4.3 Notes

(a) Kepler’s Three Laws of Planetary Motion

1. Law of Orbits (Elliptical Orbits)

Planets move in elliptical orbits around the Sun, with the Sun at one focus of the ellipse.

Special Case: A circular orbit is a special case of an ellipse where both foci coincide.

2. Law of Areas (Equal Areas in Equal Time)

A line connecting a planet to the Sun sweeps out equal areas in equal time.

This means that planets move faster when closer to the Sun (perihelion) and slower when farther (aphelion).

3. Law of Periods (Orbital Period Relation)

The square of a planet’s orbital period (T²) is proportional to the cube of its semi-major axis (r³):

T² ∝ r³

where G is the gravitational constant, and M is the mass of the central object (e.g., the Sun).

(b) Newton’s Law of Gravitation in Simple Examples

Newton’s Law of Universal Gravitation:

F = G M₁M₂ / r²

where:

• F = gravitational force between two masses

• G = gravitational constant (6.674×10⁻¹¹ Nm²kg⁻²)

• M₁, M₂​ = masses of two objects

• r = distance between their centres

Examples:

Earth and Moon:

The gravitational force keeps the Moon in orbit around the Earth.

The force is proportional to 1/r², so if the Moon were twice as far, the force would be four times weaker.

Planetary Motion:

The Sun’s gravity provides the centripetal force needed for planets to stay in orbit.

For a planet of mass mmm in a circular orbit:

GM_sunm / r² = mv² / r

This leads to Kepler’s third law

Satellites:

Artificial satellites stay in orbit because of Earth’s gravity acting as a centripetal force.

Higher orbits have longer periods.

(c) Deriving Kepler’s 3rd Law for Circular Orbits

For a planet of mass m orbiting a star of mass M in a circular orbit:

Gravitational Force Provides Centripetal Force:

GMm / r² = mv² / r

Rearrange for Orbital Speed v:

v² =GM / r

Using Orbital Period:

The period of orbit T is given by:

v = 2πr / T

Squaring both sides:

v² = 4π²r² / T²

Equating the Two Expressions for v²

4π²r² / T²= GM / r​

Rearrange to Get Kepler’s 3rd Law:

T²=4π²r³ / GM

This confirms Kepler’s law mathematically from Newton’s law of gravity.

(d) Using Orbital Data to Calculate Mass of a Central Object

Using Kepler’s third law:

T2 = 4π²r³ / GM

Rearrange to find mass M:

M = 4π²r³ / GT²

Example:

• If we know a planet's orbital period and its distance from a star, we can calculate the star’s mass.

• This method is used to determine the mass of the Sun from Earth’s orbit.

(e) Orbital Speeds in Spiral Galaxies and Dark Matter

Observations:

In spiral galaxies, stars farther from the centre orbit faster than expected based on visible matter.

According to Newton’s laws, the orbital speed v should decrease at large distances: v = GM / r ​​ where M is the mass enclosed within radius r.

Dark Matter Hypothesis:

The observed flat rotation curves (constant speed at large radii) suggest extra unseen mass (dark matter).

Dark matter is thought to form a halo around galaxies, providing extra gravitational pull.

(f) Higgs Boson and Dark Matter

Higgs Boson:

The Higgs boson, discovered in 2012, is responsible for giving mass to fundamental particles.

Relation to Dark Matter:

Some theories propose Higgs-mediated interactions could link to dark matter particles.

Alternative particles like WIMPs (Weakly Interacting Massive Particles) may interact via the Higgs field.

Research continues into whether the Higgs boson directly couples to dark matter particles.

(g) Centre of Mass and Orbital Period of Two Orbiting Bodies

Centre of Mass of Two Bodies

For two masses M₁​ and M₂​ separated by a distance d, the centre of mass R is:

R = M₁ d₁ + M₂ d₂ / M₁ + M₂

where d₁​ and d₂ are distances from the centre of mass.

Mutual Orbital Period for Circular Orbits

Both bodies orbit their common centre of mass, with orbital radii r₁​ and r₂ such that:

M₁r₁ = M₂r₂

Using Kepler’s law for a two-body system:

T² = 4π²(r₁+r₂)³ / G(M₁ + M₂)

where (r₁ + r₂)= d is the total separation.

Example:

• The Earth and Moon orbit their common centre of mass, which is inside the Earth but not at its centre.