Gravity and Motion
Prof. Andy Lawrence
Astronomy 1G 2011-12
Motion in astronomy
• Most common cause of motion in the cosmos is Gravity • Need to understand how this works and get key formulae • Observing motions then allows us to calculate masses of astronomical bodies
Astronomy 1G 2011-12
Newton's Laws of Motion
Newton’s first law: An object moves at a constant velocity if there is not net force acting upon it.
Newton’s second law: The acceleration imparted to a body is proportional to and in the direction of the force applied, and inversely proportional to the mass of the body.
dp~ note : acceleration and F = m~a = dt momentum are vectors
Newton’s third law: For every force, there is an equal but opposite reaction force.
Astronomy 1G 2011-12 Newton's Law of Gravitation
• The force that causes motion (through F=ma) is usually Gravity • Force between two bodies depends on their masses and the distance between them
GMm G is the universal Fgrav = r2 constant of gravitation
• The result is normally to cause one body to orbit the other • Look first at simple circular orbits
Astronomy 1G 2011-12
Centripetal Acceleration
Consider a body moving in a circular orbit of radius r about a centre of force. From symmetry, the speed must be constant, but the direction must be constantly changing.
and
The centripetal acceleration (acceleration towards the centre) therefore has a magnitude of (by definition of acceleration)
Astronomy 1G 2011-12
Speed of gravitational motion
Small body m orbiting large body M GMm Fgrav = at distance r feels force: r2 GM a = By F=ma causes acceleration : grav r2
v2 a = Must be same as the centripetal acceleration : cent r
So equating these gives GM v2 = r
Probably the most important equation in astronomy !
Astronomy 1G 2011-12 Example 1 : Mass of Sun
The Earth orbits the Sun with a velocity of 29 km/sec at a distance of 1.5 x 1011 m (Will see later how we know these numbers).
GM D v2 v2 = so M = Sun Earth =1.9 1030kg r sun G ⇥
Example 2 : Masses of planets
Observe the orbits of their satellites – Need the distance to the planet (see next few slides). – Max angular distance of the satellite from planet then gives radius of orbit (R=Dθ) – Period of satellite orbit ==> orbital velocity v= 2πR/T
Astronomy 1G 2011-12
Example 3 : Binary stars
• If one star is much less massive than the other, can treat it in the same way as a solar system planet : patiently observe the orbital period – need distance to star (see later) – angular separation + period ==> orbital radius and velocity
• If the star’s have comparable mass – more complicated. – Each star orbits their common centre of mass
Astronomy 1G 2011-12
Sun’s orbital motion tells us mass within Sun’s orbit: v = 220 km/s d = 8 kpc 11 ==> M = 10 MSun (100 billion solar masses)
Rotation of outermost parts of Galaxy tells us:
12 Mtotal = 10 MSun (1 trillion solar masses)
But measuring light from observed stars suggests
11 Mstars = 10 MSun
90 percent of the Galaxy's mass is in “dark matter”. Celestial Mechanics : a little more detail
Astronomy 1G 2011-12
Celestial sphere
• The ecliptic is the path the Sun follows as it appears to circle the celestial sphere. • The celestial equator is a projection of the Earth’s equator into space. - The ecliptic and the celestial equator do not coincide because the Earth’s axis is tilted relative to the ecliptic. - Seasons!! • During the course of a year, the Sun will appear to move through the 12 constellations of the zodiac. • The planets also move relative to the stars on the celestial sphere.
Astronomy 1G 2011-12
The stars are very distant; their The motion of the stars relative pattern stays fixed while the pattern as a whole rotates due to the rotation of the Earth
The observed motion of the stars will depend on the latitude from which you are observing.
Astronomy 1G 2011-12 The motion of the planets • Planets move gradually with respect to the background pattern of stars • This is because they in orbit around the Sun, at different speeds depending on their distance from the Sun, making for a fairly complicated motion as seen from Earth (one of the planets...) • Kepler showed that they move in (somewhat) elliptical orbits, but for now we will treat them as moving in circular orbits. • Comets move in very elliptical orbits, or in hyperbolic orbits, coming in to the inner solar system temporarily.
