Appendix A: A gravitomagnetic Cauchy invariant
The gravitomagnetic Cauchy invariant implies that the canonical vorticity divided by the mass density is the conserved quantity in a rotating fluid. In the weak relativistic limit the momentum conservation gives (the fully relativistic Cauchy invariant is demonstrated in reference [26])
$$\begin{aligned} \dfrac{d{\varvec{v}}}{dt}=-\dfrac{\varvec{\nabla }p}{\rho }+{\varvec{E}} +{\varvec{v}}\times {\varvec{B}}. \end{aligned}$$
(A1)
This equation can be written in the form of a diffusion equation for the vorticity. Making use of the vector relation
$$\begin{aligned} {\varvec{v}}\cdot \varvec{\nabla v}=\underset{\varvec{\omega } }{\underbrace{\left( \varvec{\nabla }\times {\varvec{v}}\right) }} \times {\varvec{v}}+\dfrac{1}{2}\varvec{\nabla }\left( {\varvec{v}} \cdot {\varvec{v}}\right) , \end{aligned}$$
(A2)
the acceleration becomes
$$\begin{aligned} \dfrac{d{\varvec{v}}}{dt}=\dfrac{\partial {\varvec{v}}}{\partial t}+{\varvec{v}}\cdot \varvec{\nabla v}=\dfrac{\partial {\varvec{v}} }{\partial t}+\varvec{\omega }\times {\varvec{v}}+\dfrac{1}{2} \varvec{\nabla }\left( {\varvec{v}}\cdot {\varvec{v}}\right) \end{aligned}$$
(A3)
and the curl gives, using the continuity equation,
$$\begin{aligned} \varvec{\nabla }\times \dfrac{d{\varvec{v}}}{dt}=\dfrac{\partial \varvec{\omega }}{\partial t}+\varvec{\nabla }\times \left( \varvec{\omega }\times {\varvec{v}}\right) =\dfrac{d\varvec{\omega } }{dt}-\dfrac{\varvec{\omega }}{\rho }\dfrac{d\rho }{dt}-\varvec{\omega }\cdot \varvec{\nabla v}.\nonumber \\ \end{aligned}$$
(A4)
This leads to the vorticity diffusion equation
$$\begin{aligned} \dfrac{d}{dt}\left( \dfrac{\varvec{\omega }}{\rho }\right) =\dfrac{\varvec{\omega }}{\rho }\cdot \varvec{\nabla v}+\dfrac{1}{\rho }\varvec{\nabla }\times \dfrac{d{\varvec{v}}}{dt}. \end{aligned}$$
(A5)
Now, assuming barotropic flow (in which the pressure p and the density \(\rho \) are directly related) the equation of motion gives, with the help of the gravitoelectromagnetic Faraday’s law,
$$\begin{aligned} \begin{array} [c]{rcl} \dfrac{d}{dt}\left( \dfrac{\varvec{\omega }}{\rho }\right) &{} = &{} \dfrac{\varvec{\omega }}{\rho }\cdot \varvec{\nabla v}-\dfrac{1}{\rho }\dfrac{\partial {\varvec{B}}}{\partial t}+\dfrac{1}{\rho }\varvec{\nabla }\times \left( {\varvec{v}}\times {\varvec{B}}\right) \\ &{} = &{} \dfrac{\varvec{\omega }}{\rho }\cdot \varvec{\nabla v}-\dfrac{1}{\rho }\dfrac{d{\varvec{B}}}{dt}-\dfrac{{\varvec{B}}}{\rho }\varvec{\nabla }\cdot {\varvec{v}}+\dfrac{{\varvec{B}}}{\rho } \cdot \varvec{\nabla v}. \end{array} \end{aligned}$$
(A6)
It follows that the quantity \(\left( \varvec{\omega }+{\varvec{B}} \right) /\rho \) satisfies a modified vorticity equation
$$\begin{aligned} \dfrac{d}{dt}\left( \dfrac{\varvec{\omega }+{\varvec{B}}}{\rho }\right) =\left( \dfrac{\varvec{\omega }+{\varvec{B}}}{\rho }\right) \cdot \varvec{\nabla v}. \end{aligned}$$
(A7)
