celestial Motion in Physics and Astronomy

The study of celestial mechanics has immense practical and theoretical applications:

• Predicting planetary positions and eclipses.
• Designing artificial satellites and space missions.
• Understanding tidal effects due to the Moon and Sun.
• Explaining astronomical phenomena such as comets and meteor showers.
• Applying orbital mechanics for interplanetary missions like Mars probes.

8.0Solved Examples on Motion of Celestial Objects

Example 1: A satellite orbits Earth at height 600 km above the surface. Find its orbital velocity.

• Radius of Earth, R=6400 km.
• Mass of Earth, M= $6\times 10_{24}$ kg.

Solution:

$$\begin{gathered} r=R+h=7000\text{km}=7\times 10^6\text{m} \\ v=\sqrt{\frac{GM}{r}} \\ =\sqrt{\frac{6.67\times 10^{-11}\times 6\times 10^{24}}{7\times 10^6}} \\ v\approx 7546\text{m/s} \end{gathered}$$

Example 2: The orbital period of a satellite is T=24 hr. Find the radius of its orbit around Earth.

Solution:

$$\begin{gathered} \text{Using Kepler’s third law:} \\ {T}^2=\frac{4{\pi }^2{r}^3}{GM} \\ \text{Rearranging,} \\ r={\left(\frac{GM{T}^2}{4{\pi }^2}\right)}^{1/3}\text{Substitute T=86400\hspace{0.17em}s, }GM=3.986\times 10^{14}{\text{m}}^3/{\text{s}}^2. \\ r\approx 4.22\times 10^7\text{m} \\ \text{This corresponds to the geostationary orbit.} \end{gathered}$$