The Expanding Universe Tuesday, January 15

Thomas Digges (16th century) proposed an infinite universe.

Infinite universe: hard to reconcile with appearance of the night sky. Night sky is dark, with stars (small in angular size) scattered across it.

Why is the darkness of the night sky paradoxical? “The night sky is dark”: called Olbers’ paradox, after astronomer who discussed the subject in 1823. Why is the darkness of the night sky paradoxical?

Only a finite number of stars on the celestial sphere. If stars were stuck on a celestial sphere or dome, darkness would not be paradoxical. Only a finite number of stars on the celestial sphere.

In an infinite universe with infinite number of stars, paradox arises. How bright do we expect the sky to be in such a universe?

ASSUMPTIONS: Suppose there are n stars per cubic parsec of the universe. In Sun’s neighborhood, n = 1/pc3 Suppose that an average star is a sphere with radius R. For Sun, R = 7×105 km = 2×10-8 pc

You are here r = radius of shell t = thickness of shell

What’s the surface area of the spherical shell? Area = 4 π r2 What’s the volume of the spherical shell? Volume ≈ area × thickness ≈ 4 π r2 t How many stars are in the shell? Number = volume × n = 4 π r2 t n

What fraction of the shell’s area is covered by stars? What’s the cross-sectional area of a single star? Cross-section = π R2 What’s the cross-sectional area of all the shell’s stars? Total = (π R2) × (4 π r2 t n) What fraction of the shell’s area is covered by stars?

What fraction of the shell’s area is covered by stars? Fraction = area of stars / area of shell = (π R2 ) × ( 4 π r2 t n ) / ( 4 π r2 ) = π R2 t n Independent of r, the radius of the shell!

But we’ve assumed an infinite number of shells! A single shell will cover only a tiny part of the sky with stars. For a shell 1 parsec thick, fraction covered = π R2 t n ≈ 10-15 But we’ve assumed an infinite number of shells!

The entire night sky should be as bright as the Sun’s surface! 1015 (one quadrillion) shells, each covering a quadrillionth of the sky with stars, will completely pave the sky with stars. The entire night sky should be as bright as the Sun’s surface!

Olbers’ Paradox for Trees: In a large enough forest, every line of sight ends at a tree.

Which of my assumptions was wrong? My conclusion – that the sky is uniformly bright – is utter rubbish. The night sky really is dark. Which of my assumptions was wrong?

Dubious assumption #1: Dubious assumption #2: The universe is infinitely large. Dubious assumption #2: The universe is eternally old.

The speed of light (c) is large but finite. c = 300,000 km/sec (186,000 miles/sec).

If the universe has a finite age, then distant stars haven’t had time to send us the message “We’re here!”

Discussing Olbers’ paradox, we assumed the universe was static (neither expanding nor contracting). This was the general assumption until the 20th century: but was it correct?

If the universe is expanding, distant galaxies will be moving away from us. If the universe is contracting, distant galaxies will be moving toward us.

Q: How can we tell if a galaxy is moving toward us or away from us? A: Look for the Doppler shift of light from the galaxy.

We can think of light as a wave traveling through space. Wave = any periodic fluctuation traveling through a medium.

Ocean wave = fluctuation in height of water. Sound wave = fluctuation in pressure. Electromagnetic wave = fluctuation in electric and magnetic fields.

Describing a wave: a λ Wavelength (λ) = distance between wave crests. Amplitude (a) = height of crests above troughs. a

The color of visible light is determined by its wavelength. Visible light: wavelengths from 400 to 700 nanometers. [1 nanometer (nm) = 10-9 meters]

The Sun’s spectrum (amount of light as a function of wavelength): Hydrogen Note the dark lines: from elements that absorb light at specific wavelengths.

Radial velocity of an object is found from the Doppler shift of its light. Radial velocity = how fast object is moving toward you or away from you.

Doppler shift: If a wave source moves toward you or away from you, the wavelength changes. Christian Doppler (1803-1853)

The reason for Doppler shifts: Wave crests are “bunched up” ahead of wave source, “stretched out” behind wave source.

(should be “violetshift”, more logically) If light source is moving toward you, wavelength is shorter (called blueshift). (should be “violetshift”, more logically) If light source is moving away from you, wavelength is longer (called redshift).

Doppler shifts are easily detected in a spectrum with dark lines.

Galaxies moving away from each other! In early 20th century, astronomers were surprised to discover that all distant galaxies are redshifted! Galaxies moving away from each other!

“The Universe is expanding.” Note: Applies only on large scales. The Solar System is not expanding; it’s held together by gravity. Milky Way Galaxy is not expanding; it’s held together by gravity.

Thursday’s Lecture: The Big Bang Model Reading: Chapter 3