Introducing a change in the dependent variables as proposed by Serrin [41],
$$\begin{aligned} \dfrac{\varvec{\omega }+{\varvec{B}}}{\rho }={\varvec{C}}\cdot \varvec{\nabla }_{0}{\varvec{r}}, \end{aligned}$$
(A8)
the vorticity equation becomes
$$\begin{aligned} \dfrac{d}{dt}\left( {\varvec{C}}\cdot \varvec{\nabla }_{0}{\varvec{r}} \right) =\left( {\varvec{C}}\cdot \varvec{\nabla }_{0}{\varvec{r}} \right) \cdot \varvec{\nabla v}. \end{aligned}$$
(A9)
Here \(\varvec{\nabla }_{0}{\varvec{r}}=\left| \partial {\varvec{r}} /\partial {\varvec{r}}_{0}\right| =\overline{\overline{{\varvec{J}}}}\) is the Jacobian dyadic of the transformation \({\varvec{r}}=\) \({\varvec{r}} \left( {\varvec{r}}_{0},t\right) \) from the Lagrangian \({\varvec{r}} _{0}\) to the Eulerian \({\varvec{r}}\) coordinates (\(\left| \overline{\overline{{\varvec{J}}}}\right| \ne 0\) and \(\varvec{\nabla } _{0}\equiv \overline{\overline{{\varvec{J}}}}\cdot \varvec{\nabla }\)). The transformation \({\varvec{r}}=\) \({\varvec{r}}\left( {\varvec{r}} _{0},t\right) \) specifies the trajectory of a fluid element (or material particle). For fixed t, it determines the transformation of the fluid element from the initial position \({\varvec{r}}_{0}\) to the position \({\varvec{r}}\) at time t. Since
$$\begin{aligned} \begin{array} [c]{rcl} \dfrac{d}{dt}\left( {\varvec{C}}\cdot \varvec{\nabla }_{0}{\varvec{r}} \right) &{} = &{} \dfrac{d{\varvec{C}}}{dt}\cdot \varvec{\nabla } _{0}{\varvec{r}}+{\varvec{C}}\cdot \varvec{\nabla }_{0}{\varvec{v}}\\ &{} = &{} \dfrac{d{\varvec{C}}}{dt}\cdot \varvec{\nabla }_{0}{\varvec{r}} +{\varvec{C}}\cdot \overline{\overline{{\varvec{J}}}}\cdot \varvec{\nabla v}\\ &{} = &{} \dfrac{d{\varvec{C}}}{dt}\cdot \varvec{\nabla }_{0}{\varvec{r}} +\left( {\varvec{C}}\cdot \varvec{\nabla }_{0}{\varvec{r}}\right) \cdot \varvec{\nabla v} \end{array} \end{aligned}$$
(A10)
the vorticity equation reduces to
$$\begin{aligned} \begin{array} [c]{ccccc} \dfrac{d{\varvec{C}}}{dt}\cdot \varvec{\nabla }_{0}{\varvec{r}}=0&\Longrightarrow&\dfrac{d{\varvec{C}}}{dt}=0&\Longrightarrow&{\varvec{C}}={\varvec{C}}\left( {\varvec{r}}_{0}\right) \end{array} \end{aligned}$$
(A11)
so that
$$\begin{aligned} \dfrac{\varvec{\omega }+{\varvec{B}}}{\rho }={\varvec{C}}\left( {\varvec{r}}_{0}\right) \cdot \varvec{\nabla }_{0}{\varvec{r}}. \end{aligned}$$
(A12)
Setting \(t=0\)
$$\begin{aligned} \dfrac{\varvec{\omega }+{\varvec{B}}}{\rho }=\dfrac{\varvec{\omega }_{0}+{\varvec{B}}_{0}}{\rho _{0}}\cdot \varvec{\nabla }_{0}{\varvec{r}}. \end{aligned}$$
(A13)
This result was obtained, for an incompressible fluid and without the gravitomagnetic field, by Cauchy [42] in 1815 and reviewed by Frisch and Villone [43]. In the absence of the gravitomagnetic field, this shows that a fluid element that is initially in irrotational motion remains in this condition throughout the flow. However, the modified vorticity equation shows that the gravitomagnetic field affects the vorticity distribution.
