Appendix A: Systematic velocity components in the proposed galactic model

In this appendix we show the expressions of the velocity components in the three systematic contributions considered in our galactic model: solar motion, differential galactic rotation and spiral arm kinematics.

A.1 Solar motion

A star with galactic longitude l and galactic latitude b has the following radial and tangential velocity components owing to solar proper motion:

 

$$v_{{\rm {r_1}}} - U_\odot \cos l \cos b - V_\odot \sin l \cos b - W_\odot \sin b$$ =

$$v_{{\rm {l_1}}} U_\odot \sin l - V_\odot \cos l$$ =

$$v_{{\rm {b_1}}} U_\odot \cos l \sin b + V_\odot \sin l \sin b - W_\odot \cos b$$ = (A.1)

where $U_\odot$, $V_\odot$ and $W_\odot$ are the components of the solar motion in galactic coordinates.

A.2 Galactic rotation

We consider axisymmetric differential rotation of our galaxy, with a rotation curve that can be developed in the solar neighbourhood as:

 

$$\Theta(\varpi) \approx \Theta(\varpi_\odot) + \left( \frac {\partial \Theta}{\partial \v... ...left( \frac {\partial^2 \Theta}{\partial \varpi^2} \right)_\odot \Delta\varpi^2$$

$$\equiv \Theta(\varpi_\odot) + a_{{\rm r}} \Delta\varpi + b_{{\rm r}} \Delta\varpi^2$$ (A.2)

where $\Delta\varpi = \varpi - \varpi_\odot$ ( $\varpi_\odot$ is the galactocentric distance of the Sun) and $\Theta (\varpi _\odot )$ is the circular velocity at the Sun's position. We note that $a_{\rm r}$allows us to calculate the A Oort constant in the Sun's vicinity:

$$A = \frac {1}{2} \left[ \frac {\Theta(\varpi_\odot)}{\varpi... ...Theta(\varpi_\odot)}{\varpi_\odot} - a_{{\rm r}} \right]\cdot$$ (A.3)

The radial and tangential velocity components of a star in the galactic plane due to differential galactic rotation are:

 

$$v_{{\rm r_2}} \Theta(\varpi_\odot) \left[\sin(l+\theta) - \sin l\right] \cos b$$ =

$$+ a_{{\rm r}} \Delta\varpi \sin(l+\theta) \cos b$$

$$+ b_{{\rm r}} \Delta\varpi^{{\rm 2}} \sin(l+\theta) \cos b$$

$$v_{{\rm l_2}} \Theta(\varpi_\odot) \left[\cos(l+\theta) - \cos l\right]$$ =

$$+ a_{{\rm r}} \Delta\varpi \cos(l+\theta)$$

$$+ b_{{\rm r}} \Delta\varpi^{{\rm 2}} \cos(l+\theta)$$

$$v_{{\rm b_2}} - \Theta(\varpi_\odot) \left[\sin(l+\theta) - \sin l\right] \sin b$$ =

$$- a_{{\rm r}} \Delta\varpi \sin(l+\theta) \sin b$$

$$- b_{{\rm r}} \Delta\varpi^{{\rm 2}} \sin(l+\theta) \sin b$$ (A.4)

where $\theta$ is the galactocentric longitude of the star.

A.3 Spiral structure kinematics

Lin's theory (Lin & Shu 1964; Lin et al. ; see also Rohlfs 1977) assumes a spiral potential of the form:

$$V_{{\rm b}} = {\cal A} \cos \psi$$ (A.5)

(${\cal A}$ is the amplitude - ${\cal A} < 0$ - and $\psi$ is the phase of the density wave) which disturbs the axially symmetric gravitational potential. The shape of the spiral arms is well represented by a logarithmic spiral:

$$q(\varpi,\theta,t) = q \, {\rm e}^{i \psi(\varpi,\theta,t)}.$$ (A.6)

The amplitude q is a slowly varying function of $\varpi$ and the phase of the spiral structure can be related to the phase at the Sun's position by the following expression:

 

$$\psi(\varpi,\theta,t) = \psi_\odot(t) + m (\Omega_{{\rm p}} t - \theta) + \frac {m \ln \frac {\varpi}{\varpi_\odot(t)}}{\tan i} \Rightarrow$$

$$\Rightarrow \psi(\varpi,\theta,t=0) = \psi_\odot + m \theta + \frac {m \ln \frac {\varpi}{\varpi_\odot}} {\tan i}$$ (A.7)

where $\Omega _{{\rm p}}$ is the angular rotation velocity of the spiral pattern, m the number of spiral arms and i the pitch angle (for trailing spiral arms, i < 0). The phase of the spiral structure at the Sun's position and the pitch angle can be determined from optical and radio indicators. Nevertheless, we point out that the maximum in the distribution of spiral arm tracers (position of the observed spiral arms) may be shifted in relation to the minimum in the perturbation potential (defined as $\psi = 0\hbox {$^\circ $ }$; see Roberts 1969).

