Measuring Galactic Pitch Angle

The Black Hole Mass Function for local spiral galaxies is measured indirectly through the

M-P relation (Berrier et. al 2013), converting galactic pitch angles to Super Massive Black

Hole (SMBH) masses. One advantage of using spiral arm pitch angle to measure SMBH

mass is that pitch angle measurement requires only imaging data as opposed to spectra,

which may be available with sufficient resolution out to higher cosmological redshifts.

Two independent techniques currently in use for measuring the spiral arm pitch angle are

through the use of a Two Dimensional Fast Fourier Transform software called 2DFFT

(Davis et. al 2012), or through template fitting in a spiral coordinate system with Spirality

(Shields et al., submitted).

The Pitch Angle Probability Distribution Function is then produced, showing the relative

frequency of different pitch angle galaxies in the sample. This Pitch Angle Distribution

Function may then be converted to a Black Hole Mass Function using the M-P relation.

Pitch Angle Sample Comparison

Sample Comparison

Wavelength Dependence

The density wave theory of spiral structure (Lin and Shu, 1964) implies

that the observed pitch angle of galaxy spiral arms should vary with the

waveband of the observations. Specifically, it is predicted that stars

form close to the density wave and will either move faster than the

density wave (if inside the corotation radius) or slower than the density

wave (outside the corotation radius). So, stars, especially older stars,

should show tighter pitch angles than the gas and dust of the star

forming region itself.

Above is shown the pitch angle comparison results for the B-band

versus 8 micron band with data from Spitzer and GALEX. The 3.6

micron measurements (not shown above), and the B-band

measurements (representing stellar populations of different ages) are

both uniformly tighter than those of the 8 micron band (representing

gas and dust), in accordance with density wave theory.

Acknowledgements and References

We acknowledge the Carnegie-Irvine Galaxy Survey whose images

were measured for this project. MATLAB was employed for plots and

calculations. Plotting and statistical analysis performed with R.

Berrier, J., Davis, B., Kennefick, D., Kennefick, J., Seigar, M., Barrows, R., Hartley, M.,

Shields, D., Bentz, M., and Lacy, C. 2013, ApJ, 769, 132

Davis, B. L., Berrier, J. C., Johns, L., Shields, D., Hartley, M., Kennefick, D., Kennefick,

J., Seigar, M., and Lacy, C. 2014, ApJ, 789, 124

Davis, B. L., Berrier, J. C., Shields, D. W., Kennefick, J., Kennefick, D., Seigar, M. S.,

Lacy, C. H. S., & Puerari, I. 2012, ApJS, 199, 33

Ho, L. C., Li, Z.-Y., Barth, A. J., Seigar, M. S., & Peng, C. Y. 2011, ApJS,197, 21

Lin, C. C. and Shu, F.H. 1964, ApJ 140, 646.

Seigar, M.S., Kennefick, D., Kennefick, J., Lacy, C.H.S., 2008, ApJ, 678, L93

Shields, D. W. et al. (2015), Submitted.

R Core Team. 2013, R: A Language and Environment for Statistical Computing, R

Foundation for Statistical Computing, Vienna, Austria

The Pitch Angle Distribution Function of Local Disk Galaxies and its Role in Galactic Structure

Fusco, M. S., Imani, H. P., Davis, B. L., Shields, D. W., Kennefick, D., Kennefick, J.

Abstract

A recent study by Davis et al. (2014) has presented a pitch angle

distribution function of local galaxies from the Carnegie-Irvine

Survey (CGS ) and used it to calculate the local super massive black

hole mass function (BHMF) for those galaxies. For reasons of

completeness their sample was limited in both luminosity and

volume. We now present an analysis of the dimmer galaxies

excluded from the previous sample. This subset consists of spiral

galaxies from the CGS with Absolute B-Band Magnitude greater

than -19.12 and limiting luminosity distance (redshift independent)

less than 25.4 Mpc (z=0.00572). These parameters yield a sample set

of 74 spiral galaxies with 51 measurable pitch angles. Combining the

new subset with the sample from Davis et al., gives us a sample of

galaxies limited only in Luminosity-Distance rather than both

Luminosity-Distance and Absolute Magnitude. Not surprisingly the

newer subset is morphologically distinct from the earlier magnitude

limited sample, with more Sc and Sd classified galaxies, suggesting

smaller central bulges. But interestingly the pitch angle distribution

is not particularly dissimilar to that found by Davis et al. for the

brighter sample. The dispersion relation for spiral density waves, as

discussed in Davis et al. (2015), suggests that if the Sc and Sd

galaxies really have smaller bulges, they could still have broadly

similar pitch angles if their disks had larger gas densities. A study of

the link between these quantities would be essential to constructing a

complete BHMF, which would include the low mass end of the mass

function.