Appendix B: Green’s function for a thin galactic disk
In cylindrical coordinates with azimuthal symmetry, Poisson’s equation for the gravitational potential \(\phi \) takes the form
$$\begin{aligned} \dfrac{1}{R}\dfrac{\partial }{\partial R}\left( R\dfrac{\partial \phi }{\partial R}\right) +\dfrac{\partial ^{2}\phi }{\partial Z^{2}}=4\pi G\rho , \end{aligned}$$
(B1)
where \(\rho \) is the mass density distribution. The solution can be written in terms of the Green’s function \({\mathcal {G}}(R,Z;R^{\prime },Z^{\prime })=1/\left| {\varvec{r}}-{\varvec{r}}^{\prime }\right| \):
$$\begin{aligned}&\phi \left( R,Z\right) =-2\pi G\displaystyle \int _{-\infty }^{\infty } dZ^{\prime }\nonumber \\&\quad \times \displaystyle \int _{0}^{\infty }dR^{\prime }R^{\prime }{\mathcal {G}} (R,Z;R^{\prime },Z^{\prime })\rho \left( R^{\prime },Z^{\prime }\right) . \end{aligned}$$
(B2)
The Green’s function can be expressed in terms either of a toroidal function \(Q_{-1/2}\left( \chi \right) \) (Legendre function of the second kind, degree \(-1/2\), order zero and type three) or a complete elliptic integral of the first kind \(K\left( m\right) \) [44,45,46]
$$\begin{aligned} \dfrac{1}{\left| {\varvec{r}}-{\varvec{r}}^{\prime }\right| }= & {} \dfrac{1}{\pi \sqrt{RR^{\prime }}}Q_{-1/2}\left( \chi \right) \nonumber \\= & {} \dfrac{1}{\pi \sqrt{RR^{\prime }}}\sqrt{\dfrac{2}{1+\chi }}K\left( \dfrac{2}{1+\chi }\right) , \end{aligned}$$
(B3)
where the argument \(1\le \chi <\infty \) of the toroidal function is
$$\begin{aligned} \chi \left( R,Z;R^{\prime },Z^{\prime }\right) =\dfrac{R^{2}+R^{\prime 2}+(Z-Z^{\prime })^{2}}{2RR^{\prime }}, \end{aligned}$$
(B4)
and the squared modulus \(0\le m\le 1\) of the elliptic function is
$$\begin{aligned} m\left( R,Z;R^{\prime },Z^{\prime }\right) =\dfrac{4RR^{\prime }}{\left( R+R^{\prime }\right) ^{2}+(Z-Z^{\prime })^{2}}, \end{aligned}$$
(B5)
so that
$$\begin{aligned} \begin{array} [c]{ccc} m=\dfrac{2}{1+\chi }&\qquad \text {and}&\qquad \chi =\dfrac{2-m}{m}. \end{array} \end{aligned}$$
(B6)
In terms of the elliptic integral
$$\begin{aligned}&{\mathcal {G}}(R,Z;R^{\prime },Z^{\prime })=\dfrac{2}{\pi \sqrt{\left( R+R^{\prime }\right) ^{2}+(Z-Z^{\prime })^{2}}}\nonumber \\&\quad \times K\left( \dfrac{4RR^{\prime }}{\left( R+R^{\prime }\right) ^{2}+(Z-Z^{\prime })^{2}}\right) . \end{aligned}$$
(B7)
The radial and axial gradients of the Green’s function are given by
$$\begin{aligned} \dfrac{\partial {\mathcal {G}}(R,Z;R^{\prime },Z^{\prime })}{\partial R}= & {} -\dfrac{1}{2\pi R}\sqrt{\dfrac{m}{RR^{\prime }}}\left[ K\left( m\right) \right. \nonumber \\&\left. +\dfrac{1}{2}\left( \dfrac{R}{R^{\prime }}-\dfrac{2-m}{m}\right) \dfrac{mE\left( m\right) }{1-m}\right] ,\nonumber \\ \dfrac{\partial {\mathcal {G}}(R,Z;R^{\prime },Z^{\prime })}{\partial Z}= & {} -\dfrac{1}{4\pi }\left( \dfrac{m}{RR^{\prime }}\right) ^{3/2}\dfrac{E\left( m\right) }{1-m}\left( Z-Z^{\prime }\right) ,\nonumber \\ \end{aligned}$$
(B8)
where \(E\left( m\right) \) denotes the complete elliptic integral of the second kind.
