Tully-Fisher Lab
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Northern Virginia Community College *
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Course
142
Subject
Astronomy
Date
Dec 6, 2023
Type
docx
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Lab: The Tully-Fisher Relation (adapted from a lab by Georgia State University)
In this lab, you will explore the relationship between the rotation rate
of spiral galaxies and
their intrinsic luminosity
, and how to utilize that relation for distance measurement
.
Part 1. Emission Line Widths
The extent of HI
in spiral galaxies usually reaches much further past the regions occupied by
bright stars. So, it’s a great tool for revealing how big spirals actually are. We can also trace out
the galaxy’s rotation using the HI emission line. As the galaxy is rotating, one side is always
coming towards you and other side is always going away from you. So, on top of the redshift of
the whole emission line that’s caused by how fast the galaxy is receding from you, one side is
receding just a bit faster and one side just a bit slower, so the emission line is broadened
depending on how fast the galaxy is rotating. The faster it rotates, the broader the emission
line will be. The width of the line, then, is a measure of the galaxy’s total rotational velocity.
Because all this motion is radial, the fastest velocities are on the edges, and if the galaxy is
edge-on to our viewpoint, we get spikes or horns on the edges of the profile.
1.
In Figure 1, there are 5 spectra that exhibit the HI emission lines from different spiral galaxies.
Velocity is on the x-axis, flux is on the y-axis. In order to measure how fast each galaxy is
rotating, we need to measure the width of the emission lines. To do this:
a.
Measure the highest point of flux for both horns and calculate their average.
b.
Divide your average by 2 and draw a horizontal line through the emission line at
that average value on the y-axis.
c.
Mark where the horizontal line intersects the sides of the emission line.
d.
From the intersection points, draw straight, vertical lines down to the x-axis.
Read off the velocity values of where the vertical lines intersected the x-axis.
e.
Subtract the larger velocity value from the smaller value. Your difference is the
observed rotation rate of that galaxy, or the width of the emission line (
W
50
).
f.
Record your W
50 values in the first column of Table 1
.
g.
Repeat this process for the other 4 galaxies’ emission lines and record your
calculations in Table 1.
Figure 1: HI emission lines from 5 nearby spiral galaxies. Galaxies are numbered 1 - 5 from top left to bottom right.
Part 2. True Rotation Rates
Because each galaxy is at a different inclination relative to our point of view, the width of the
emission line does not accurately describe how fast the galaxy is actually rotating, as shown in
Figure 2
. So, we need to adjust our measured widths depending on the orientation of the
galaxy.
Figure 2: Examples of observing a spiral galaxy at different inclinations
To correct for the galaxies’ inclination, we use the following equation (Eq 1):
where W
50
is your measured line width from question 1 and i is the inclination of each galaxy,
which is listed for you in the 2nd column of Table 1
.
2. Calculate W
50(
corrected
) for each galaxy and record your answers in column 3 of Table 1.
Show Calculations here:
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Part 3. From Rotation to Brightness
The most important application of the HI emission line width is its use as an extragalactic
distance measurement method on one of the furthest-reaching rungs of the cosmic distance
ladder. The Tully-Fisher relation
(named after Brent Tully and Richard Fisher) is a correlation
between a galaxy’s rotation rate and its intrinsic luminosity (how bright it actually is), much like
the correlation between a Cepheid variable star’s period and its absolute magnitude (Leavitt’s
Law, or The Period-Luminosity relation). So, the Tully-Fisher relation allows you to calculate
each galaxy’s true luminosity using your W
50
(
corrected
) calculations. The relation is shown in
Equation 2
:
where L
is the galaxy’s luminosity in units of solar luminosities.
3. Calculate luminosities for each galaxy in scientific notation and record them in column
4 of Table 1
.
Show Calculations here:
Now that you have the galaxies’ luminosities, you can calculate their absolute magnitudes
as follows in Equation 3
:
where M
is the galaxy’s absolute magnitude and L
is your calculated luminosity from Question 3.
4. Now calculate the absolute magnitudes of your galaxies and record your results in column 5 of
Table 1
.
Show calculations here:
Part 4. Find the Distance!
You now have ½ the pieces you need to calculate a distance. You have absolute magnitudes,
and in order to calculate a distance modulus, you also need apparent magnitudes! These are given to you in Table 1 in the column labeled m
.
You can now calculate distances using the distance modulus equation.
5. Use the distance modulus equation to calculate distances for all 5 galaxies and record
your results in the last column of Table 1.
Show your calculations here:
Table 1
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