The mean peculiar velocities due to the spiral arm perturbations on the velocity field are, in the gas approximation, the following:

 

$$\Pi_1 \frac {k {\cal A}}{\kappa} \frac {\nu}{1 - \nu^2 + x} \cos\psi \equiv \Pi_{{\rm b}} \cos\psi$$

$$\Theta_1 - \frac {1}{2} \frac{k {\cal A} \varpi}{\Theta} \frac {1}{1 - \nu^2 + x} \sin\psi \equiv -\Theta_{{\rm b}} \sin\psi$$ (A.8)

$\Pi_1$ is positive towards the galactic anti-center and $\Theta_1$ is positive towards the galactic rotation. In tightly wound spirals (i.e., those with $\vert\tan i\vert \ll 1$) the amplitudes $\Pi _{\rm b}$ and $\Theta _{\rm b}$ are slowly varying quantities with the galactocentric distance. k is the radial wave number (for trailing spiral arms, k < 0):

$$k = \frac {\rm d}{\rm d \varpi} \left( \frac {m \ln \frac... ...pi}{\varpi_\odot}}{\tan i} \right) = \frac {m}{\varpi \tan i}$$ (A.9)

$\nu$ is the dimensionless rotation frequency of the spiral structure, expressed in terms of the epicyclic frequency ($\kappa$):

$$\nu = \frac {m}{\kappa} \left( \Omega_{{\rm p}} - \frac {\Theta}{\varpi} \right)$$ (A.10)

(notice that $\nu < 0$ in the region with $\Omega_{{\rm p}} < \Omega = \Theta/\varpi$, i.e. inner to the corotation circle). Furthermore:

$$\kappa^2 = \frac {2 \Theta^2}{\varpi^2} \left( 1 + \frac {\varpi}{\Theta} \frac {\rm d \Theta}{\rm d \varpi} \right)$$ (A.11)

and x is the stability Toomre's number (Toomre 1969) defined as:

$$x = \frac {k^2 a^2_{{\rm o}}}{\kappa^2}$$ (A.12)

where $a_{{\rm o}}$ is the velocity dispersion of the gas particles. Since the velocity amplitudes $\Pi _{\rm b}$ and $\Theta _{\rm b}$ depend on this velocity dispersion, we introduce a dimensionless parameter ($f_\odot$) that relates the velocity amplitudes of the Sun to those of the sample stars:

 

$$\Pi_{{\rm b \odot}} = \frac {1 - \nu^2 + x_{{\rm stars}}}{1 - \nu^2 + x_\odot} \Pi_{{\rm b}} \equiv f_\odot \Pi_{{\rm b}}$$

$$\Theta_{{\rm b \odot}} = \frac {1 - \nu^2 + x_{{\rm stars}}}{1 - \nu^2 + x_\odot} \Theta_{{\rm b}} \equiv f_\odot \Theta_{{\rm b}}.$$ (A.13)

The so-called Lindblad resonances are defined as:

$$\Omega_{{\rm p}} = \Omega \pm \frac {\kappa}{m}$$ (A.14)

where the - sign corresponds to the inner resonance and the + sign to the outer one. In the region between both resonances ($\vert\nu\vert < 1$), $\Theta _{\rm b}$ is always positive and $\Pi _{\rm b}$ has a sign that depends on the sign of $\nu$.

The amplitude of the spiral potential can be expressed as:

$${\cal A} = \frac {\kappa \Pi_{{\rm b}}}{k} \frac {1 - \nu^2 + x}{\nu}\cdot$$ (A.15)

The maximum value of the radial force owing to this spiral potential is:

$$F_{{{\rm r}}1}^{{\rm max}} = k \vert{\cal A}\vert$$ (A.16)

whereas the radial force due to the axisymmetric field is:

$$F_{{{\rm r}}0} = \frac {{\rm d}V_0}{\rm d\varpi} \approx \Omega_\odot^2 \varpi_\odot.$$ (A.17)

Therefore, the ratio between both quantities is:

$$f_{{\rm r}} = \frac {F_{{{\rm r}}1}^{{\rm max}}}{F_{{{\rm r}... ...{\Omega_\odot^2 \varpi_\odot} \frac {1 - \nu^2 + x}{\nu}\cdot$$ (A.18)

The contributions in the radial and tangential velocity components of a star due to the spiral arm perturbation velocities $\Pi_1$ and $\Theta_1$are the following:

