Extragalactic Astronomy & Cosmology

First-Half Review

[4246] Physics 316

Jane Turner [4246] PHY 316 (2003 Spring)

Kepler’s laws are empirical rules designed to describe the motionof the planets around the Sun.  Newton’s laws of motion andgravity, on the other hand, form a whole theory of dynamics thatapplies equally well to familiar objects on Earth as well asplanets.   Kepler’s laws can be mathematically derived fromNewton’s laws.

Kepler vs Newton

Jane Turner [4246] PHY 316 (2003 Spring)

Newton’s Laws

An inertial frame is one in which under the influence of no forces, an object will remain at rest or in uniform motion

Newton formulated a theory of mechanics & gravity that explained the solar system with remarkable accuracy!

Realized that gravity responsible for the motion of the Moon and planets.

Newton’s law of universal gravitationEvery mass attracts every other mass Force drops off with square of distanceKepler’s laws are a direct consequence of Newton’s law of gravity

Jane Turner [4246] PHY 316 (2003 Spring) Lecture 3

Law of Universal Gravitation

Law of universal gravitation

F = GMm/d2 Remember this one!

Gravitational force follows an inverse square law- doubling separation between two objects, grav attraction drops x 4

Jane Turner [4246] PHY 316 (2003 Spring) Lecture 3

Newtons form of Keplers Third Law

Newton also generalized Kepler’s third law as

P2=42 R3 /G(M1+M2)

Allowed Kepler’s Laws to be applied to moons and (much later) binarystars and extrasolar planets.

Remember this one!

) Plutos moon Charon orbits Pluto every 6.4 days with a semimajor axis of 19,700 km.Calculate the combined mass of Pluto and Charon

Working with Charon’s orbit around Pluto we havePCharon

2=4 2 RCharon3/G(MPluto +MCharon)

MPluto +MCharon=4 2 RCharon3 / G PCharon

2

=4 2 (1.97 10x 7) 3/ 6.67 10x -11 x (6.4 24 3600x x ) 2

=1.48 x 1022 kg

Gravitational constant G = 6.67 x 10-11 m3 kg-1 s-2

Jane Turner [4246] PHY 316 (2003 Spring)

Cosmic Distance Ladder

Jane Turner [4246] PHY 316 (2003 Spring)

Special Relativity-what is it?

In SR the velocity of light is special, inertial frames are special. Anything moving at the speed of light in one reference frame will move at the speed of light in other inertial frames.

SR satisfies Maxwells Equations, which replaced inverse square law electrostatic force by set of equations describing the electromagnetic field

SR necessary to get calculation correct where velocities ~ c

When velocities << c Newtonian mechanics is an acceptable approximation

Jane Turner [4246] PHY 316 (2003 Spring)

Summary of Formulae

Relativistic Doppler z + 1 = √[ (1+v/c)/(1-v/c) ]

Relativistic Addition of Velocities

lengthmoving =lengthrest [1-(v2/c2)]

Lorentz Contraction

timemoving = timerest [1-(v2/c2)]Time Dilation

Lorentz Factor or

Mass massmoving =massrest/[1-(v2/c2)]

Jane Turner [4246] PHY 316 (2003 Spring)

Example

Mass massmoving =massrest/[1-(v2/c2)]

An object has a mass 1g at rest. What is its mass when traveling at v=0.9999c?

massmoving =1g/[1-(0.9999c2/c2)]=70.71g !!

Jane Turner [4246] PHY 316 (2003 Spring)

SPACE-TIME DIAGRAMS

“LightCone”

light beam follows a world line ct=x , using x versus ct - this is a line at 450

object traveling at v<c has worldline > 450

Jane Turner [4246] PHY 316 (2003 Spring)

SPACE-TIME DIAGRAMS

“LightCone”

Inertial Observers

Accelerated Observer

Jane Turner [4246] PHY 316 (2003 Spring)

SPACE-TIME DIAGRAMS

“LightCone”

Jane Turner [4246] PHY 316 (2003 Spring)

The Special Theory of Relativity is restrictive in the sense that itcannot be used to study gravitational phenomena.  The GeneralTheory of Relativity (which contains the Special Theory as asubset) can be used to study gravity. More precisely, the Special Theory of Relativity assumes theexistence of inertial frames and specifies how one inertial frame isrelated to another.   The General Theory of Relativity, in addition,tells us which frames are inertial (i.e., the free-falling frames), andhow the frames “mesh” together (i.e., space-time curvature).

