# How to interpret skewness and kurtosis plots of galaxy snapshots?

What I did:- I am analysing some simulation snapshots of the Milky Way and it is modelled as a dark matter halo – bulge – disc system. I produced some skewness (using scipy.stats.skew() in python) and kurtosis (using scipy.stats.kurtosis) plots of line of sight velocity (LOSV)that I am attaching here. To produce these plots I simply extracted the $$v_x,v_y,v_z$$ components of the velocities of the stars and I picked the $$z$$-axis as the line of sight. I overlaid a grid on the galaxy snapshot, and at different inclinations ($$\theta=0,45,60,90$$ degrees) of the galaxy I plotted the skewness and kurtosis in the grid cells ($$v_z$$ is the $$z$$-component of the velocities). [![enter image description here][1]][1]

My question: I want to now understand and interpret my plots. What can I deduce from the plots for the different angles of inclinations? My model galaxy has a halo, bulge at centre and disk. What can we tell about the losv distributions or the components of the galaxy? I'm new to astrophysics so these kind of plots and concepts are new to me. Any help is much appreciated, thanks so much!

• What are you asking here? Do you know what skew and kurtosis represent in statistics? Or are you trying to discern actual parameters of a galaxy? Commented Nov 18, 2021 at 19:18
• Hi @Carl, I understand that skewness is a measure of symmetry of vlos (in this case) and kurtosis a measure of how heavy vlos distribution tails are compared to a normal distribution...what I want to know is what the skewness and kurtosis plots might imply about vlos or about the parameters of galaxies. Commented Nov 19, 2021 at 6:32
• Ahhh, for us rookies, vlos --> $V_{los}$ , vertical line of sight Commented Nov 19, 2021 at 13:02
• @Carl, sorry, I meant V_los as in line of sight velocity (edited on my post). Commented Nov 19, 2021 at 22:26

To begin with, I would suggest also plotting the line-of-sight velocity and velocity dispersion maps for each inclination. In particular, you should see a clear anti-correlation between the velocity and the skewness, especially in the edge-on case. This is a signature of rotation-dominated kinematics and a flat or slowly changing rotation curve, due to the line-of-sight vetor intersecting (nearly) circular orbits at different angles outside the tangent radius:

Edited to add: The idea is that your line of sight (dashed green line) is tangent to the point A, and so the projected lines-of-sight velocity = the rotation velocity at that radius. The line of sight intersects stellar orbits at larger radii at increasingly large angles, so the projected velocity along the line of sight gets smaller and smaller (if the stellar velocity curve is flat or decreasing or even mildly increasing). Assuming a locally Gaussian distribution of stellar velocities at each radius, the contributions from radii $$> A$$ show up at smaller velocities (curves in the right-hand figure). Assuming that stellar density decreases at larger radii, the contributions from larger radii are smaller (lower heights in the right-hand figure). The net effect will be a shape with negative skew. For the other side of the galaxy, where velocities are negative (blueshifted), you will get the mirror situation, with a positively skewed LOSVD.

(One small suggestion would be to plot positive quantities with read and negativity quantities with blue -- the opposite of what you're currently doing -- since then positive velocities correspond directly to redshifts and negative velocities to blueshifts.)

The strong pixel-to-pixel variations in the outer part of the disk in your plots is almost certainly just due to low S/N (since the number density of particles is low at large radii).

Note that the standard approach in analyzing galaxy stellar kinematics is to model the line-of-sight velocity distribution using Gauss-Hermite polynomials, where the first- and second-order terms correspond to the mean velocity ($$V$$) and the velocity dispersion ($$\sigma$$), and the third- and fourth-order terms ($$h_{3}$$ and $$h_{4}$$) to the skewness and kurtosis, respectively. (See, e.g., van der Marel & Franx (1993) and Gerhard (1993).) I point this out mainly in case you want to compare your results with published analyses of other models and real galaxies.

• thanks! Right, I can see the anti correlation for the edge on case for mean velocity and skewness. I did not understand this part you mentioned:'' This is a signature of rotation-dominated kinematics and a flat or slowly changing rotation curve, due to the line-of-sight vector intersecting (nearly) circular orbits at different angles outside the tangent radius.'' Also, do you have any suggestion on how to calculate h3 and h4 moments from the line of sight velocities? I just got skewness and kurtosis from scipy functions. Commented Nov 26, 2021 at 6:53
• @Jerome The basic idea is that you fit the LOSVD profile with a model consisting of the first 4 Gauss-Hermite moments, as specified in the papers I linked to. Commented Nov 27, 2021 at 11:48
• @Jerome I've updated my answer with a figure and accompanying explanation which attempts to describe the part you didn't understand. Commented Nov 27, 2021 at 12:19
• Thanks so much, Peter! I also wanted to know more about the last paragraph you wrote regarding modelling Los velocity as Gauss Hermite distribution and then finding higher order moments. I basically have the los velocities as an array of values. From there, I just computed kurtosis and skewness using scipy functions in python. So, how do I calculate h3 and h4 like you are saying? How's that different to the way I got skewness and kurtosis? Commented Nov 27, 2021 at 20:50
• @Jerome it's an exercise in "curve-fitting". Your data needs to be in the form of N(V) -- that is, an array of velocity-bin values (e.g. [-200,-190, ..., 190, 200]) and a matching array of counts per bin. You then attempt to fit the data with a function of five variables: an overall amplitude (not very important), a central velocity, a velocity dispersion ($\sigma$), h3, and h4, as specified in the papers I linked to. The simplified version is assuming h3 = h4 = 0, in which case you're fitting the data with a pure Gaussian (center = $V$, dispersion = $\sigma$). Commented Nov 28, 2021 at 12:30