Our Place in the Cosmos

Our Place in the Cosmos Lecture 7 Gravity - Ruler of the Universe

Gravity • Gravity rules the Universe • It holds objects like the Sun and Earth together • Sun’s gravity determines motion of the planets of the Solar System • Gravity binds stars into galaxies and galaxies into clusters • In this lecture we will follow Newton’s lines of reasoning in arriving at his law of gravity

What is Gravity? • Gravity is a force between any two objects due to their masses • It is a “force at a distance” - two objects do not need to come into contact for them to exert a gravitational force on one another • As with the law of inertia, our understanding of gravity begins with Galileo Galilei

Acceleration due to Gravity • Galileo observed that all freely falling objects accelerate towards the Earth at the same rate regardless of their mass • A marble and a cannonball dropped at the same time from the same height will hit the ground simultaneously • The gravitational acceleration near the Earth’s surface is usually indicated by the symbol g and has a measured value of about 10 m/s2 • An object dropped from rest will be moving at 10 m/s after 1 second, 20 m/s after two and so on (neglecting air resistance)

Isaac Newton • Newton realised that if all objects fall with the same acceleration, then the gravitational force on an object must be determined by its mass • Recall that Newton’s 2nd law says acceleration = Force/mass • Since all objects have the same acceleration, then the gravitational force divided by mass must be the same for all objects • A larger mass feels a larger gravitational force: Fgrav = mg • Note that gravitational mass is the same as inertial mass - this equivalence is the basis for GR

Weight vs Mass • Weight is the gravitational forceFgacting on an object • An object’s weight thus depends on its location, whereas its mass does not • On the Earth’s surface, weight is equal to mass times g, the acceleration due to gravity • It is incorrect (but common) to say that an object “weighs 2 kg” • A 2 kg mass actually weighs about 2 kg x 10 m/s2 or 20 kg m/s2 or 20 Newtons (20 N)

Gravitational Force • As with every other force, any gravitational force has an equal and opposite force (Newton’s 3rd law) • Drop a 20 kg cannonball and it falls towards the Earth • At the same time Earth falls towards the cannonball! • We do not notice the Earth’s motion in this case because the Earth is so much more massive than the cannonball • Each object feels an equal and opposite force but acceleration equals force divided by mass

Gravitational Force • Newton realised that if doubling the mass of an object doubles the gravitational force between it and the Earth, then doubling the mass of the Earth would do the same • Thus the gravitational force experienced by an object is proportional to the product of the mass of the object times the mass of the Earth: Fg = something x mass of Earth x mass of object • Since objects fall towards the centre of the Earth, Fgis an attractive force acting along a line between the two masses

Gravitational Force • But why, reasoned Newton, should this law of gravity apply only to the Earth? • Surely the gravitational force between any two masses m1 and m2 should be given by the product of the masses: Fg = something x m1 x m2 • Above reasoning follows from Galileo’s observations of falling objects and Newton’s laws of motion • But what is the “something” in the above equation?

Inverse Square Law • Kepler had already reasoned that since the Sun is at the focus of planetary orbits, then it must be exerting some influence over the planets’ motion • He also reasoned that this influence weakens with distance - why else does mercury orbit so much faster than Jupiter or Saturn? • The area of a sphere increases with the square of its radius (A = 4r2) • Thus Kepler reasoned that the Sun’s influence should decrease with the square of distance

Inverse Square Law • Kepler’s proposal was interesting but not a scientific theory as he lacked a good idea as to the true source of the influence and also lacked the mathematical tools to predict how an object should move under such an influence • Newton had both - he realised that gravity should act between the Sun and the planets, and that the gravitational force was probably Kepler’s “influence” • In this case, the “something” in Newton’s expression for gravity should diminish with the square of the separation between two objects

Newton’s Universal Law of Gravitation • Gravity is a force between any two objects, and has the following properties • It is an attractive force acting along a straight line between the objects • It is proportional to the product of the masses of the objects m1 x m2 • It decreases with the square of the separation r between the objects • Fg = G x m1 x m2 / r2 universal gravitational constant

Weakness of Gravity • It is now possible to measure the value of the gravitational constant G using sensitive equipment: G = 6.67 x 10-11 N m2 / kg2 • The force between two bowling balls placed 1 foot apart is Fg  4 x 10-8 N, about the same as the weight of a single bacterium! • Gravity is only noticeable in everyday life because the Earth is so massive

Acceleration due to Gravity • For an object of mass m, Newton’s 2nd law of motion says Fg= mg • Universal law of gravitation saysFg= G Mm/R2 • Equating these two expressions givesmg= G Mm/R2 • The massm appears on both sides and so may be divided out to giveg= G M/R2 • Thus the acceleration g due to gravity is independent of the mass of the object - as observed by Galileo!

