4.3 - Orbits and The Wider Universe

Kepler’s First Law of Planetary Motion

Each planet moves in an ellipse with the Sun at one focus

Kepler’s second law of planetary motion

The line joining a planet to the centre of the Sun sweeps out equal areas in equal times

Kepler’s third law of planetary motion

T² , the square of the period of the planet’s motion, is proportional to r³ , in which r is the semi-major axis of its ellipse. [For orbits which are nearly circular, r may be taken as the mean distance of the planet from the Sun.]

Derivation of Kepler’s 3rd Law

Equation for location of centre of mass

r₁ = (M₂/M₁ + M₂)d

Equation for orbital period

T = 2π√(d³/G(M₁+M₂)

Approximation of masses in circular orbits

M₁ » M₂

M₁ + M_{2 =} M₁

Dark matter

Matter which we can’t see, or detect by any sort of radiation, but whose existence we infer from its gravitational effects

Discovery made based on orbital speeds of objects in spiral galaxies

Implies the existence of dark matter

Explain the Doppler Effect

Observing the spectra of stars revealed that the wavelength of spectral lines changes. This due to the apparent motion of the object. The velocity of this motion can therefore be calculated using the equation; Δλ/λ = v/c

Doppler Effect equation

Δλ/λ = v/c

v = radial velocity (ms^-1)

c = speed of light (3×10⁸ ms^-1)

Δλ = observed (shifted) wavelength (m)

λ = emitted (unshifted) wavelength (m0

Radial velocity of a star

The component of a star’s velocity along the line joining it and an observer on the Earth

Use of the graph of velocity to determine mass

Galactic radial velocity

The mean component of a galaxy’s velocity along the line joining it and an observer on Earth

How Hubble’s constant is determine

Measurements of the radial velocity of galaxies plotted against the distance to the galaxy gave a straight line

v ∝D the gradient of the line, the proportionality constant is the Hubble’s constant

How 1/H₀ approximates the age of the universe

The radial velocity has been approximately constant since the begining of the universe so that D=vT

Hence, v = H₀ x vt and = 1/H₀

Derivation of critical density of a ‘flat’ universe