What should the power-law slope be in three dimensions?

[will-henney]

          Comment on the power law slopes:   I think we should maybe be using `2 + m` instead of `3 + m` for the slope of the power spectrum

But @JavGVastro should check this in the literature

Originally posted by @will-henney in https://github.com/JavGVastro/PhD.Paper/issues/16#issuecomment-1416284854

[will-henney]

Here are some relevant passages from Medina et al 2014:

From section 2.3.4 Projection smoothing:

[will-henney]

Combining the concepts from the above quotes, we have that

m(2d) = m(3d) + 1 + δκ

as we wrote in our paper, and therefore

κ(3d) = -n = 3 + m(3d) = 2 + m(2d) - δκ

So the conclusions are:

1. For the constant density case we should have δκ = 0 (full projection smoothing), so that we need to use a power-spectrum slope of 2 + m(2d). For example, for our average case of m(2d) = 1, we should use a slope of 3
2. For the case with strong density fluctuations, we should have δκ ≈ -1, so that we need to use a power-spectrum slope of 3 + m(2d)

[will-henney]

We ended up adjusting the 3D power law by hand in order to get m(2d) = 1. @JavGVastro please summarize here the resulting 3d power laws m(3d) that we ended up using for the three cases of sigma_E = 0, 1, and 2

[JavGVastro]

$\sigma_E = 0$ (green)

$\kappa = 2.0 + 0.85$

$\sigma_E = 1$ (red)

$\kappa = 3.0 + 0.30$

$\sigma_E = 2$ (blue)

$\kappa = 3.0 + 0.55$

[will-henney]

[Edited: added the images, which it seems do not work when you reply to an issue via email]

This is great. It looks like the constant emissivity case is consistent with kappa = 3, which is what we expect with full projection smoothing: δκ = 0. Then we get δκ = -0.3 for sigma_E = 1 and δκ = -0.55 for sigma_E = 2

Strangely, the projection smoothing is reduced for steeper values of κ, even for the constant density case.

We should summarize this in the Appendix, replacing the following

with n = m + 2 - δκ (I think, assuming m is m2d and n is kappa)

[will-henney]

I have rewritten the intro to Appendix A to deal with this. It now reads:

@JavGVastro please check over this and see if you agree

[will-henney]

I have made some more changes to this part. I have added in some extra references to Mivile-Deschenes and to the various Esquivel papers on the subject.

It now reads

In order to create a 2D plane-of-sky velocity map, we first create 3D cubes of fluctuating velocity and emissivity with make_3dfield, then transform to a PPV (position-position veloc- ity) cube using the make_ppv command from turbustat, assum- ing thermal broadening with FWHM of 20 km s−1, as appropriate for a hydrogen line at 𝑇 ≈ 104 K. Finally, the first velocity moment map is obtained by integrating along the velocity axis. For the emissivity field, we assume a log-normal distribution with RMS fractional width 𝜎𝐸/𝐸0 = 1.0, which is typical of our sources (see footnote 8 and Figure 12 above). We further assume that there is no correlation between the emissivity and velocity fluctuations and that their spatial correlation length and power law indices are equal.10 The 3-dimensional 2nd-order structure function slope is re- lated to the power law index as 𝑚3D = 𝑛3D − 3, while the equivalent relation in two dimensions is 𝑚2D = 𝑛2D − 2 for separations smaller than the line-of-sight depth of the emitting region. So long as the emissivity fluctuations are weak and uncorrelated with the veloc- ity fluctuations, then 𝑛2D = 𝑛3D (Miville-Deschênes et al. 2003; Levrier 2004), but if these conditions are not satisfied, then the structure function of the velocity centroids does not purely reflect the velocity power spectrum (Brunt & Mac Low 2004; Esquivel & Lazarian 2005; Ossenkopf et al. 2006; Esquivel et al. 2007) and one has 𝑚2D = 𝑛3D −2+𝛿𝜅 where 𝛿𝜅 is a correction factor that accounts for the contribution to the structure function from column density fluctuations and cross terms (see section 5.2.2). Brunt & Mac Low (2004) found that 𝛿𝜅 is a function of the Mach number for hydrodynamic turbulent box simulations. In prin- ciple, we could calculate 𝛿𝜅 for the fractional Brownian motion models that we use here by using the analytic machinery of Esquivel et al. (2007), but we prefer instead to use a more empirical approach. We vary 𝑛3D until the resultant structure function slope matches a particular value in the range 𝑚2D = 0.8 → 1.2 that encompasses our observational results.11 At each step, we estimate 𝑚2D by fitting our idealized model structure function (equation (10), without the seeing and noise terms) to the structure function of the simulated velocitymap.Inthecaseofauniformemissivityfield(𝜎𝐸/𝐸0 =0), we find 𝛿𝜅 ≈ 0, which is maximal projection smoothing, exactly as expected for a case that mimics incompressible turbulence (see Appendix of Miville-Deschênes et al. 2003). As the amplitude of the emissivity fluctuations increase, we find that 𝛿𝜅 becomes in- creasingly negative, with 𝛿𝜅 ≈ −0.3 for 𝜎𝐸/𝐸0 = 1 (the case that we illustrate here) and 𝛿𝜅 ≈ −0.55 for 𝜎𝐸/𝐸0 = 2. This is qualita- tively consistent with the results in Figure 12a of Brunt & Mac Low (2004), although a quantitative comparison is hard to make, since they are not holding 𝑚 constant in the way we do here.