Lab9- PHYS1403.901
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Collin County Community College District *
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1403
Subject
Astronomy
Date
Apr 3, 2024
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docx
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HUBBLE'S CONSTANT
OBJECTIVES
In this exercise, the students will learn about the expanding universe, Hubble’s Law, and how we compute the age of the universe. The students will determine their own value for Hubble's constant and compare this to results obtained by scientists. They will then use this result to calculate a maximum age for the universe.
EQUIPMENT
The students will need to provide their own scientific calculators. Other material that is required will be provided and includes: Hubble Space Telescope images of Cepheid variables in distant galaxies, apparent and absolute magnitudes of the target stars and the velocity of the target galaxy.
INTRODUCTION
During the 1770's, Charles Messier made the first catalog of nebulae ('clouds' in Latin), not because
he had a particular interest in them, but because he didn't want to mistake them for comets, which he was more interested in. Today, Messier's catalog of 110 'Messier Objects' is a famous listing of some of the most
spectacular objects in the night sky. These objects are frequently denoted with a capital M and a number, such as M1 (the Crab Nebula).
Astronomers came to realize that these nebulae could be divided up into a number of distinct groups, two of the primary groups being the planetary and spiral nebulae. By the early 20th century a debate arose concerning the nature of these nebulae with some astronomers claiming that they were all within our galaxy, the Milky Way, while others believed that only the planetary nebulae were. This second group of astronomers believed that the spiral nebulae were actually other galaxies, far outside the limits of the Milky Way. But seen through the telescope
-
s of the late 18th and 19th centuries, all the nebulae looked like vague, fuzzy patches of light, sometimes mixed with stars. It would take much superior telescopes to achieve the necessary resolution to answer this question.
The controversy over this issue came to a head shortly after World War 1, and in 1920, the National Academy of Sciences sponsored a "debate, which actually consisted of two consecutive lectures by Heber Curtis and Harlow Shapley, about the nature and extent of the Milky Way_ Curtis insisted that the nebulae must
lie far outside the Milky Way, while Shapley argued that all nebulae lie within the Milky Way.
Shapley based his conclusion on the work of another astronomer, Adriaan van Maanen. Van Maanen claimed to have measured the rotation of the spiral nebulae by photographing them at times spaced
several years apart. His observations, if true, would imply that most spiral nebulae take a few hundred thousand years to rotate once, If so, then the spiral nebulae could not possibly be as large as the Milky Way,
for it is easy to show that the outer edges of the nebulae would have to be traveling at a speed in excess of the speed of light. Thus Shapley, relying on van Maanen's observations, concluded that the spiral nebulae must be far smaller than the Milky Way, and must lie within the outer reaches of our own galaxy, which he believed included everything. But van Maanen's analysis of his photographs was simply incorrect, Today, we know that spiral nebulae (actually, spiral galaxies) do rotate, but with rotation times of hundreds of millions of years.
The debate was finally settled by the American astronomer Edwin Powell Hubble (1889 - 1953). Hubble used the 2.5 meter (100 inch) telescope of the Mount Wilson Observatory overlooking Los Angeles, then the largest telescope in
the world, to study galaxies through night after night of long duration photography. In 1923, Hubble found that several of the spiral nebulae, including the Andromeda nebulae, contained periodic variable stars, called Cepheid variables. Cepheids are pulsating stars whose intrinsic brightness can be determined by their pulsation period. Comparing its real brightness to its apparent brightness allows us to calculate how far away the star is from us, and therefore, how far away the parent galaxy is. Because these stars had periods that matched those of well-studied stars in the Milky Way, Hubble concluded that the stars had the same luminosities. Then, by using the inverse-square law, he compared the apparent
brightness of the two stars and obtained the relative distances. This comparison put the Andromeda Galaxy well beyond the
outer reaches of our galaxy. Hubble made the assumption that all Cepheids work the same and that the light from these stars had not been absorbed by dust. Both of these assumptions were wrong and the value that he obtained, about 700,000 light years, was about one third of the value calculated today, about 2.1 million light years. Nonetheless, his results demonstrated conclusively that Andromeda was outside the Milky Way, and was a galaxy, not a nebula.
Hubble continued his work, and working with Milton Humason, made estimates of the distances to a number of galaxies. In 1929, when he combined these distances with the Doppler shift measurements of the velocities of galaxies, he came to the remarkable conclusion that the universe is expanding! He was able to come to this conclusion by finding a general trend that showed the more distant a galaxy was, the higher its velocity away from us was.
Hubble concluded that each galaxy has some random velocity of its own relative to other galaxies, and that this random velocity can add or subtract a relatively small amount to or from the overall trend_ This explains, in part, the scatter about the straight line in his plot. The rest of the scatter is due to inaccuracies in determining distances. But the trend represents Hubble's greatest contribution to astronomy. If galaxies' recession velocities increase in proportion to the galaxies' distance from us, then the entire universe must be expanding!
