Hubble's Law and the Age of the Universe-
In the pre-lab, you plotted the distance from Los Angeles as a function of time for a hypothetical drive to Las Vegas. We can make a similar plot for the distance between two distant galaxies as a function of time for an expanding universe.
The Hubble Law is v = H0d. The constant H0 ("H naught") tells us about the rate of expansion in the Universe. Measuring H0 is tricky work, but the best value (for the real universe, verse, not our toy model!) appears to be about H0 = 70 km/s per Mpc (i.e., 70 km s-1 Mpc-1, where 1 Mpc = 106 pc.)
(1) Rearrange the familiar "distance equals velocity times time" formula to solve for the velocity. Express your new formula as a multiplication of two factors.
(2) Your answer to the previous problem should look a lot like the Hubble Law. What is playing the part of H0 in question (1)?
(3) Notice that the units of H0 are a velocity (i.e., distance/time) over a distance. One pc is 3.26 light years, so you could multiply this all out, but we'll save you the effort: 1 pc 3.1 x 1013 km. Convert the units of H0 so all the distances are in kilometers, and then cancel out any common factors. Be sure to have the correct units on your converted H0!
(4) Reread your answer to question (2), then compute the reciprocal of H0; this is known as the "Hubble time." Express your answer in years. (Scientific notation will help, and remember from Lab 1 that there are 3.15 x 107 sec/yr).
(5) What does the time in question (4) represent? What assumption is required?
(6) Let's consider the (more likely) case that the rate of expansion is changing over time. We'll revisit the example from the pre-lab to help understand these concepts.
(a) Draw a plot that has "distance from Los Angeles" on the vertical axis and "time" on the horizontal axis. Draw a straight line beginning in Los Angeles (think about where that is on the plot) and passing through Las Vegas (280 miles) 4 hours later at 5:00 pm.
(b) What is the slope of this line? (Make sure you have the right units!) Based on the units, what does the slope of this line represent?
(c) Describe what the plot would look like for a car parked in Los Angeles all day.
(d) If I am standing by the freeway in Las Vegas at 5pm and see you zoom by at 70 mph, do I know when you left Los Angles? Can I make a good guess? Explain!
(e) Draw a line on your plot in part (a) that shows your distance from Los Angeles as a function of time if you have been slowing clown the entire time since you left. Remember, I am standing by the freeway and see you zoom through Las Vegas at 5pm at 70 mph! (Hint: if you're going 70 mph now and are slowing down, how fast were you going earlier?)
(f) If you have been slowing down the entire way since leaving Los Angeles, was your time en route actually more or less than my guess in part (d)?
(g) This trip to Vegas is an analogy for the Universe. The common gravity of mass pulling on all the other mass in the Universe leads us to expect that the expansion of the Universe has been slowing down the whole time since the Big Bang (when "time = 0"). If the Universe has been decelerating since the Big Bang, how does the age of the Universe compare to the "Hubble time" (defined as 1/H0)? Explain your answer.
Sketch a plot of the distance (R) between two distant galaxies as a function of time, if the has been expanding at a constant rate. Indicate the present time as "now" and label the Hubble Time as T0. Remember that the Hubble Time is a time interval, not a particular paint in time.
Sketch a curve showing the distance between the two galaxies under the assumption that the expansion has been steadily decelerating. Label the time of the Big Bang on your plot, and indicate the age of the Universe (also a time interval) under this assumption. If the Universe has been steadily decelerating, is it older or younger than the Hubble Time?
Why must the two curves you drew have the same slope at the time "now"?
Why does the Hubble Law imply that the Universe has a finite age (i.e., that the Universe is not infinitely old right now)?