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4.4. Implementation of a Quadrupole Flow

In the discussion of VELMOD in Section 2, it was assumed that the IRAS-predicted velocity field, for the correct value of betaI, is as good a model as can be obtained. However, there can be additional contributions to the local flow field from structures beyond the volume surveyed (R leq 12,800 km s-1), as well as from shot noise-induced and Wiener filter-induced differences between the true and derived density fields beyond 3000 km s-1 but within the IRAS volume (cf. Appendix B).

Fortunately, the nature of this contribution is such that we can straightforwardly model its general form and thus treat it as a quasi-free parameter (see below) in the VELMOD fit. Let us write the error in the IRAS-predicted velocity field due to incompletely sampled fluctuations as verr(r). Because the total peculiar velocity field, v + verr, must satisfy equation (4), and because v does so by construction (eq. [3]), it follows that verr must have zero divergence. Moreover, if we suppose that verr corresponds to the growing mode of the linear peculiar velocity field, it must have zero curl well. These properties will be satisfied if verr is given by the gradient of a velocity potential phi that satisfies Laplace's equation. Such a potential may be expanded in a multipole series, each term of which vanishes at the origin (where, by construction, verr must itself vanish).

The leading term in the resulting expansion of verr is a monopole, verr(0)(r) = Ar, or Hubble flow-like term. However, such a term is degenerate with the zero point of the TF relation (Section 3.3) and thus is undetectable. The next term in the expansion is a dipole, verr(1) = B, or bulk flow independent of position. Like the monopole term, however, the dipole term is undetectable, because we work in the frame of the Local Group. Whatever bulk flow is generated by distant density fluctuations is shared by the Local Group as well. The leading term in the expansion of verr(r) to which our method is sensitive is therefore a quadrupole term. Such a term represents the tidal field of mass density fluctuations not traced by the IRAS galaxies. We may write the quadrupole velocity component as

Equation 19 (19)

where curlyVQ is a 3 × 3 matrix. In order for both the divergence and the curl of vQ(r) to vanish, curlyVQ must be a traceless, symmetric matrix. Consequently, it has only five independent elements, two diagonal and three off-diagonal.

We could allow for the presence of such a quadrupole in VELMOD by treating these five elements as free parameters. However, this is a dangerous procedure, because the modeled quadrupole would then have the freedom to fit the quadrupole already present in the IRAS velocity field, which is generated by observed density fluctuations. We wish to allow for the external quadrupole, but we do not want it to fit the beta-dependent quadrupolar component of the IRAS-predicted velocity field. In other words, we want the external quadrupole to be that required for the true value of betaI, which we do not know a priori, rather than the "best-fit" value at any given betaI. This problem would indeed be very serious if inclusion of the quadrupole made a large difference in the derived value of betaI. Fortunately, however, it does not. As we show below, we obtain a maximum likelihood value betaI = 0.56 when the quadrupole is not modeled. When we treat all five components of the quadrupole as free parameters for each betaI, we obtain betaI = 0.47. (12) Because the best-fit quadrupole is relatively insensitive to betaI, we can estimate the external quadrupole by averaging the fitted values of the five independent components obtained for betaI = 0.1, 0.2, ..., 1.0. In this way, we "project out" the betaI-independent part of the quadrupole. In our final VELMOD run, we use this average external quadrupole at each value of betaI. Throughout, we ignore the very small effect that this quadrupole might have on the derived IRAS density field.

In Figure 4, this quadrupole field is plotted on the sky in Galactic coordinates for a distance of 2000 km s-1. The inflow due to the quadrupole, which occurs near the Galactic poles, is of greater amplitude than the outflow, which occurs at low Galactic latitude. The quadrupole reaches its maximum amplitude at l appeq 165°, b appeq 55°, in the direction of the Ursa Major cluster, as well as on the opposite side of the sky. In Section 5, when we plot VELMOD residuals on the sky with and without the quadrupole, the need for the quadrupole field shown in Figure 4 will become clear. Indeed, we will show in Section 5 that the VELMOD fit is statistically acceptable only when the quadrupole is included. Table 2 tabulates the numerical values of the independent elements of curlyVQ that generate this flow. The rms value of this quadrupole over the sky is 3.3%, pleasingly close to the value we expect from theoretical considerations (Appendix B).