But why does this mean the universe is expanding? Why can't we explain this observation with the conclusion that all galaxies are simply moving away from us and we are the center of the universe? The answer to this lies in the principle of mediocrity, which states that there is no special significance to the point in space we occupy, or any other point either. In other words, the principle of mediocrity states that all points are equal and that there is no center of the universe.
Therefore, if we were able to travel to another galaxy, we would still find that all other galaxies were moving away from us, To envision this, imagine a loaf of raisin bread dough with the raisins spread uniformly throughout the dough. As the dough expands, the distance between the raisins would increase, So, if you could sit on any of the raisins, the other raisins would all appear to be moving away from you, and this would be true no matter which raisin you picked.
Hubble used these observations to obtain his law for the expansion of the universe, now called Hubble's Law. This law states that the velocity at which a galaxy is receding from us is equal to its distance from us times
some constant, called the Hubble constant. In algebraic form, Hubble's Law is
(1)
v = H
0
d
where v is the receding velocity, d is the distance between us and the other galaxy, and H
o
is Hubble's constant, which represents how fast the universe is expanding, We can obtain the velocity easily using the Doppler shift of the galaxy, which provides us with the galaxies velocity to within 1%. Therefore, if we know the value of Hubble's constant, we can find how far away the galaxy is from us.
Ironically, Hubble was not actually the first to come to this conclusion. In 1916 Albert Einstein presented his General Theory of Relativity which contained equations that dealt with the large scale structure of the universe.
In 1917, the Dutch astronomer Willem de Sitter pointed out that a consequence of these equations was that the universe was expanding. Because there was no observational evidence to indicate that this was true, Einstein introduced a special term, the universal constant, in his equations to make the universe static. After Hubble showed the universe really was expanding, Einstein removed that special term and called it the greatest scientific mistake of his life.
The problem with Hubble's Law is that we are unable to obtain a precise value for Hubble's constant. This is because determining a reliable value requires obtaining reliable measurements for the distances to enough
galaxies to make a plot. Obtaining these measurements is very difficult and the result is that values for Hubble's
constant have varied wildly. One camp has measured the value to be between 40 and 50 kilometers per second per megaparsec, while another camp has it between 80 and 100 kilometers per second per megaparsec, where a megaparsec is a million parsecs, or about 3.26 million light-years. As a result, 70 kilometers per second per megaparsec is currently the most commonly used value. This, we believe, provides us with the means to measure the distance to other galaxies with an accuracy of plus or minus 25%.
But if the universe is expanding, we could reverse this expansion in our minds and find that there is some point in time when all of the universe was located at one point. This is when the expansion began and is called the "Big Bang". Hubble's constant is in units of distance per time per distance, or if we simplify, one over units of time. So, if we invert Hubble's constant we will get units of time and 1/H, will yield the amount of time, in seconds, since the Big Bang, assuming the universe has expanded at a constant rate. We are sure the expansion
rate is slowing down, so this will actually provide the maximum age of the universe. This amount of time is called the Hubble time. The importance of the value of Hubble's constant is that the higher the value, the younger the universe.
It was thought that the Hubble Space Telescope could be used to determine the distance to nearby galaxies with great accuracy, thus allowing us to obtain a definitive value for Hubble's constant. The idea was to use the telescope to measure the distance to galaxies in the Virgo cluster. These galaxies are far enough away that their own motion is less than their motion due to the expanding universe, and at the same time they are still close enough for us to see individual stars with the space telescope. Specifically, we are interested in seeing Cepheid variable stars. Once we observe Cepheids, we can determine the distance. Combining this distance with the galaxies redshift would then yield Hubble's constant.
Unfortunately, the mirror on the space telescope was flawed and it was not possible to make the necessary measurements, Before the space telescope could be repaired, a learn from Indiana University equipped a telescope on Mauna Kea in Hawaii with new technology optics that permitted them to see three Cepheids in a galaxy in the Virgo cluster. With this data, they determined that Hubble's constant has a value of 87 plus or minus 7 kilometers per second per megaparsec, Since then, NASA researchers have used the space telescope to measure the distance to another galaxy in the Virgo cluster and obtained a value of 80 plus or minus
17 kilometers per second per megaparsec, which is in agreement with the Indiana results.
The NASA team was able to obtain these results by determining the period of the Cepheid variables they observed, By measuring the period, they were able to obtain the absolute magnitude of the stars, M. Then, by measuring the apparent magnitude, m, they can determine a value, which we'll call 'x', which can be found with the equation
(2) x= ( m – M + 5) / 5
The importance of this is that it can now be used to find the distance to the star by using the equation
(3) d = 10
x
These results have presented astronomers with a difficult problem. The ages of the oldest stars are calculated to be about 13 billion years old. But these values for Hubble's constant yield a value for the Hubble time of less than that. This would mean that the oldest stars in the universe are older than the universe itself.
