The Memphis Astronomical Society Presents A SHORT COURSE in ASTRONOMY

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Memphis Astronomical SocietyMemphis Astronomical SocietyPresentsPresents

A SHORT COURSEA SHORT COURSEinin

ASTRONOMYASTRONOMY

CHAPTER 12CHAPTER 12

C O S M O L O G YC O S M O L O G Y

Dr. William J. BuslerDr. William J. BuslerAstrophysical Chemistry 439Astrophysical Chemistry 439

COSMOLOGYCOSMOLOGYI. Measuring Cosmological Distances

A. Knowledge of how to measure distances by parallax is assumed.

(See Chapter 9.)

The annual parallax of a star depends upon its distance from the Solar System: The farther away

it is, the smaller the angle of parallax.

• Usually, the star is observed from two points in the Earth’s orbit 6 months apart.

• This provides a baseline of 2 astronomical units (186 million miles) for triangulation.

• This drawing is not to scale! The parallax angle for even the nearest stars is less than 1 second of arc (1/3600 of one degree).

• Suppose a star is at a distance such that its parallax is exactly 1" either side of its average position (i.e., 2" total) when viewed from diametrically opposite points in the Earth’s orbit.

• This distance is known as one parsec (from parallax-second).

• We can easily calculate the distance of the star, using trigonometry:

• tan = ;

• tan 00'1" =

• tan 2.778 10-4 =

• 4.848 10-6 =

• x = = 1.92 1013 miles

= 19.2 trillion miles = 3.261 light-years = 1 parsec.

Opposite

Adjacent

adjacentopposite

xmiles103.9 7

xmiles103.9 7

xmiles103.9 7

6

7

10848.4miles103.9


B. Cepheid Variables:

(Recall their significance in stellar evolution and their position on the HR diagram.)

Hertzsprung-Russell Diagram

|

O B A F G K M

AbsoluteMagnitude

•

•

•

•

•

•

•

- 7

- 5

-3

- 1

+1

3

5

7

9

11

13

15

Puppis

O5, -7.1

SpicaB1, -3.3

SiriusA1, 1.45 Sun

G2, 4.84

EridaniK2, 6.15

61 CygniK7, 8.3

Barnard’s StarM5, 13

•

•

•

•

••

-4.5-3.6

Cephei

G2

BetelgeuseM2, -5.6

ArcturusK2, -0.3

Sirius BA5, 10

ProcyonF5, 2.7

MAIN SEQUENCE

BLUE GIANTS RED SUPERGIANTS

WHITE DWARFSRED

DWARFS

GIANTS

Spectral Type

F5

Unstable Variable



• Periods vary from a few hours to about 100 days.• Brightness increases faster than it decreases.• The distance to relatively near Cepheids can be

determined by parallax.• Then, their absolute magnitudes (luminosities) can

be calculated.

Light Curve of a Cepheid Variable


B. Cepheid Variables:• There is a relationship between the luminosity of a

Cepheid variable and its period.• Once this relationship has been determined, the

graph can be used to obtain the luminosity of any Cepheid merely by measuring its period, which is relatively easy to do.

• Then, knowing its absolute magnitude (and of course its visual magnitude), its distance can be calculated.

Cepheid Variable Period-Luminosity Relation

L

P



Note: There are actually two slightly different graphs; one applies to Population I stars (in the

spiral arms of galaxies); the other to Population II stars (in galactic nuclei and globular clusters).

• In 1911, Henrietta Swan Leavitt, working at Harvard, used Cepheid variables to measure the distance to the Smaller Magellanic Cloud.

Henrietta LeavittHenrietta Leavitt1868 -1868 - 19211921

• In 1916, Harlow Shapley measured distances to Cepheids in the globular clusters surrounding the Milky Way.

• He assumed (correctly) that the common center of the clusters was the center of the Milky Way.

• Then, he determined our distance from the center of the galaxy (33,000 light-years). Harlow ShapleyHarlow Shapley

1885 -1885 - 19721972

EdwinEdwinPowellPowellHubbleHubble

1889 -1889 - 19531953


B. Cepheid Variables:• In 1924, Edwin Hubble (100-inch telescope, Mt.

Wilson) used Cepheids to measure the distance to M31 (the “Andromeda Spiral Nebula”).

• The distance he obtained, 1 million light-years, proved that this was a galaxy like our own (Herschel had called them “island universes”).

• In 1952, this distance was corrected to 2.5 million light-years when the two types of Cepheids were distinguished.


C. RR Lyrae Variables (and other similar types):

• These stars have a very distinctive light curve, and are easily recognized.

Light Curve of an RR Lyrae Variable Star

M

time


C. RR Lyrae Variables (and other similar types):

• All of them have the same absolute magnitude (0.6), so their distances can quickly be determined.

• Note that both Cepheids and RR Lyrae stars are among the most luminous of stars, so they are relatively conspicuous, even in other galaxies.

• They are known as the “yardsticks of the Universe” (especially Cepheids).


D. The “Average Brightness of Galaxies” Assumption:

• It is useful to assume that the average brightness of all the galaxies in one cluster of galaxies is about the same as the average brightness in another cluster.

• This allows us to estimate the distance to a cluster of galaxies which is so remote that no Cepheids or RR Lyrae stars can be observed.


D. The “Average Brightness of Galaxies” Assumption:

Example: In the Virgo cluster of galaxies, theaverage-brightness method leads to a distance

estimate of 16.5 times the distance to M31,or 38 million light-years.