Consider the value of the potential along the equatorial plane (\(Z=0\)) for a vertically symmetric equilibrium
$$\begin{aligned} \phi \left( R,0\right)= & {} -8G\displaystyle \int _{0}^{\infty }dZ^{\prime }\displaystyle \int _{0}^{\infty }\dfrac{\rho \left( R^{\prime },Z^{\prime }\right) R^{\prime }}{\sqrt{\left( R+R^{\prime }\right) ^{2}+Z^{\prime }{} ^{2}}}\nonumber \\&\quad \times K\left( \dfrac{4RR^{\prime }}{\left( R+R^{\prime }\right) ^{2}+Z^{\prime }{}^{2}}\right) dR^{\prime }. \end{aligned}$$
(B9)
The potential at the field position \(\left( R,0\right) \) must be calculated by integration along the source position \(\left( R^{\prime },Z^{\prime }\right) \). Changing variables, the integration along \(R^{\prime }\) can be divided in two branches along m:
$$\begin{aligned} \begin{array} [c]{rclll} R^{\prime } &{} = &{} \left( \dfrac{2-m-\sqrt{4\left( 1-m\right) -m^{2}n^{2}} }{m}\right) R &{} &{} \text {first integration branch}\\ R^{\prime } &{} = &{} \left( \dfrac{2-m+\sqrt{4\left( 1-m\right) -m^{2}n^{2}} }{m}\right) R &{} &{} \text {second integration branch} \end{array} \end{aligned}$$
(B10)
where \(n=Z^{\prime }/R\). Note that
$$\begin{aligned} \dfrac{\partial R^{\prime }}{\partial m}=\pm \dfrac{2R}{m^{2}}\left( \dfrac{2-m\mp \sqrt{4\left( 1-m\right) -m^{2}n^{2}}}{\sqrt{4\left( 1-m\right) -m^{2}n^{2}}}\right) , \end{aligned}$$
(B11)
where the upper and lower signs correspond to the first and second integration branches, respectively. In the first integration branch the variable \(0\le m\lesssim 1\) runs from zero to one in the radial range \(R^{\prime }/R\lesssim 1\) (for \(Z^{\prime }\simeq 0\)) corresponding to the bulge of the density profile. In the second integration branch \(1\gtrsim m>0\) the radial range \(R^{\prime }/R\gtrsim 1\) corresponds to the disk region with \(R^{\prime }\) extending to infinity. The transformed integral becomes
$$\begin{aligned} \phi \left( R,0\right)= & {} -8GR\displaystyle \int _{0}^{\infty }dZ^{\prime }\displaystyle \int _{0}^{2\left( \sqrt{1+n^{2}}-1\right) /n^{2}} dm\nonumber \\&\quad \times \dfrac{K\left( m\right) }{m^{2}\sqrt{4\left( 1-m\right) -m^{2}n^{2}}}\nonumber \\&\quad \times \left[ \rho \left( \dfrac{2-m-\sqrt{4\left( 1-m\right) -m^{2}n^{2}} }{m}R,nR\right) \right. \nonumber \\&\quad \times \left( \dfrac{2-m-\sqrt{4\left( 1-m\right) -m^{2}n^{2}} }{m}\right) ^{3/2}\nonumber \\&\quad +\rho \left( \dfrac{2-m+\sqrt{4\left( 1-m\right) -m^{2}n^{2}}}{m}R,nR\right) \nonumber \\&\quad \times \left. \left( \dfrac{2-m+\sqrt{4\left( 1-m\right) -m^{2}n^{2}}}{m}\right) ^{3/2}\right] .\nonumber \\ \end{aligned}$$
(B12)
The upper limit in the integration over m has the limiting value (\(n=Z^{\prime }/R\))
$$\begin{aligned} \underset{n\rightarrow 0}{\lim }\left( \dfrac{2\left( \sqrt{1+n^{2}}\right) -1}{n^{2}}\right) =1. \end{aligned}$$
(B13)
The above transformed integral expression of \(\phi \left( R,0\right) \) is exact. Now, for a thin disk the mass density distribution can be assumed of the form
$$\begin{aligned} \rho \left( R,Z\right) \simeq \rho \left( R,0\right) \exp \left( -\dfrac{Z^{2}}{2\Delta \left( R\right) ^{2}}\right) . \end{aligned}$$