Special vs General Relativity

Jane Turner [4246] PHY 316 (2003 Spring)

General Relativity

General Relativity is a geometrical theory concerning the curvature of Spacetime

Gravity is the manifestation of the curvature of Spacetime Gravity is no longer described by a gravitational "field" /”force” but is a manifestation of the distortion of spacetime

Matter curves spacetime; the geometry of spacetime determines how matter moves

Energy and Mass are equivalent Light is energy, and in general relativity energy is affected by gravity just as mass is

Jane Turner [4246] PHY 316 (2003 Spring)

The Equivalence Principlethe effects of gravity are exactly equivalent to the effects of acceleration

thus you cannot tell the difference between being in a closed room on Earth and one accelerating through space at 1g

any experiments performed would produce the same results in both cases

Jane Turner [4246] PHY 316 (2003 Spring)

Geometries

Our homogeneous & isotropic universe can have one of 3 types of geometry

Jane Turner [4246] PHY 316 (2003 Spring)

“Straight Lines” in Curved Spacetime

Can examine the geometry of spacetime by looking at the orbits of bodies around large masses

- Earths motion around the Sun, not under the force of gravity but following the straightest possible path in curved spacetime (curved due to Suns large mass

Jane Turner [4246] PHY 316 (2003 Spring)

The Metric EquationA metric is the "measure" of the distance between points in a geometry

For close points r2= fx2+ 2g x y + hy2 - metric equationso for any 2 points sum the small steps along the path- integrate!

A general spacetime metric is

s2= c2t2 -ctx-x2 - for coordinate x, , depend on the geometry

Einstein took the spacetime metric, homogeneity, isotropy, local flatness absoluteness of speed of light. Machs idea and the reduction to Newtonian solutions for small gravity ->

Jane Turner [4246] PHY 316 (2003 Spring)

One-line description of the Universe

G=8GT

c4

G, T tensors describing curvature of spacetime & distribution of mass/energy G constant of gravitation labels for the space & time components

This one form represents ten eqns!

geometry = matter + energy

Jane Turner [4246] PHY 316 (2003 Spring)

GR - Gravitational Redshift

We discussed how gravity affects the energy of light

Light traveling up ‘against’ gravity loses energy, ie the frequency gets longer (larger)

GRAVITATIONAL REDSHIFT!

Time between peaks increases -time passes more slowly under strong gravity

GRAVITATIONAL TIME DILATION

Jane Turner [4246] PHY 316 (2003 Spring)

Redshift Review

Redshift refers to the lengthing of the wavelength of light atobservation compared with its length at emission. Redshifts can beproduced by several processes. One is simply due to relativemotion in the sources (the simple Doppler shift). Another isrelativistic time dilation. Still another process is the result of aphoton losing energy as it climbs out of a gravitational field. Thislatter effect is a gravitational redshift and it is mathematicallyexpressed in the spacetime metric produced by the gravitatingbody. The term cosmological redshift is reserved for the redshiftsproduced by the overall expansion of space. It is important to keepthe various causes of redshift distinct.

Doppler redshift Cosmological redshift Gravitational redshift

Jane Turner [4246] PHY 316 (2003 Spring)

GR Tests: Light Bending

Eddington’s measurements of star positions during eclipse of 1919 were found to agree with GR, Einstein rose to the status of a celebrity

Jane Turner [4246] PHY 316 (2003 Spring)

GR-Light Bending

Light bending can be most dramatic when a distant galaxy lies behind a very massive object (another galaxy, cluster, or BH)

Spacetime curvature from the intervening object can alter different light paths so they in fact converge at Earth - grossly distorting the appearance of the background object

Jane Turner [4246] PHY 316 (2003 Spring)

GR Tests: Planetary orbits

GR predicts the orbits of planets to be slightly different to Newtonian physics

Long been know there was a deviation of Mercurys orbit vs Newtonian-prediction

Einstein delighted to find GR exactly accounted for the discrepancy

Jane Turner [4246] PHY 316 (2003 Spring)

GR Tests: Gravitational Waves

Supernova explosion may cause them

Massive binary systems cause them & thus lose energy resulting in orbital decay-decays detected! Taylor & Hulse in 1993 -indirect support of GR

Changes in mass distribution/gravitational field which changes with time produces ripples in spacetime-gravitational waves

WeakPropagate at the speed of lightShould compress & expand objects they pass by

Jane Turner [4246] PHY 316 (2003 Spring)

Tests of GR: Line Profiles