This document describes an activity involving Hubble's law and the use of Cepheid variables as standard candles to measure distances to galaxies. It contains the following key points:
1) Cepheid variables have a direct relationship between their period and luminosity, allowing their distance to be estimated. Images show a Cepheid in galaxy M100 with an estimated period of [blank days].
2) Galaxies' recessional velocities can be measured via redshift and are plotted against distance. A linear relationship is found, described by Hubble's law as v=Hd, where H is the Hubble constant.
3) Estimating H from the graph gives a value of [blank] km/s/Mpc, consistent with current estimates
1. Activity 10
Hubble’s Law
Please print your name and sign next to it (only those present).
Leader: (B)____________________________ ____________________________
Explorer: (C)____________________________ ____________________________
Skeptic: (D)____________________________ ____________________________
Recorder: (A)____________________________ ____________________________
Learning Objectives
1. Understand how Cepheid variables are used to estimate galactic distances from the period-
luminosity curve.
2. Know how the recessional velocities of galaxies are determined.
3. Interpret the velocity versus distance plot for galaxies.
Part I: Cepheid Variables
The photograph at right is a NASA
Hubble Space Telescope image of
a variable star in galaxy M100
(Courtesy of NASA). You can see
that the brightness of the star
changes with time (see the date on
each image). These variable stars
help astronomers calculate
distances to other galaxies. Before
1910, Harvard astronomer
Henrietta Leavitt began measuring
the brightness of stars in a class
known as Cepheid variables—
bright stars with masses 5 to 20
times that of our Sun. Variable stars are so named because their luminosities, and therefore their
apparent magnitudes, vary with time. Cepheid variables—or simply, Cepheids—are periodic
variables that cycle through a complete bright-dim-bright cycle in times ranging from days to
months, although they are typically in the range of weeks. What makes Cepheids so important to
astronomers is that there turns out to be a direct relationship between a Cepheid’s period and its
average luminosity—the longer the period, the brighter the star. The figure on the next page
presents data for twelve different Cepheids. Although there is some scatter in the data, it is clear
that knowledge of the period does allow one to predict the luminosity.
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2. 1. Use the graph at right to
determine which star takes
longer to go through one
complete bright-dim-bright
cycle, Star A or Star B.
Answer: __________
2. Which of the two stars, A or
B, has the greater number for
its average absolute
magnitude? Explain your
answer.
3. Use the series of HST images of the Cepheid variable in the galaxy M100 (previous page) to
estimate the period of the star (the period is the number of days for one bright-dim-bright
cycle).
Answer: __________
4. From the Period-Luminosity graph (above), estimate the luminosity of the star.
Answer: __________
5. Imagine another galaxy—call it SBGC98—contains a Cepheid with the same period as the
one in M100. If the Cepheid in SBGC98 appears brighter in the sky than the Cepheid in
M100, which galaxy is closer? Explain your reasoning.
6. In answering Problem 3 you effectively estimated the distance to the galaxy SBGC98 (except
for doing some math) using a Cepheid variable as a standard candle. This process only
requires two measurements to be made on a specific Cepheid to find its distance. What are
these two measurements?
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1 2 3 5 10 20 30 50 100
Luminosity
(solar units)
Period (days)
3000
100
10000
1000
300
6000
600
Average Luminosity for 12 Different Cepheids
Star A
Star B
3. Activity 10
Part II: Red Shift
Another property of galaxies
—significantly easier to
measure than distance—is
recessional velocity (how fast
it is moving away from us).
When a light source is
receding, the observed
wavelengths of standard
emission and absorption lines no longer appear where you would expect. The fact that the
source is moving away causes the positions of the lines to shift to longer wavelengths, an effect
known as red shifting. This phenomenon, similar to the changing pitch of a passing siren, is
called the Doppler effect.
The table at right gives
the distances and
recessional velocities
for five different
galaxies. The galaxies
are actually members
of galactic clusters
whose names are taken
from the constellations
in which they appear. Indicate the position of each galaxy on the graph below with a small X.
Then, using a ruler, draw a straight line through the data.
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Normal absorption
spectrum
“Red-shifted”
spectrum due to
moving source.
CLUSTER
GALAXY IN
DISTANCE IN Mpc
(millions of parsecs)
RECESSIONAL
VELOCITY (km/s)
Virgo 19 1210
Hydra 1200 61,200
Bootes 770 39,300
Ursa Major 300 15,000
Corona Borealis 430 21,600
Recessional
Velocity
(km/s)
10000
20000
30000
40000
50000
60000
70000
200 400 600 800 1000 1200
Distance (Mpc)
4. 7. Summarize the relationship between recessional velocity and distance.
8. By measuring the red shift in the spectral lines from a newly discovered galaxy it is
determined that the galaxy is moving away from us at 30,675 km/s. From your graph,
estimate the distance to this new galaxy. Briefly explain your procedure.
Part III: Determining Hubble’s Constant
The fact that your data can be reasonably well fit with a straight line means that we can represent
the relationship in the simple form
v = H d
where H is called the Hubble constant and is usually expressed in units of km/s/Mpc. That is, H
tells us the speed, in km/s, with which a galaxy at 1 Mpc is receding. A galaxy at 1000 Mpc
would then be receding at 1000H, and so on.
9. From your graph, estimate the recessional velocity of a galaxy at a distance of 1000 Mpc.
10. What value does this give you for H (just divide by 1000)?
11. The Hubble constant, H, is a measure of the rate at which the universe has been expanding
since the big bang to get to its present size. If H had been larger, would the universe have
taken more or less time to reach its present size? Explain your answer.
12. Current estimates for H range from 60 to 75 km/s/Mpc. Which end of this range, 60 or 75,
predicts the older universe. Explain your answer.
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