AST 105 College of Southern Nevada Astronomy Worksheet

Instructions for Hubble’s Law LabFor this lab we are going to show how to use the velocity of a galaxy to measure the distance of
a galaxy from earth. This relies on the fact that the universe is expanding. Hubble’s law tells us
that the further a galaxy is from earth, the faster it is moving away from earth. So in order to
perform this lab we’ll be using spectroscopy.
Recall that when we did spectroscopy earlier in the semester we stated that all hydrogen atoms in
the universe will emit the same discreet wavelengths of light. In the visible part of the spectrum
those wavelengths are 656.2nm, 486.1nm, 434.0nm, and 410.1nm. First, on the line, sketch in
where those lines should appear. Next we’re going to show how those wavelengths will change
when those hydrogen atoms are moving away from us, such as when they are in a galaxy moving
away from us. In this case, we’re going to consider a galaxy moving away from Earth at 106 m/s.
To do this we’ll use this equation
𝑪𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝑾𝒂𝒗𝒆𝒍𝒆𝒏𝒈𝒕𝒉 = 𝑹𝒆𝒔𝒕 𝑾𝒂𝒗𝒆𝒍𝒆𝒏𝒈𝒕𝒉 ×
𝒗𝒆𝒍𝒐𝒄𝒊𝒕𝒚
𝒔𝒑𝒆𝒆𝒅 𝒐𝒇 𝒍𝒊𝒈𝒉𝒕
For each wavelength, calculate the amount the wavelength will shift due to the motion of the
galaxy. To do this, we’ll plug in 106m/s for the velocity, 3x108m/s for the speed of light, and
each of the 4 wavelengths given above.
Step 1: Since the velocity of the galaxy and speed of light doesn’t change we can calculate this
pretty quickly
109 𝑚/𝑠
= .003
3 × 10> 𝑚/𝑠
Step 2: For each wavelength, multiply the rest wavelength by .003 to get the change in
wavelength. Recall that 1 nm = 10-9m.
656.2 × 10CD 𝑚 × .003 = 2.0 × 10CD 𝑚
Step 3: We add the change in wavelength to the rest wavelength to get the resulting
wavelength.
656.2 × 10CD 𝑚 + 2.0 × 10CD 𝑚 = 658.2 × 10CD 𝑚
This is the new wavelength of the hydrogen as a result of it being in a galaxy moving away from
the earth. We’ll do this for the remaining 3 wavelengths to get the spectrum of a galaxy moving
away from earth at 106m/s. What this means is if we look at hydrogen in a distant galaxy, by
observing how much the lines are shifted from their rest wavelengths, we can measure velocity.
Now that we’ve shown that there is a relationship between velocity and the wavelengths we
observe in spectroscopy, we’ll now show how we can use velocity to calculate the distance to a
galaxy. In the table we have 6 galaxies. Each galaxy has a measured velocity and distance. The
velocity we can measure by measuring the shift of hydrogen. Distance we can use main
sequence fitting like we did last week, standard candles which we may do a bit later, or a
number of other techniques. But now that we have a sample of galaxies, lets plot them. On the
graph paper in your lab, make a plot with distance on the x axis and velocity on the y axis. You’ll
want to do this plot landscape so you can use as much of the paper as possible. Make the graph
as big as you can. See the example below. Notice I use the entire paper, you should too.
Velocity
Distance
Go ahead and plot the six data points. You may want to label each data point in case you make
a mistake so we can fix it easier.
Next, we’re going to figure out the best fit line to model this data. Since the motion of a galaxy
is a combination of the motion due to the expanding universe as well as gravitational
interactions with it’s local environment, we can’t simply connect all the dots to get an estimate
for the expansion rate of the universe. Rather we need to pick a best fit line to describe the
average motion of the galaxies. If the universe weren’t expanding, it would stand to reason that
galaxies would move in random directions much like gas molecules in a balloon. When the
universe is expanding, those random motions will still be there but the net motion should be
away from earth. So we need to compensate for this random motion by plotting a best fit line
for the data.
