Kepler's Law: Galaxies And Their Elliptical Orbits

Kepler's laws of planetary motion describe how planetary bodies orbit the Sun. They state that planets move in elliptical orbits with the Sun as a focus, a planet covers the same area of space in the same amount of time no matter where it is in its orbit, and a planet’s orbital period is proportional to the size of its orbit. While Kepler's laws were developed in the context of planetary orbits, they can be applied to model other natural objects like stars or comets, as well as man-made devices like rockets and satellites in orbit. However, it is not clear if Kepler's laws apply to galaxies.

Kepler's First Law

> The orbit of every planet is an ellipse with the sun at one of the two foci.

Mathematically, an ellipse can be represented by the formula:

> r = p / (1 + ε cos θ)

Where p is the semi-latus rectum, ε is the eccentricity of the ellipse, r is the distance from the Sun to the planet, and θ is the angle to the planet's current position from its closest approach, as seen from the Sun.

For an ellipse, 0 < ε < 1; in the limiting case of ε = 0, the orbit is a circle with the Sun at the centre. At θ = 0°, perihelion, the distance is minimum (rmin = p / (1 + ε)), and at θ = 180°, aphelion, the distance is maximum (rmax = p / (1 - ε)).

The semi-major axis, a, is the arithmetic mean between rmin and rmax:

> a = (rmax + rmin) / 2 = p / (1 - ε^2)

The semi-minor axis, b, is the geometric mean between rmin and rmax:

> b = sqrt(rmax * rmin) = p / sqrt(1 - ε^2)

The semi-latus rectum, p, is the harmonic mean between rmin and rmax:

> p = ((rmax^-1 + rmin^-1) / 2)^-1 = rmax * rmin = b^2

The eccentricity, ε, is the coefficient of variation between rmin and rmax:

> ε = (rmax - rmin) / (rmax + rmin)

The area of the ellipse is:

> A = πab

The special case of a circle is ε = 0, resulting in r = p = rmin = rmax = a = b and A = πr^2.

Kepler's Second Law

The Second Law can be used to show that the velocity of a planet changes as it moves along its orbit. When a planet is closest to the Sun, it travels faster, and when it is farthest away, it travels slower. This is because the angular momentum of the planet remains constant. The point of nearest approach to the Sun is called perihelion, and the point of greatest separation is called aphelion.

Kepler's Third Law

The law can be summarised as follows: "The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit." In other words, the square of the time a planet takes to orbit the Sun (its period, P) is directly proportional to the cube of the mean distance from the Sun (d).

The equation for Kepler's Third Law is P² = a³, meaning the period of a planet's orbit (P) squared is equal to the size of the semi-major axis of the orbit (a) cubed when expressed in astronomical units.

This law allows us to compare the orbital period and radius of a planet's orbit with those of other planets. It calculates the harmonies of the planets by taking the ratio of the squares of the periods (T²) to the cubes of their average distances from the Sun (R³).

Thanks to Kepler's Third Law, if we know a planet's distance from its star, we can calculate the period of its orbit, and vice versa. This is possible because the distance between Earth and the Sun (approximately 92,960,000 miles or 149,600,000 kilometres) and one Earth year (365 days) are known.

As a planet's distance from the Sun increases, the time it takes to orbit the Sun also increases rapidly. For example, Mercury, the closest planet to the Sun, completes an orbit in 88 days, whereas Saturn, the sixth planet from the Sun, takes 10,759 days.

This law, in combination with Kepler's second law, has enabled astronomers to derive the masses of stars in binary systems, which is vital for understanding the structure and evolution of stars.

The impact of Kepler's laws on the discovery of dark matter

Kepler's laws of planetary motion describe how planetary bodies orbit the Sun. Kepler's laws state that:

• Planets move in elliptical orbits with the Sun as a focus.
• A planet covers the same area of space in the same amount of time no matter where it is in its orbit.
• A planet’s orbital period is proportional to the size of its orbit (its semi-major axis).

These laws were formulated by Johannes Kepler in the 17th century, based on the astronomical observations of Tycho Brahe. Kepler's laws replaced the previously assumed circular orbits and epicycles in the heliocentric theory of Nicolaus Copernicus with elliptical orbits and explained how planetary velocities vary.

In addition, velocity dispersion estimates of elliptical galaxies and globular clusters also suggest the presence of non-luminous matter. By applying Kepler's laws, the mass distribution in these systems can be calculated, and the discrepancy between the predicted and observed velocity dispersions implies the existence of dark matter.

Furthermore, gravitational lensing observations of galaxy clusters provide another line of evidence for dark matter. By measuring the mass of a cluster through gravitational lensing and comparing it to estimates based on the brightness and number of galaxies, it was found that the gravity effect of the visible galaxies was too small to account for the fast orbits, thus indicating the presence of unseen mass.

In summary, Kepler's laws have had a significant impact on the discovery and understanding of dark matter. By describing the motion of planetary bodies and allowing for the calculation of mass distributions, Kepler's laws provided crucial evidence for the existence of dark matter in the form of non-luminous matter that influences the dynamics of galaxies and galaxy clusters.

The limitations of Kepler's laws

Kepler's laws of planetary motion describe how planetary bodies orbit the Sun. However, they have certain limitations. Firstly, Kepler's laws assume that the orbit of a planet is a perfect ellipse, but in reality, planets do not move in exact ellipses. Most planetary orbits are nearly circular, with eccentricities close to zero. For example, the eccentricity of Earth's orbit is only 0.0167, making it almost a perfect circle.

Another limitation of Kepler's laws is that they do not take into account the gravitational interactions between planets. These interactions cause planets to speed up and slow down slightly as their positions align, which is not accounted for in the laws. Additionally, Kepler's laws were formulated based on observations of the Solar System and may not apply universally to other celestial bodies or galaxies.

Furthermore, Kepler's laws assume a heliocentric model of the Solar System, with the Sun at the centre. This assumption may not hold true for other planetary systems or galaxies, where the central body may not be a star, or the orbits may not be centred on a single body.

While Kepler's laws provide a good summary of the overall motion of the planets in our Solar System, they are not exact, and there are alternative models that may be more accurate in certain cases. Kepler's laws also do not consider the impact of relativity, which can affect the motion of bodies in space, as demonstrated by Einstein's general theory of relativity.

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