Lab 6: The Tully-Fisher Relation

Name:

Overview

In this activity, 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.

Objectives

After completing this activity students will be able to:

· Recognize the profile of an HI emission line from a spiral galaxy

· Measure the width of an HI emission line

· Calculate galaxy luminosity by using the Tully-Fisher relation

· Calculate distances using the absolute magnitude from the Tully-Fisher relation

**Note: If a question is labeled “ THOUGHT QUESTION” we are looking for you to show critical thinking/justification in your answer, not a “correct” answer**

Definitions

Here are some terms from lecture that we will be using today in lab:

· diagram showing rotating galaxy and blueshifted and redshifted signal HI – neutral hydrogen; the most abundant element in spiral galaxies, and the best tracer for spiral galaxy size

· Emission Line – an increase in flux at a specific wavelength or frequency in a spectrum

· W 50 – the width of an emission line at 50% of the flux

· Redshift – the amount of increase in wavelength (or decrease in frequency) of a photon due to its velocity away from the observer

· Blueshift – the amount of decrease in wavelength (or increase in frequency) of a photon due to its velocity towards the observer

· Radial Velocity – the measure of the amount of a celestial object’s velocity that is in the line of sight of the observer

· Noise – low-level fluctuations of energy in a spectrum or image due to weak radiation from space and background energy produced by electronics in the instrument

· Radio Frequency Interference – RFI; large energy spikes in a spectrum caused by near-Earth objects that emit in the radio portion of the spectrum (GPS, satellites, etc.)

· Apparent magnitude – m; the apparent brightness of a celestial object as measured from Earth.

· Parsec – equal to 3.26 light-years; the distance an object must be to have a parallax angle of 1 arcsecond.

· Absolute magnitude – M; the brightness of a celestial object as measured from 10 parsecs.

· Distance Modulus – m-M; the difference between an objects apparent and absolute magnitudes.

Part 1. The HI Emission Line

HI is cold gas that is very abundant in spiral galaxies. Each atom of hydrogen will eventually emit a photon in the radio portion of the electromagnetic spectrum. So, the more photons we detect (i.e., the higher the signal of the emission line), the more gas there is in the galaxy we’re looking at. Less gas = less photons = weaker signal.

We can also trace out a 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, as 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 rotation. 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 horns on the edges of the profile.

In Figure 1, there are 5 HI emission lines from different spiral galaxies. Velocity is on the x-axis, flux is on the y-axis. EACH GRAPH HAS DIFFERENT TICK MARK VALUES!!! To measure rotation, we need to measure the width at ½ flux 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. WATCH OUT FOR RFI!

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 ( W50).

f. Record your W50 values in the first column of table 1.

g. Repeat this process for the other 4 galaxies’ emission lines and record you calculations in table 1.

1

HI emission plot called ‘Adam’ HI emission plot called ‘Val’ HI emission plot called ‘Michael’

MICHAEL

VAL

ADAM

PsBattle: Shocked Batman meme : photoshopbattles HI emission plot called ‘Ben’ HI emission plot called ‘Christian’

BEN

CHRISTIAN

Figure 1: HI emission lines from 5 nearby spiral galaxies

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.

Diagram showing how HI emission varies based on inclination between 0 and 90 degrees

Figure 2: Examples of observing a spiral galaxy at different inclinations

To correct for the galaxies’ inclination:

where W50 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.

WE RECOMMEND THIS WEBSITE FOR YOUR CALCULATIONS: https://www.wolframalpha.com/

· An example using Equation 1, you would type in: 200/sin(80), WHERE 200 IS W50 AND 80 IS i

· IF YOU USE A SCIENTIFIC CALCULATOR/YOUR PHONE, CHECK TO MAKE SURE YOU ARE IN DEGREES, NOT RADIANS. IF YOU GET A NEGATIVE NUMBER, YOU ARE IN RADIANS, STOP IT, BIG NO-NO!

2. Calculate W50( corrected) for each galaxy and record your answers in column 3 of table 1. Round off your answer to 2 decimal places.

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 W50( corrected) calculations. The relation is:

where L is the galaxy’s luminosity in units of solar luminosities.

3. Calculate luminosities for each galaxy IN SCIENTIFIC NOTATION, ROUNDED OFF TO 2 DECIMAL PLACES, and record them in column 4 of Table 1.

Now that you have the galaxies’ luminosities, you can calculate their absolute magnitudes as follows:

where M is the galaxy’s absolute magnitude and L is the luminosity value from Question 3.

AGAIN, WE RECOMMEND THIS WEBSITE: https://www.wolframalpha.com/

· Example using Equation 3 : -2.5log10(1e10) + 4.82, WHERE 1e10 (1 x 1010) IS L

· IF YOU USE A SCIENTIFIC CALCULATOR/YOUR PHONE, MAKE SURE TO USE LOG BASE 10!

4. Calculate the absolute magnitudes of your galaxies and record your results in column 5 of table 1. Round off your answer to 2 decimal places.

Part 4. 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. The writer of this lab was nice enough to give those to you too! The galaxies’ m values are listed in column 6 of table 1. You can now calculate distances using the distance modulus equation:

Note that D will be in parsecs, so divide your distance by 1,000,000 in order to convert from parsecs to megaparsecs (Mpc).

5. Use the distance modulus equation to calculate distances for all 5 galaxies and record your results in the last column of Table 1.

Table 1

Galaxy

W50

(km/s)

i

(°)

W50( corrected)

(km/s)

L

(L☉)

M

(mag)

m

(mag)

D

(Mpc)

Adam

53.3

12.61

Michael

64.1

12.82

Val

45.8

14.00

Christian

71.5

15.28

Ben

88.2

14.95

6. THOUGHT QUESTION: Some of the profiles have different levels of flux in the two horns, why do you think that is? (Hint: the horns are produced from rotation, but the flux level of each horn depends on how many photons each side of the galaxy has. Where are the photons coming from? Why would the amount change?)

7. THOUGHT QUESTION: Some of the emission lines exhibit strange shapes between the two horns, what do you think might cause these? (Hint: the horns come from the sides of the galaxy, so where does the middle part of the emission line come from? Why would the flux be changing here?)

8. THOUGHT QUESTION: A number of the emission lines contain strong flux and are well above the noise, and some are barely distinguishable from the noise. List some (at least 2) reasons why this may be. (Hint: flux depends on the number of photons we detect. Again, where are the photons coming from? Also, what would cause us to detect less photons, i.e. a weaker signal?)