Mass of the Earth • Rearranging the last expression for g, we find M=gR2/G • Everything on RHS may be measured • g by acceleration of falling objects • R by altitude of celestial pole with latitude • G via lab experiments • We find M 6 x 1024 kg • Newton inverted this argument to estimate a value for G by assuming that Earth has the same density as typical rocks

Gravity and Orbits • Newton speculated that Kepler’s solar “influence” on the planets’ orbits is gravity, but a good physical theory should be testable • He lacked the sensitive apparatus to measure gravitational forces directly, but he was able to show that his law of gravity predicted that the planets should orbit the Sun just as Kepler’s empirical laws described • Newton was thus able to explain Kepler’s laws • Gravity is just one example of a physical law that was first tested by astronomical observations

Predicting Orbits • A full prediction of planets’ orbits requires use of the branch of mathematics known as calculus that Newton invented for the purpose • However, we can still gain a conceptual understanding of how orbits come about by a series of thought experiments • These are experiments that are not executable in practice, but that still give us a good conceptual grasp of a physical problem

Falling around the Earth • In this thought experiment we fire a cannonball horizontally from a height of a few metres and neglect air resistance • As cannonball travels horizontally, it also falls towards the ground • The faster we fire the ball, the further it travels before hitting the ground • As the ball travels further and further, the ground starts to curve away from underneath it • If we fire the ball fast enough, it will maintain a constant height above the ground and complete an orbit of the Earth

Captions

Falling around the Earth • An object orbiting the Earth is literally “falling around it” - it is always falling towards the Earth’s centre • First man-made satellite to orbit the Earth was the Sputnik I satellite launched in 1957 • Astronauts float around the cabin of an orbiting spacecraft for the same reason: both the spacecraft and the astronaut are in free fall - according to Newton’s law of gravity both accelerate towards the Earth at the same rate

Astronaut falling freely around the Earth Shuttle and astronaut experience same gravitational acceleration Both are independent satellites sharing the same orbit

Orbital Velocity • How fast must Newton’s cannonball move to orbit the Earth? • An object moving round in a circle requires a centripetal force to prevent it from flying off in a straight line (Newton’s 1st law) • For a ball on a string, the string provides the force, for an object in orbit it is gravity • For a satellite on a circular orbit, force required for uniform circular motion = force provided by gravity

Circular Velocity • One can show that centripetal acceleration for an object moving in circle of radius r at velocity v is given by a = v2/r • By Newton’s 2nd law of motion, the centripetal force is given by F = ma = mv2/r • If object of mass m is orbiting a body of much larger mass M, centripetal force is provided by gravityFg= G Mm/r2 • Equating these forces, mv2/r = GMm/r2 • Mass m cancels out, leaving v2circ = GM/r

Sun’s Mass • Any satellite moving on a stable circular orbit must be travelling at the circular velocity vcirc • Circular velocity at Earth’s surface is about 8 km/s • Earth’s orbit about the Sun is almost circular with a speed of about 30 km/s [determined from stellar aberration] • We also know radius of Earth’s orbit [1 AU  1.5 x 1011 m] • We can then invert the formula for circular velocity to estimate the Sun’s mass: M  2 x 1030 kg

“Harmony of the Worlds” • We can now predict the period for a circular orbit • Period P = circumference of orbit/circularvelocityP = 2r/[G M/r] • Square each side and rearrange to giveP2 = 42/(G M) x r3 • Newton was thus able to predict Kepler’s 3rd law for circular orbits • Kepler’s laws provide an empirical test of Newton’s theory  Newton’s theory helps us understand Kepler’s laws

Mass Estimates • Newton’s form of Kepler’s 3rd law can be rearranged to readM = 42/G x (A3/P2) • This formula is used throughout astronomy to make mass estimates • It still holds when mass of orbiting object is comparable to central mass • In this case each object orbits about their common centre of mass and M above is the total mass of the system

Summary • Starting with Galileo’s observation that objects fall at the same rate, Newton predicted a gravitational force between all masses that was proportional to the product of the masses and inversely proportional to the square of their separation • He showed that this simple model could explain Kepler’s three laws of planetary motion and could be used to estimate masses of astronomical objects • However, we still don’t really know exactly what gravity is

Our Place in the Cosmos