Objectives: Graphically determine the Hubble constant and from that determine the age of the Universe.

Background: Edwin Hubble made one of the most important discoveries of modern astronomy in 1929 when he determined that the spectra of most galaxies were red-shifted. Since a Doppler red shift occurs when an object is receding from the observer, Hubble concluded that the Universe is expanding. He further discovered that the farther away a galaxy is from us, the greater its red shift and hence it must be moving away at a greater velocity than nearby galaxies. This is known as Hubble's Law and states that galaxies are receding from us at a speed that is proportional to their distance.

This relationship between velocity and distance is written mathematically as:

v = H x d

v = velocity, d = distance, H = Hubble constant

The value of H, the Hubble constant can be used to determine the age of the Universe as follows: an object that travels at a given velocity (v) for a given time (t) will cover a distance (d) equal to its velocity multiplied by time.

d = v x tsolving for 'v' by dividing both sides by t and cancelling gives:

v = d/tsince v = H x d (Hubble's Law, from above) then:

H x d = d/t we can cancel out the 'd's by multiplying both sides by 1/d

H = 1/t now solve for 't' by multiplying both sides by t and cancelling, then dividing both sides by H and cancelling (a two step process) to give:

t = 1/H or, in other words, the age of the Universe (t) (since it started expanding) is the inverse of the Hubble constant.

Procedure: show your working on the worksheet otherwise you will not receive full credit.

STEP 1 Plot the data in the table below (Data Table of Distance and Recessional Velocities of Galaxy Clusters) either on graph paper or on the graph paper provided or using Excel or similar program. Use dots to indicate each galaxy cluster interms of distance and velocity (40 pts.).

STEP 2 Use a ruler or straight edge to draw a best fit line through the data points. Do not connect the dots. A best fit line is a straight line that keeps an equal number of data points above and below the line an equal distance. I prefer that you do this by hand to get a better 'feel' for how to do it but, if you used Excel it is okay to let it fit a best fit line. (10 pts.)

STEP 3 Determine the slope of the line, this is the Hubble constant. You must do this yourself and show your calculation to get the points. The slope is determined as follows: Slope = rise/run = change in velocity/change in distance

To obtain the slope from your plotted line chose two locations on your line that are far apart. Write down the velocity value and the distance value for each location. Subtract the smaller velocity value from the larger velocity value (for example 200 km/s minus 20 km/s = 180 km/s). This is the 'change in velocity'. Put that number at the top of a fraction and subtract the smaller distance value from the larger distance value, (for example 100x10^20 minus 10x10^20 km = 90x10^20 km) and put that number on the bottom of the fraction. This is the 'change in distance'. Dividethe fraction, giving you a value for H (in my example this would be 180 km/s/90x10^20 km = 2x10^-20, note the negative sign in front of the exponent. The example values I have given here are just to help you - they are not the correct values, you need to get the correct values from your graph. Show your work and enter your answer on the answer sheet (below).

Note the units for H (the slope) = velocity (km/s)/distance (km) leaving 1/s ("per second").

Be careful: the distance scale on the horizontal (x) axis is in 10²⁰ km that is 100 quintillion km! (20 pts.)

STEP 4 Remember that the age of the Universe is 1/H so take the inverse of the value for H you calculated in STEP 3 (1 divided by H). Our units are now 1/1/s so the 1/1 part cancels and we have an age of the Universe in seconds. That is a lot of seconds! (10 pts.)

STEP 5 Convert your answer from STEP 4 to years. Divide by the number of seconds in a year. (10 pts.)

STEP 6 Convert your answer to STEP 5 to billions of years. You just calculated the age of the Universe (in billions of years) from 13 sets of recessional velocity and distance data. (10 pts.)

Data Table of Distance and Recessional Velocities of Galaxy Clusters

Cluster Name	Distance (km x 1020)	Velocity (km/s)
Virgo	2.4	490
Pegasus	18.1	3300
Pisces	33.0	8000
Cancer	52.4	9600
Perseus	58.3	11300
Coma	66.2	12700
Hercules	72.9	17600
Ursa Major I	114.6	22100
Leo	119.7	26600
Geminin	150.2	33100
Corona Borealis	166.8	27500
Bootes	179.9	35200
Ursa Major II	200.6	42600

Worksheet - Hubble's constant and the Age of the universe

STEPS 1 and 2: Do not forget to submit your graph with best fit line.

STEP 3 Answer: The slope of the line on your graph (the Hubble constant) = (H) = ___________________________________________ 1/s

Show all work here:

STOP. Double check, because you are dividing a quite large number by an absolutely enormous number (distance in km x 10²⁰ remember) your answer should be a very small number (a big negative exponent).

STEP 4 Answer: Age of the universe in seconds = 1/H = _________________________s

Show all work here:

STEP 5 Answer: Age of the universe in years = ___________________________ yrs.

Show work here:

STEP 6 Answer: Age of the universe in billions of years = _________________ billion yrs.

Show work here:

Do not forget to submit your plot of the data with best fit line from STEPS 1 and 2.