To do this, we start by taking the average value of the velocity and distance values. So add up
all 6 velocities and divide by 6. Repeat this with the distances and write them in the space
provided on the top of page 2. Do not include any decimal values, just whole numbers will be
sufficient.
Next, in the first table on page 2, we want to take the velocity of each galaxy and subtract the
average value. So for galaxy A, this would be:
31376 − 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 =
You’ll repeat this for Galaxies A-F. We will label this whole column “Y.” We’ll repeat this for the
distance.
432 − 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 =
This column will be column “X.” Record these values in row A in the table on the second page.
Y
31376-average Velocity
X
432- Average distance
In the next table we’re going to take the value for column X times the value for column Y. So for
galaxy A, for example, take the two values you just calculated above and multiply them
together. That’ll go in the first column in the second table. Next take the value you got for
measured distance- average, square it and write it in the second column in the second table.
Repeat this for each galaxy.
Next, add up all the values of X x Y you calculated and write this value in the space provided
beneath second table. Do the same thing for the values of X2, and write this value in the space
provided beneath the second table. Now, you’ll take the value you got for the sum of X x Y and
divide it by the value you got for the sum of X2. This value is the slope of the best fit line.
Recall the equation for a straight line is y=(slope)x + y intercept, or y=mx + b. In this case b is
zero, y is velocity, and x is distance. Let’s assume the slope you calculated is 80 (This is not the
actual value, just an example so do not use it). The equation for the line is then
𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 80 × 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
Let’s pick an easy value of distance to work with, 1000 Mpc. The velocity of this imaginary
galaxy using this line would be 80 times 1000 Mpc which would give us a value of 80,000 km/s.
So on your graph you would put a dot at the point where 1000 Mpc and 80,000km/s intersect.
Then draw a line from the origin of your graph through the dot you just made. This is the best
fit line for the data.
Now that we have an equation for the expanding universe, we can use the fact that there is a
direct relationship between distance and velocity. So if we were given a velocity, we can solve
for the expected distance to the galaxy. Since we know how to measure velocity using the
doppler effect, this gives us a fairly easy way of measuring the distance to even very far away
galaxies. So if a galaxy has a velocity of 100km/s, we can do a bit of algebra and get
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 =
𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 100 𝑘𝑚/𝑠
=
80
80
You should now have your entire lab completed. At this point, you are ready to submit. Please
make sure you submit all 3 pages (the two lab pages plus the graph).
Lab Exercise: Spectroscopy 2 Hubble’s Law
The goal of this lab is to demonstrate the use of spectroscopy in conjunction with the Doppler Effect to
study the expansion of the universe. We’ll draw on the skills acquired during the spectroscopy lab.
Doppler Effect
Hydrogen’s visible spectrum has lines at 656.2nm, 486.1nm, 434.0nm, and 410.1nm. On the graph
below, sketch in where these lines should appear.
400nm
550 nm
700nm
The Doppler effect states that wave emitting objects in motion will have a shift in the wavelengths they
emit as viewed from an outside observer. The equation for this is:
Wavelength change=rest wavelength×
velocity
speed of light
Suppose an object is moving away from earth at speed of 1×10 6 m/s. What would the observed
spectrum be?
Suppose you did this calculation for the following table of galaxies
Galaxy
Velocity (km/s)
Distance Mpc
A
31376
432
B
70084
867
C
59838
921
D
62252
989
E
63264
1038
F
92988
1191
On the graph paper provided, plot these values.
Find the average values of the velocity and distance measurements
Velocity_________ Distance _________
For each galaxy, take the measured velocity minus the average and do the same for distance. We’ll call
these values X and Y
Galaxy
(measured velocity – average)
(measured distance – average)
A
B
C
D
E
F
Now for each galaxy, take X x Y and X2 and sum the values
Galaxy
X2
XxY
A
B
C
D
E
F
Sum: ______________________
The slope of your best fit line will be =
∑ ( X ×Y )
∑ ( X 2)
____________________
=
With your instructors help draw your best fit line on your graph.
Imagine you measured the velocity of a galaxy to be 100 km/s. Using your graph, how far away is this
galaxy in km?