Appendix A: Correction for the velocity gradient

We assume a well-resolved velocity field so that second and higher order derivatives of the velocity field can be neglected. We also assume an HI disk of negligible thickness so that all lines of sight cross the disk at a single radius, implying the one-dimensional linear situation sketched in Fig. 6. We choose the x-axis along the velocity gradient, with the zeropoint at the center of the Gaussian beam. The intensity-weighted mean velocity over the beam corresponds to that of position x₀, which is not necessarily the center of the beam.

Figure 6: Definition of symbols used in the calculation of the broadening of the line profile by a velocity gradient over the synthesized beam. The bar at the position xi,vi indicates the velocity dispersion $\sigma _i$ at that position. The center of the beam is x=0 by definition.

We now divide the beam into many (N) lines of sight, each with a large number of identical elements (M) with velocities v_ik, $k=1 \ldots M$ at position x_i, $i=1 \ldots N$. This definition includes an implicit integration over the coordinate perpendicular to the velocity gradient. The elements are identified with individual HI clouds of very small intrinsic velocity dispersion. At every position x_i we define the mean velocity v_i as $v_i={1 \over M} \sum_k v_{ik}$, which is related to the intensity-weighted mean velocity v₀ and the velocity gradient $\nabla v$ through

$$v_i - v_0 = (x_i-x_0) \nabla v.$$

The velocity dispersion of the elements at position x_i is $\sigma_i={1 \over {M-1}} \sum_k (v_{ik}-v_i)^2$. Substitution of $v_i=v_0+(x_i-x_0) \nabla v$ and evaluation of the cross-product yields

$$\sigma^2_i={1 \over {M-1}} \sum_{k=1}^M \{(v_{ik}-v_0)^2 -2(v_{ik}-v_0)(x_i-x_0)\nabla v +(x_i-x_0)^2(\nabla v)^2 \} \cdot$$

The third term is independent of k. For the second term we may write

$$-2 (x_i-x_0)\nabla v \sum_{k=1}^M(v_{ik}-v_0) -{{2M}\over{M-1}}(v_i-v_0)(x_i-x_0)\nabla v$$ =

$$-{{2M}\over{M-1}}(x_i-x_0)^2(\nabla v)^2.$$ =

Therefore, we have

$$\sigma^2_i= {1 \over {M-1}} \sum_{k=1}^M (v_{ik}-v_0)^2-{{M}\over{M-1}}(x_i-x_0)^2(\nabla v)^2.$$

The intensity-weighted mean velocity dispersion over the beam is

$$\langle \sigma^2 \rangle ={ {\sum_{i=1}^N w_i \sigma^2_i} \over {\sum_{i=1}^N w_i}}$$

with weight $w_i={\rm e}^{-x^2/b^2} I_i$. Therefore, with $M \gg 1$

$$\langle \sigma^2 \rangle = {{\sum_{i=1}^N w_i \sum_{k=1}^M (v_{ik... ...}} -(\nabla v)^2{ {\sum_{i=1}^N w_i (x_i-x_0)^2} \over {\sum_{i=1}^N w_i}}\cdot$$

This equation is of the general form

$$\langle \sigma^2 \rangle = \sigma^2_{\rm obs} - \Omega_I b^2 (\nabla v)^2,$$

where $\sigma_{\rm obs}$ is the observed dispersion of a local line profile, corrected for the instrumental spectral resolution. The second term is the line broadening due to the velocity gradient over the beam. Note that the velocity gradient $\nabla v$ is a function of position if the galaxy is not in solid-body rotation, necessitating use of a model velocity field constructed from the rotation curve in order to calculate $\nabla v$ at every position. The coefficient $\Omega_I > 0$ is a weighted mean of the intensity distribution over the beam. If we assume a constant intensity (i.e. I_i=I₀) over the Gaussian beam e ^-(x2/b2), we have $\Omega_I = {1 \over 2}$:

$$\langle \sigma^2 \rangle = \sigma^2_{\rm obs} - {1 \over 2} b^2 (\nabla v)^2.$$

The error introduced by the assumption of a constant intensity can be estimated by calculating the correction for simple analytical intensity distributions. For any assumed intensity distribution

$$x_0={{\sum_{i=1}^N w_i x_i} \over {\sum_{i=1}^N w_i}}$$

and $\langle \sigma^2 \rangle$ can be calculated. The coefficients $\Omega _I$ are given for three types of intensity distribution in Table 6.

Figure 7: The effect of an intensity gradient $I(x)=1+0.2\cdot x$ (dashed line; a = 0.2 in Table 6) over the beam (thin curve). The thick curve is the product of the intensity and the beamshape function with an arbitrary scaling. The vertical solid line marks the position x0 = 0.1. The small difference between the thin and the thick curve (scaling in intensity is free) is the reason that the value of $\Omega _I$ is not sensitive to an intensity gradient over the synthesized beam. In this case, $\Omega _I=0.490$ although I(x=1)/I(x=-1)=1.5.

Shallow intensity gradients do not make much of a difference. To first order, the effect of an intensity gradient is to shift the distribution of w_i in the direction of the intensity gradient. Since the beam function falls off rapidly for large x, only large gradients produce a significant difference with constant intensity. The greatest effect on the correction for the velocity gradient is brought about by the symmetric distribution. If the emission is highly concentrated towards the center of the beam, the velocity gradient has no effect. On the other hand, if the emission is concentrated in the wings of the beam, the effect of the velocity gradient is maximal. If the intensity does not change more than 50% over the beam, the error in $\Omega _I$ introduced by the assumption of constant intensity is of the order of 10%.

   

Table 6: Values of $\Omega _I$ for a gradient, a minimum and an jump in the intensity distribution at the center of the synthesized beam.

	Int. gradient	Int. minimum	Int. jump
a	I(x)=1+ax	I(x)=1+ax2	I(x)=1-aH(x)
0.0	0.500	0.500	0.500
0.1	0.498	0.548	0.497
0.2	0.490	0.591	0.487
0.5	0.439	0.700	0.421
1.0	0.349	0.833	0.182

Note: scale factor a defines the magnitude of the intensity change over the beam. The position x is in units of beamsize b. At positions where the indicated functional forms are negative, the intensity was set to zero. The symbol H(x) is used for the heaviside function $H(x)={x \over {\mid x \mid}}\cdot$