COSMOLOGYCOSMOLOGYII. The Hubble Relationship

A. The Doppler Effect• Well-known for sound sources: When a sound

source such as a train or siren is approaching, the pitch of the sound is higher than when the source is receding.

• This is caused by the “bunching together” (or “stretching out”) of the sound waves, so that more (or fewer) waves hit the ear each second.


A. The Doppler Effect• The same principle holds for light waves:

• If a yellow star is stationary, the light waves it emits are neither bunched together nor stretched out; therefore, it appears yellow.



• If the same yellow star is approaching rapidly, its light waves will be bunched together; therefore, it appears blue.



• If the same yellow star is receding rapidly, its light waves will be stretched out; therefore, it appears red.


A. The Doppler Effect• When a star (or galaxy) is approaching us or

receding from us, the light waves emitted as a particular color (e.g., yellow) are bunched together or stretched out.

• This shifts the object’s light (both the maximum and all spectral lines) toward the blue or red part of the spectrum.


• By measuring the shift in a spectral line () with respect to the same line from a stationary source in a laboratory (0), a remarkably simple equation lets us calculate the object’s velocity (v) towards or away from us:

Note: this equation is valid only when v << c; other-wise, a more complicated equation must be used.

Spectrum of the Quasar 3C 273. Note the red-shift of the three hydrogen lines due to its recession speed

of 0.16 c.

Red-shifting of spectral lines from an object receding at about 0.2 c; z = / 0 = 0.2.


• When a galaxy’s speed of recession (v) is an appreciable fraction of the speed of light (c), a more complicated equation must be used:

• Instead of ,

we must now use .1c/v1c/v1

Spectral lines red-shifted by z = / 0 = 1.0. Note: Speed is not c, but 0.60 c.


B. The Hubble Diagram:• By 1931, Hubble (and others) had used Cepheid

variable stars to measure distances to numerous galaxies; spectroscopists had also measured their speeds of recession.

• Hubble noticed that a graph of velocity vs. distance gave a linear plot, indicating a direct relationship: v D, or v = kD.

The Hubble Diagram: Velocity of Recession vs. Distance of Galaxies

(-) v

D

•

• •

••

•••

•


B. The Hubble Diagram:• The proportionality constant k is now known as H0,

the Hubble constant; it equals v/D for individual galaxies. This is also the slope of the diagram.

• For example, the Virgo cluster of galaxies lies at a distance of 38 million light-years, and is receding at a velocity of 710 miles/second. A simple calculation shows that


B. The Hubble Diagram:

• This value may also be expressed as

• Present values for H0 range from 50 to 100 ,

or 10 to 20 , with most astronomers

leaning toward the lower values.

Variation of the Hubble “Constant” over time


C. Using the Hubble Diagram:

Once the Hubble diagram was developed, it became possible to estimate the distances to very remote galaxies by measuring their redshifts, calculating their recession velocities, and reading their distances off the Hubble diagram (or calculating D = v/H0).

The Hubble Diagram: Velocity of Recession vs. Distance of Galaxies

(-) v

D

•

• •

••

•••

•


C. Using the Hubble Diagram:

Example: A galaxy has a recession speed of 9500 mi/sec (15,300 km/sec). How far away is it? (Assume H0 = 50 km/sMpsc.)

Spectra of galaxies

showing red-shifts.

Distances are based on a

value of H0 of 50 km/sMpc.

COSMOLOGYCOSMOLOGYIII. Cosmological Implications of the

Hubble RelationshipA. The Big Bang and the Expanding Universe:

• Why are all the galaxies receding from us (and therefore from one another), with the most remote receding the fastest?

• Most likely, the explosive formation of the Universe (the Big Bang) caused the proto-galaxies (see Chapter 11) to fly away from one another; the fastest ones have now traveled the farthest.



• If the Hubble diagram is truly a straight line, the Universe is “open” – the galaxies are not slowing down and can be expected to sail on forever.

• If, however, the line has an upward curvature, a slowing down of the expansion of the Universe is indicated, possibly caused by invisible matter or the mass of neutrinos.



• This might cause the Universe to stop expanding and eventually fall back in on itself (the Big Crunch).

• Might there be recurrent Big Bangs?

Open or closed Universe?

q = 0: no deceleration; open Universe.

q = 1/2: some deceleration; marginally open Universe.

0<q<1/2: some deceleration; open Universe.

Q > 1/2: full deceleration; closed (collapsing) Universe.

Open or closed Universe?

q = 0: no deceleration; open Universe.

q = 1/2: some deceleration; marginally open Universe.

0<q<1/2: some deceleration; open Universe.

q > 1/2: full deceleration; closed (collapsing) Universe.

q < 0: expansion of Universe accelerating!

qo < 0


Hubble RelationshipB. The Age of the Universe:• It is well known that distance = velocity time.• If we solve for time, t = D/v.• Recall that the slope of the line on the Hubble

diagram is v/D, or H0.

• Thus, t = 1/H0.


Hubble RelationshipB. The Age of the Universe:• Since t = 1/H0, and if we use a value of 50

km/sMpsc for H0, t = 19.5 billion years.

• A value of 100 km/sMpsc for H0 would lead to t = 9.8 billion years.

• These values for t are the “Hubble age” of the Universe; the higher values agree with other experimental data.

T H E E N DT H E E N D

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The Memphis Astronomical Society Presents A SHORT COURSE in ASTRONOMY