Figure 4

Figure 4. The external quadrupolar velocity field used in the VELMOD analysis, plotted in Galactic coordinates. The open symbols indicate negative radial velocities, and the stars indicate positive radial velocities. The amplitude of the quadrupole is shown for a distance r = 2000 km s-1. As indicated by eq. (19), the quadrupole flow increases linearly with distance at a given position on the sky. The maximum amplitude of the quadrupole at this distance is 147 km s-1, which occurs at l appeq 165°, b appeq 55°, as well as on the opposite side of the sky.


Quantity Value Comments

curlyVQ(x, x) 37 km s-1 At 2000 km s-1 (cf. eq. [19])
curlyVQ(y, y) 36 km s-1 At 2000 km s-1 (cf. eq. [19])
curlyVQ(x, y) 15 km s-1 At 2000 km s-1 (cf. eq. [19])
curlyVQ(x, z) 113 km s-1 At 2000 km s-1 (cf. eq. [19])
curlyVQ(y, z) -24 km s-1 At 2000 km s-1 (cf. eq. [19])
sigmav 125 km s-1
wLG,x -30 km s-1
wLG,y -10 km s-1
wLG,z 30 km s-1
bA82 10.36 ± 0.36 10.29 ± 0.22 (Mark III value)
AA82 -5.96 ± 0.09 - 5.95 ± 0.04 (Mark III value)
sigmaTF,A82 0.464 ± 0.026 0.47 ± 0.03 (Mark III value)
bMAT 7.12 ± 0.22 6.80 ± 0.08 (Mark III value)
AMAT -5.75 ± 0.09 - 5.79 ± 0.03 (Mark III value)
sigmaTF,MAT 0.453 ± 0.013 0.43 ± 0.02 (Mark III value)
betaI 0.492 ± 0.068 With quadrupole
betaI0.563 ± 0.074 Without quadrupole
betaI 0.489 ± 0.084 A82 data only
betaI 0.498 ± 0.107 MAT data only
betaI 0.453 ± 0.093 0 < czLG leq 1350 km s-1
betaI 0.495 ± 0.133 1350 < czLG leq 2150 km s-1
betaI 0.573 ± 0.142 2150 < czLG leq 3000 km s-1
betaI 0.521 ± 0.050 wLG = 0; sigmav fixed to 250 km s-1
betaI 0.491 ± 0.045 wLG = 0; sigmav fixed to 150 km s-1
betaI 0.544 ± 0.071 With quadrupole; 500 km s-1 smoothing
betaI 0.635 ± 0.083 Without quadrupole; 500 km s-1 smoothing
betaI 0.510 ± 0.038 With quadrupole; TF parameters fixed at Mark III values
betaI 0.517 ± 0.039 Without quadrupole; TF parameters fixed at Mark III values

We did not do a likelihood search in parameter space to find formal error bars on quantities other than betaI. Error estimates for the TF parameters come from averaging over the mock catalog VELMOD runs (see Table 1).

When both the quadrupole and the Local Group random velocity vector are modeled, the radial peculiar velocity u(r) that enters into the likelihood analysis (see eq. [9]) is given by

Equation 20 (20)

We emphasize again that while the three components of the Local Group random velocity wLG are treated as free parameters in VELMOD, the five independent parameters of curlyVQ are not, with the exception of a single run that we used to obtain and then average their fitted values at each betaI. In the final run, from which we derive the estimate of betaI quoted in the abstract, the quadrupole velocity field shown in Figure 4 was used at each value of betaI.

12 This value differs from the value of 0.49 quoted in the abstract because ultimately we will not allow the components of the quadrupole to be free parameters at each value of betaI. Back.

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