 

$$v_{{\rm {r_3}}} = - \Pi_1 \cos (l + \theta) \cos b + \Pi_{1\odot} \cos l \cos b + \Theta_1 \sin (l {+} \theta) \cos b {-} \Theta_{1\odot} \sin l \cos b$$

$$= - \Pi_{{\rm b}} \cos \psi_\odot \left( \cos \left[ m \left( \th... ...ot}}{\tan i} \right) \right] \cos(l + \theta) {-} f_\odot \cos l \right) \cos b$$

$$- \Pi_{{\rm b}} \sin \psi_\odot \sin \left[ m \left(\theta - \fra... ...n i} \right) \right] \cos(l + \theta) \cos b - \Theta_{{\rm b}} \sin \psi_\odot$$

$$\times \left( \cos \left[ m \left( \theta {-} \frac {\ln \frac {\... ...ot}}{\tan i} \right) \right] \sin(l + \theta) {-} f_\odot \sin l \right) \cos b$$

$$+ \Theta_{{\rm b}} \cos \psi_\odot \sin \left[ m \left(\theta - \... ...n \frac {\varpi}{\varpi_\odot}}{\tan i} \right) \right] \sin(l + \theta) \cos b$$

$$v_{{\rm {l_3}}} \Pi_1 \sin (l + \theta) - \Pi_{1\odot} \sin l +\Theta_1 \cos (l + \theta) - \Theta_{1\odot} \cos l$$ =

$$\Pi_{{\rm b}} \cos \psi_\odot \left( \cos \left[ m \left( \theta ... ...varpi_\odot}}{\tan i} \right) \right] \sin(l + \theta) - f_\odot \sin l \right)$$ =

$$+ \Pi_{{\rm b}} \sin \psi_\odot \sin \left[ m \left( \theta - \fr... ...t}}{\tan i} \right) \right] \sin(l + \theta) - \Theta_{{\rm b}} \sin \psi_\odot$$

$$\times \left( \cos \left[ m \left( \theta - \frac {\ln \frac {\va... ...varpi_\odot}}{\tan i} \right) \right] \cos(l + \theta) - f_\odot \cos l \right)$$

$$+ \Theta_{{\rm b}} \cos \psi_\odot \sin \left[ m \left( \theta - \frac {\ln \frac {\varpi}{\varpi_\odot}}{\tan i} \right) \right] \cos(l + \theta)$$

$$v_{{\rm {b_3}}} = \Pi_1 \cos (l + \theta) \sin b - \Pi_{1\odot} \cos l \sin b - \Theta_1 \sin (l + \theta) \sin b + \Theta_{1\odot} \sin l \sin b$$

$$= \Pi_{{\rm b}} \cos \psi_\odot \left( \cos \left[ m \left( \thet... ...}}{\tan i} \right) \right] \cos (l {+} \theta {-} f_\odot \cos l \right) \sin b$$

$$+ \Pi_{{\rm b}} \sin \psi_\odot \sin \left[ m \left(\theta - \fra... ...n \frac {\varpi}{\varpi_\odot}}{\tan i} \right) \right] \cos(l + \theta) \sin b$$

$$+ \Theta_{{\rm b}} \sin \psi_\odot \left( \cos \left[ m \left( \t... ...}}{\tan i} \right) \right] \sin(l {+} \theta) {-} f_\odot \sin l \right) \sin b$$

$$- \Theta_{{\rm b}} \cos \psi_\odot \sin \left[ m \left(\theta - \... ... \frac {\varpi}{\varpi_\odot}}{\tan i} \right) \right] \sin(l + \theta) \sin b.$$ (A.19)

A.4 Systematic velocity field in the proposed galactic model

The systematic radial and tangential velocity components of a star in our galactic model are given by:

$$v_{{\rm r}} = v_{{\rm r_1}} + v_{{\rm r_2}} + v_{{\rm r_3}} = \sum\limits_{j=1}^{10} a_j f_j^{{\rm r}}(R,l,b)$$

$$v_{{\rm l}} = v_{{\rm l_1}} + v_{{\rm l_2}} + v_{{\rm l_3}} = \sum\limits_{j=1}^{10} a_j f_j^{{\rm l}}(R,l,b)$$

$$v_{{\rm b}} = v_{{\rm b_1}} + v_{{\rm b_2}} + v_{{\rm b_3}} = \sum\limits_{j=1}^{10} a_j f_j^{{\rm b}}(R,l,b)$$ (A.20)

where the constants a_j contain combinations of the kinematic parameters that we wish to determine:

 

$$a_1 = U_\odot$$

$$a_2 = V_\odot$$

$$a_3 = W_\odot$$

$$a_4 = \Theta(\varpi_\odot)$$

$$a_5 = a_{{\rm r}}$$

$$a_6 = b_{{\rm r}}$$

$$a_7 = \Pi_{{\rm b}} \cos \psi_\odot$$

$$a_8 = \Pi_{{\rm b}} \sin \psi_\odot$$

$$a_9 = \Theta_{{\rm b}} \sin \psi_\odot$$

$$a_{10} = \Theta_{{\rm b}} \cos \psi_\odot$$ (A.21)

and f_jⁱ(R,l,b) are functions of the heliocentric distance and the galactic longitude and latitude:

