Lab 6

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 t solving for 'v' by dividing both sides by t and cancelling gives: v = d/t since 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 your own graph paper or on the graph paper provided or using Excel or similar program. Use dots to indicate each galaxy cluster in terms of distance and velocity .? 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.

STEP 3 Determine the slope of the line, this is the Hubble constant. showing the calculation to get the points. The slope is determined as follows: Slope = rise/run = change in velocity/change in distance? slope = 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'. Divide the 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. 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! 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! STEP 5 Convert the answer from STEP 4 to years. Divide by the number of seconds in a year. STEP 6 Convert the 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. 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

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 : Age of the universe in seconds = 1/H = _________________________s?

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

STEP 6 Age of the universe in billions of years = _________________ billion yrs.? Do not forget to submit your plot of the data with best fit line from STEPS 1 and 2.