## Overview This topic covers Kepler’s laws of planetary motion and circular orbits of satellites, planets and stars. It explores the evidence for the existence of dark matter and examines the Doppler shift of spectral lines. It discusses how the Hubble constant relates galactic radial velocity to distance and how a knowledge of the Hubble constant can give an approximation to the age of the universe. ## Mathematical Skills There are a number of opportunities for the development of mathematical skills in this unit. These include estimating results; making order of magnitude calculations; understanding and using the symbols: =, <, <<, >>, >, ∝, ≈, ∆; changing the subject of an equation, including non-linear equations; solving algebraic equation, including quadratic equations; translating information between graphical, numerical and algebraic forms. ## How Science Works There are opportunities within this topic for learners to use theories, models and ideas to develop scientific explanations; to use knowledge and understanding to pose scientific questions, define scientific problems, present scientific arguments and scientific ideas; to analyse and interpret data to provide evidence, recognise correlations and causal relationships; to know that scientific knowledge and understanding develops over time; to communicate information and ideas in appropriate ways using appropriate terminology. Learners can be given the opportunity to understand how scientific knowledge and understanding developed over time and how the Big Bang Theory developed until its acceptance. Learners can use astronomical data to show how the orbital speeds of objects in spiral galaxies implied the existence of dark matter and evaluate the role of the scientific community in validating the recently discovered Higgs boson. ### Learners should be able to demonstrate and apply their knowledge and understanding of: (a) Kepler's three laws of planetary motion (b) Newton's law of gravitation 1 2 2 M M F G r = in simple examples, including the motion of planets and satellites (c) how to derive Kepler's 3rd law, for the case of a circular orbit from Newton's law of gravity and the formula for centripetal acceleration (d) how to use data on orbital motion, such as period or orbital speed, to calculate the mass of the central object (e) how the orbital speeds of objects in spiral galaxies implies the existence of dark matter (f) how the recently discovered Higgs boson may be related to dark matter (g) how to determine the position of the centre of mass of two spherically symmetric objects, given their masses and separation, and calculate their mutual orbital period in the case of circular orbits (h) the Doppler relationship in the form v c    = (i) how to determine a star's radial velocity (i.e. the component of its velocity along the line joining it and an observer on the Earth) from data about the Doppler shift of spectral lines (j) the use of data on the variation of the radial velocities of the bodies in a double system (for example, a star and orbiting exo-planet) and their orbital period to determine the masses of the bodies for the case of a circular orbit edge on as viewed from the Earth (k) how the Hubble constant (H0 ) relates galactic radial velocity (v) to distance (D) and it is defined by 0 v H D = (l) why 0 1 H approximates the age of the universe (m) how the equation 2 0 3 8 c H G   = for the critical density of a 'flat' universe can be derived very simply using conservation of energy