Astronomy 1G 2011-12
Kepler’s laws
By observing the heavens, Johannes Kepler derived 3 empirical laws about the motion of bodies (planets) in the solar system. Kepler’s first law: The orbit of each planet is an ellipse with the Sun at one focus. Kepler’s second law: The radius vector to a planet sweeps out equal areas in equal time. Kepler’s third law: The squares of the sidereal periods of the planet’s orbits are proportional to the cubes of the semimajor axes of their orbits (P2=ka3).
Note 1 The Sun is at the focus of the ellipse, not at its centre.
Note 2 The figure is exaggerated; planetary orbits are only slightly elliptical. Astronomy 1G 2011-12
Planetary orbits and scale of solar system
• Using circular orbit approximation, it is easy to derive Kepler's third law :
GM • For planet orbiting the Sun at distance D : v2 = Sun D
2⇡D • The orbital period is : P = v
P 2GM • So the distance to the planet is given by D3 = 4⇡2
• Note if we know MSun we can work out all the planet distances... but to calculate MSun we need at least one planet distance ...
Astronomy 1G 2011-12 Scale of solar system
• Earth-Sun distance usually given symbol a and is referred to as the "Astronomical Unit (AU)" : a = 1 A.U. • Everything else in Astronomy is relative to this • Key measurement is the distance to Venus. Historically measured by parallax but modern measurement by radar
When Venus is at its greatest elongation (the furthest from the Sun when viewed from Earth), we can use trigonometry to determine the distance from the Sun to the Earth – Radar gives distance, d, to Venus. – a = d/cos(e)
Elliptical orbits An ellipse has polar equation where r, θ are distance and angle as seen from the focus, and a is the semimajor axis -the average distance from the Sun to the planet. The eccentricity e is the ratio of the centre-focus distance CF to the semi- major axis
The sum r+r' (see figure) is constant and equal to 2a. Classically, this was taken as the definition of an ellipse and leads to the "two pins plus string" method for drawing one.
Note also that if the Sun is at F :
perihelion (closest approach) : rperi = a – ae = a(1-e)
aphelion (furthest) : rap = a + ae = a(1+e)
Finally, note that the eccentricity can also be expressed as : b2 where b is the semi-minor axis e = 1 2 r a
Astronomy 1G 2011-12
Types of orbit
Orbits caused by gravity are always one of a family of curves :
• Circle : e=0 Most planets, satellites and asteroids • Ellipse 0 Most comets have orbits that are close to parabolic, • Parabola e=1 coming in from large distances Some comets have hyperbolic • Hyperbola e>1 orbits and will eventually leave the solar system completely Astronomy 1G 2011-12 Conservation of Angular Momentum A body orbiting at a radius r with a mass m and speed v has an angular momentum L = ~r p~ = m(~r ~v ) ⇥ ⇥ Differentiating the above gives dL dp~ = ~v p~ + ~r = ~r F~ dt ⇥ ⇥ dt ⇥ ✓ ◆ For a central force like gravity, r and F are parallel so dL/dt = 0. The angular momentum, L, of an orbiting body is constant. Note this is true in a vector sense; i.e. L stays in the same direction as well as the same size - so the orbit stays in a fixed plane. But why are all the planets in the same plane as each other ? Tells us something about history of solar system... Astronomy 1G 2011-12 Angular Momentum in the Solar System • Earth : M=6x1024 kg v=3x104 m s-1 D=1.5x1011 m → L= M.v.D = 2.7 x 1040 kg m2 s-1 • Sun : M=2x1030 kg v=2x103 m s-1 R=7x108 m shape factor=0.4 Rotation period → L= 0.4 x M.v.R = 1.1 x 1042 kg m2 s-1 =25.3 days • Jupiter : M=1.9x1027 kg v=1.3x104 m s-1 D=7.8x1011 m Jupiter year = 11.86 years → L= M.v.D = 1.9 x 1043 kg m2 s-1 20 times as much as the Sun • Adding up planets, they contain 97% of A.M. in solar system • Sun contains most of the mass • but planets contain most of the angular momentum • a big clue to the formation of the solar system Astronomy 1G 2011-12