 

$$f_1^{\rm r} = - \cos l \cos b$$

$$f_2^{\rm r} = - \sin l \cos b$$

$$f_3^{\rm r} = - \sin b$$

$$f_4^{\rm r} = \left[ \sin(l + \theta) - \sin l \right] \cos b$$

$$f_5^{\rm r} = \Delta \varpi \sin (l + \theta) \cos b$$

$$f_6^{\rm r} = \Delta \varpi^2 \sin (l + \theta) \cos b$$

$$f_7^{\rm r} = {-} \left( \cos \left[ m \left( \theta {-} \frac {\... ... {\tan i} \right) \right] \cos (l {+} \theta) {-} f_\odot \cos l \right) \cos b$$

$$f_8^{\rm r} = - \sin \left[ m \left( \theta - \frac {\ln \frac {\varpi}{\varpi_\odot}} {\tan i} \right) \right] \cos (l + \theta) \cos b$$

$$f_9^{\rm r} = {-} \left( \cos \left[ m \left( \theta {-} \frac {\... ...}} {\tan i} \right) \right] \sin (l {+} \theta) - f_\odot \sin l \right) \cos b$$

$$f_{10}^{\rm r} \sin \left[ m \left( \theta - \frac {\ln \frac {\varpi}{\varpi_\odot}} {\tan i} \right) \right] \sin (l + \theta) \cos b$$

$$f_1^{\rm l} = \sin l$$

$$f_2^{\rm l} = - \cos l$$

$$f_3^{\rm l} = 0$$

$$f_4^{\rm l} = \cos(l + \theta) - \cos l$$

$$f_5^{\rm l} = \Delta \varpi \cos(l + \theta)$$

$$f_6^{\rm l} = \Delta \varpi^2 \cos(l + \theta)$$

$$f_7^{\rm l} = \cos \left[ m \left( \theta - \frac {\ln \frac {\varpi}{\varpi_\odot}} {\tan i} \right) \right] \sin(l + \theta) - f_\odot \sin l$$

$$f_8^{\rm l} = \sin \left[ m \left( \theta - \frac {\ln \frac {\varpi}{\varpi_\odot}} {\tan i} \right) \right] \sin(l + \theta)$$

$$f_9^{\rm l} = - \cos \left[ m \left( \theta - \frac {\ln \frac {\varpi}{\varpi_\odot}} {\tan i} \right) \right] \cos(l + \theta) - f_\odot \cos l$$

$$f_{10}^{\rm l} = \sin \left[ m \left( \theta - \frac {\ln \frac {\varpi}{\varpi_\odot}} {\tan i} \right) \right] \cos(l + \theta)$$

$$f_1^{\rm b} = \cos l \sin b$$

$$f_2^{\rm b} = \sin l \sin b$$

$$f_3^{\rm b} = - \cos b$$

$$f_4^{\rm b} = - \left[ \sin(l + \theta) - \sin l \right] \sin b$$

$$f_5^{\rm b} = - \Delta \varpi \sin (l + \theta) \sin b$$

$$f_6^{\rm b} = - \Delta \varpi^2 \sin (l + \theta) \sin b$$

$$f_7^{\rm b} = \left( \cos \left[ m \left( \theta {-} \frac {\ln \... ... {\tan i} \right) \right] \cos (l {+} \theta) {-} f_\odot \cos l \right) \sin b$$

$$f_8^{\rm b} = \sin \left[ m \left( \theta - \frac {\ln \frac {\varpi}{\varpi_\odot}} {\tan i} \right) \right] \cos (l + \theta) \sin b$$

$$f_9^{\rm b} = \left( \cos \left[ m \left( \theta {-}\frac {\ln \f... ... {\tan i} \right) \right] \sin (l {+} \theta) {-} f_\odot \sin l \right) \sin b$$

$$f_{10}^{\rm b} = - \sin \left[ m \left( \theta - \frac {\ln \frac {\varpi}{\varpi_\odot}} {\tan i} \right) \right] \sin (l + \theta) \sin b.$$ (A.22)

In our resolution procedure the parameters a_j are computed following an iterative scheme (extensively explained in Fernández 1998) and, from these, the kinematic parameters $U_\odot$, $V_\odot$, $W_\odot$, $a_{\rm r}$, $b_{\rm r}$, $\psi _\odot$, $\Pi _{\rm b}$, $\Theta _{\rm b}$ and $f_\odot$ are derived.

With regard to the free parameters of our model, different values for m, i, $\varpi_\odot$, $\Theta (\varpi _\odot )$ were considered (see Sect. 7).