Figure 1: Left: Luminosity Distance vs. absolute B-band magnitude for CGS

survey galaxies. Davis et. al 2014 sample (green); Dim galaxy sample (purple).

Right: Mass-Pitch Angle Relation for directly measured black hole masses in

Berrier et al. 2013.

Discussion and Future Work

The latest sample of CGS spiral galaxies have a pitch angle

distribution function (and corresponding SMBH mass distribution)

remarkably similar to that of Davis et al. 2013. There is some

indication of an increased abundance of lower mass black holes in the

sample.

Recent work has also shown a dependence of galactic spiral arm pitch

angle on the waveband of the observations, strong evidence in

confirmation of density wave theory. Further research into the

variation in pitch angle as a function of the age of the stellar

population is of interest.

The performance of pitch angle measurement techniques as a function

of galaxy morphology, luminosity, and redshift, is yet to be explored.

Such studies, along with studies on selection effects and biases, should

greatly reduce the errors in our measurements going forward.

Arkansas Center for

Space and Planetary Sciences

Fayetteville, Arkansas

Presented at

STScI:

What Shapes

Galaxies?

April, 2016

0

5

10

15

20

25

30

35

40

45

50

a ab b bc c cd d m

Pe

rce

nt

Hubble Type

Relative Frequency of Hubble Types

Vol. Limited Sample

Dim Sample

Figure 2: Left: Hubble types from each sample. The Volume limited sample

contains primarily spirals of medium tightness (b, bc, c), while the Dim

sample favors less bulge dominated spirals (cd, d). Therefore there exists a

morphological difference between the two selections of galaxies.

Right: Number of arms for the two samples is similar, with two armed

galaxies being by far the most abundant. It should be noted, however, that the

Dim sample has several difficult to measure, flocculent galaxies in addition to

grand design spirals.

0

10

20

30

40

50

60

70

80

1 2 3 4 5 6

Pe

rce

nt

m

Relative Frequency of Fourier Modes (arm number)

Vol. Limited Sample

Dim Sample

Figure 3: Left: The Pitch Angle of a spiral is defined as the angle between the tangent of the curve at a given

radius and the tangent of a circle at the same radius. For logarithmic spirals, pitch angle is constant as radius

varies. Middle and Right (from Davis et. al 2012): Galaxy M51 pitch angle measurement example using the

2DFFT software. For M51a, the m=2 Fourier mode (2 arm spiral) is clearly dominant, with a stable positive

chirality pitch angle of +16.26°±3.20°.

Figure 4: Bandwidth Optimized Kernel Density

Estimation of the Pitch Angle Distribution Function for

the two samples. The functions largely agree at lower

pitch angles and out to the peak of the function, but

diverge at the tail of the distribution. There is some

indication that the Dim Sample (purple, solid) has a

higher probability of galaxies with higher pitch angles

than the Volume Limited Sample (green, dotted).

Although the dimmer sample and the

volume limited sample are

morphologically distinct, their pitch angle

distributions are strikingly similar. Only a

possible tendency towards very loose

spirals at the distribution’s tail betrays any

sign of the shift to higher P one might

expect in a sample dominated by Sc and Sd

hubble types. This bump in the tail of the

distribution is small, and measurement

errors are known to be larger in higher P

regimes, so the strength of this assertion

remains weak as of yet.

The fundamental plane result reported in

Davis et al 2014 suggests that the smaller

bulges associated with these later type

galaxies could be compatible with tighter

pitch angles if they had low gas densities in

their disks, a possibility which would be

interesting to follow up.

Moving forward, correcting our sample for

the effects of incompleteness will lead to

better constraint of the Black Hole Mass

Function in this lower mass regime.

M-P Relation