5.4
Structure function and observed velocity disper-
sion
Often a structure function of the velocity field is used instead of the power spectrum, which is defined as
SF(∆x) =h(V(x+ ∆x)−V(x))2i, (5.7)
where averaging is over a number of pairs of points in space separated by distance ∆x. The line-of-sight velocity dispersion can be linked to the structure function. Indeed, since the emissivity peaks at the center of the cluster and declines with the radius, the largest contribution to the total flux and to the line-of-sight velocity dispersion at distanceRfrom the center comes from the region, the size of which is ∝R.
The structure function and the observed velocity dispersion can be related to the PS (see Appendixes 5.10.3 and 5.10.4):
SF(x) = 2 Z +∞ −∞ P1D(kz)(1−cos 2πkzx)dkz (5.8) and hσ2(R)i= Z ∞ −∞ P1D(kz)(1−PEM(kz))dkz, (5.9)
where P1D is an expectation value of the 1D velocity PS and PEM is a PS of normalized
emissivity along the line of sight. Fig. 5.3 shows the integrands in eq. 5.8 and eq. 5.9 multiplied bykand 2krespectively for the line of sight near the cluster center (black curves) and at a projected distanceR= 800 kpc from the center (red curves). Extra factor of 2 for
σ2(R) is introduced to compensate for the factor of 2 in front of the expression 5.8 for the
structure function. It is more clear if one considers the limits of these equations at large x
and R. When x→ ∞ then cos 2πkzx oscillates with high frequency over relevant interval
of k and mean value of 1−cos 2πkzx is ∼1. When R→ ∞ the emissivity distribution is
very broad and PEM is almost a δ-function. Therefore,
lim x→∞(SF(x)) = 2 Z ∞ −∞ P1D(kz)dkz (5.10) and lim R→∞(σ 2(R)) = Z ∞ −∞ P1D(kz)dkz. (5.11)
From Fig. 5.3 it is clear that the integrands in eq. 5.8 and eq. 5.9 are very similar, suggesting that observed 2σ2(R) should correlate well with the structure function.
5.4 Structure function and velocity dispersion 129
Figure 5.5: Left column: structure function (see analytical expression 5.41), line-of-sight velocity dispersion multiplied by factor 2 (eq. 5.42 and eq. 5.46) and their relation for different models of cored power law power spectrum (eq. 5.37). Parameters for power spectrum (slope and break frequency) are shown in the top left corner. Parameters of the
β− model are given in the bottom right corner.
Right column: the same as left column, but calculated by averaging over 100 statistical realizations of velocity field. The uncertainties in single measurement of the velocity dis- persion are shown with dotted curves.
The choice of parameters is discussed in Section 5.2.
The structure function and the velocity dispersion (eq. 5.41 and 5.42 respectively) are plotted in Fig. 5.5 in left column. We fixed parameters of the β−model of the cluster and varied the slope α and break km of the power spectrum model (eq. 5.2). The relation of SF and 2σ2 is shown in the left bottom panel in Fig. 5.5. For a given R, 2σ2(R) is used
for x−axis, while the SF is plotted as a function of leff(R), where leff is an effective length
along the line of sight, which provides dominant contribution to the line flux. leff is found
from the condition that
Rleff 0 n 2 e( √ R2+l2)dl R∞ 0 n2e( √ R2+l2)dl ≈0.5. (5.12)
Relation between leff and projected distance depends on the β− model of galaxy cluster
as shown in Fig. 5.4
We then made multiple statistical realizations of the PS for a simpleβ−model of galaxy cluster with β = 0.6 and rc = 10 kpc to estimate the uncertainties. The size of the box is 1 Mpc3 and resolution is 2 kpc. We assume that the 3D PS of the velocity field has a
cored power law model (eq. 5.37) with slope α and break wavenumber (injection scale) at km. We made 100 realizations of a Gaussian field with random phases and Gaussian- distributed amplitudes in Fourier space. Taking inverse Fourier transform, we calculated one component of the 3D velocity field (component along the line of sight) in the cluster. Structure function and the line-of-sight velocity dispersion are evaluated using resulting velocity field. Right column in Fig. 5.5 shows velocity dispersion along the line of sight and structure function averaged over 100 realizations. The expected uncertainty in single measurement of the velocity dispersion is shown with dotted curves. One can see that the overall shape and normalization of SF and 2σ2 are the same as predicted from analytical
expressions (left column in Fig. 5.3), however, there are minor differences (especially at small R) due to limited resolution of simulations. Relation between 2σ2 and SF is in a
good agreement with expectation relation, however the uncertainty in measured velocity dispersion (due to stochastic nature of the velocity field) is significant (see Section 6).
Figure 5.6: Velocity dispersion and VRMS as functions of distance from the cluster center
with uncertainties due to stochastic nature of the velocity field (left panels) and their ratio from eq. 5.15 (right panel). VRMS(R) and σ(R) are averaged over a ring at distance R
from the cluster center. Errors show the characteristic uncertainties in one measurement. The slope of assumed velocity PS is -11/3. Break wavenumber km of the PS is shown on the right panel. Color coding is the same in both panels.