Critics of these values point out that the Virgo cluster is very large and the galaxies that these teams used are far removed from the center of the cluster and are thus much closer to us, while the velocity value is for the cluster center. This would have the effect of inflating their calculated value for Hubble's constant.
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Dispute over the value of Hubble's constant is not new. In fact, Hubble himself originally calculated a value of 500 kilometers per second per megaparsec, which yielded an age for the universe of about 2 billion years. Understandably, geologists were in great disagreement with this value because they already knew the Earth was older than that. Work is continuing and, hopefully, within a decade or so, we will have a good value of Hubble's constant that everyone will agree on. The value in this is that Hubble's constant is one of the most important numbers in cosmology. With it, we are able to estimate the size and age of the universe, determine the intrinsic brightness and masses of stars in nearby galaxies, examine those same properties in more distant galaxies and galaxy clusters, deduce the amount of dark matter present in the universe, obtain the scale size of faraway galaxy clusters, and serve as a test for theoretical cosmological models.
QUESTIONS
1.
What does Hubble's constant represent? This equation represents hoe fast the universe is expanding. 2.
What did Hubble use to prove that the Andromeda Galaxy is outside of the Milky Way? By using the inverse-square law, for comparing the apparent brightness of two stars, to be able to obtain their relative distances. Then, with his comparison put the Andromeda Galaxy beyond the outer reaches of our galaxy. 3.
What do we find if we reverse the idea of an expanding universe?
The point where everything started to expand which is called the Big Bang. The expansion rate will be slowing down and will provide the age of the universe too. 4.
What was it that Einstein called the biggest scientific mistake of his life. When Einstein wanted to explain the universal constant, he made an equation to prove that the universe was static which is not true because the universe is expanding. 5.
What is Hubble time? The maximum age of the universe is the amount of time called the Hubble time. Meaning that if the value is higher the universe at time was younger.
139
EXERCISE
1. Using Figure 2, estimate the period, in days, of the Cepheid variable in M100. The whole cycle
is not visible, so you will have to estimate how long a complete cycle is based on the pictures present. Do not assume that one-half of a cycle takes one-half of the period. Don't get frustrated if the answer doesn't come easily. Make your best effort and a small amount of error will have no appreciable effect on your final calculations.
This figure, 'Cepheid Variable Star in Galaxy M100’, was created with support to Space Telescope Science Institute (STScl), operated by the Association of Universities for Research in Astronomy, Inc., from NASA contract NAS5-26555, and is reproduced with permission from AURA/STScl. It can be obtained free of charge from STScI at hftp://rnarvel.stsci.
edu/EPA/Latest.
html. 55 days
2. Using the Cepheid luminosity-period plot in Figure 1 and your results in step #1, estimate the absolute magnitude, M, of the Cepheid variable. M= -6
3.
Using equation (2), your results from question #2 (
x= ( m – M + 5) / 5
) for M, and using m = 24.9, determine a value for 'x'. (24.9 -- -6 + 5) / 5 = x = 7.18
4.
Using your results from step #3 and equation (3), determine a value for the distance to M100. This value will be in parsecs, divide by 10
6
to get the answer in megaparsecs. (10
7.18
)/ (10
6
) = 15.135 megaparsecs
5.
The velocity of M100 is calculated to be 1375 krn/sec. Divide this value by your results for step #4 to obtain your estimated value for H
o
. In comparison, the NASA team, headed by Dr. Wendy Freedman of Carnegie Observatories, used these images to calculate a value of 80 km/sec/M pc for H
o
. (1375 / 15.126) = H
0
= 90.843 km/sec/M pc
6.
A megaparsec is equal to 3.1 x 10
19
kilometers. So the value of H
o
can be simplified by dividing by 3.1 x 10
19
kilometers per megaparsec. What result do you obtain when you use your result from step #5? (90.843 km/sec/M pc)/ (3.1 x 10
19
)/ = 2.930 x 10
-18
7.
Now, take 1/
H
o
to obtain the age of the universe in seconds. (1/
2.931 x 10
-18
) = 3.412 x 10
17
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8.
There are 3.15 x 10
16
seconds in a billion years. So divide your results from step #7 by this value to rind the
maximum age of the universe in billions of years. (
3.412 x 10
17
/ 3.15 x 10
16
) = 10.831 billion years
9.
Repeat steps #6 and #7 using the NASA calculated value for H
0
. This value was provided in step #5. #6 with value of NASA- (80 / (3.1 x 10
19
)) = 2.581 x 10
-18
. #7 with the value of NASA- (1 / (2.581 x 10
-18
)) = 3.874 x 10
17
10.
Repeat step #8 using the results from step #9 to obtain the NASA team's calculated maximum age of the universe in billions of years. (3.874 x 10
17
) / (3.15 x 10
16
) = 12.3 billion years Figure 1: Period vs Brightness
Figure 2: Cepheid Variable Brightness Fluctuations