(B14)
In this approximation all cross-sections of the disk have the same form, but the characteristic width \(\Delta \left( R\right) \) varies with R. For small values of \(\Delta \left( R\right) \) the dominant term of the integration over \(Z^{\prime }\) is given by the Laplace approximation
$$\begin{aligned}&\phi \left( R,0\right) \simeq -4\sqrt{2\pi }GR\displaystyle \int _{0}^{1} dm\dfrac{K\left( m\right) }{2\sqrt{m\left( 1-m\right) }}\nonumber \\&\qquad \times \left[ \left( \dfrac{2-m-2\sqrt{1-m}}{m}\right) ^{3/2}\Delta \left( \dfrac{2-m-2\sqrt{1-m}}{m}R\right) \right. \nonumber \\&\qquad \times \rho \left( \dfrac{2-m-2\sqrt{1-m}}{m}R,0\right) \nonumber \\&\qquad +\left( \dfrac{2-m+2\sqrt{1-m}}{m}\right) ^{3/2}\Delta \left( \dfrac{2-m+2\sqrt{1-m}}{m}R\right) \nonumber \\&\qquad \times \left. \rho \left( \dfrac{2-m+2\sqrt{1-m}}{m}R,0\right) \right] . \end{aligned}$$
(B15)
In the original integration over \(R^{\prime }\) the thin disk approximation gives the simpler expression
$$\begin{aligned} \phi \left( R,0\right)\simeq & {} -4\sqrt{2\pi }G\displaystyle \int _{0}^{\infty }\dfrac{R^{\prime }\Delta \left( R^{\prime }\right) \rho \left( R^{\prime },0\right) }{R+R^{\prime }}\nonumber \\&\times K\left( \dfrac{4RR^{\prime }}{\left( R+R^{\prime }\right) ^{2}}\right) dR^{\prime }, \end{aligned}$$
(B16)
but the transformed integration removes the Green’s function singularity by replacing the upper limit in the m integration by \(1-\epsilon \), where \(\epsilon \) can be taken as small as required by the precision of the calculation (the lower limit can also be replaced by \(\epsilon \) in order to avoid the divergence in the integrand for \(m\rightarrow 0\)).
In general, the integration along the disk branch ends at
$$\begin{aligned} m_{\min }=\dfrac{4RR_{\max }}{\left( 1+n^{2}\right) R^{2}+2RR_{\max }+R_{\max }^{2}}<1, \end{aligned}$$
(B17)
where \(R_{\max }\) corresponds to the maximum extension of the disk defined by \(\rho \left( R_{\max },Z^{\prime }\right) \simeq 0\). Along the equatorial plane the integration path is defined by \(Z^{\prime }=0\) and
$$\begin{aligned} \begin{array} [c]{ccc} m_{\min }=\dfrac{4RR_{\max }}{\left( R+R_{\max }\right) ^{2}}<1&\,&\text {for }Z^{\prime }=0. \end{array} \end{aligned}$$
(B18)
Therefore, for a thin disk the full integration over m can be divided along the two branches as follows
$$\begin{aligned}&\phi \left( R,0\right) \nonumber \\&\quad \simeq -4\sqrt{2\pi }GR\displaystyle \int _{\epsilon }^{1-\epsilon }dm\dfrac{K\left( m\right) }{2\sqrt{m\left( 1-m\right) } }\nonumber \\&\qquad \times \left( \dfrac{2-m-2\sqrt{1-m}}{m}\right) ^{3/2}\nonumber \\&\qquad \times \Delta \left( \dfrac{2-m-2\sqrt{1-m}}{m}R\right) \rho \left( \dfrac{2-m-2\sqrt{1-m}}{m}R,0\right) \nonumber \\&\qquad -4\sqrt{2\pi }GR\displaystyle \int _{m_{\min }}^{1-\epsilon }dm\dfrac{K\left( m\right) }{2\sqrt{m\left( 1-m\right) }}\nonumber \\&\qquad \times \left( \dfrac{2-m+2\sqrt{1-m}}{m}\right) ^{3/2}\nonumber \\&\qquad \times \Delta \left( \dfrac{2-m+2\sqrt{1-m}}{m}R\right) \rho \left( \dfrac{2-m+2\sqrt{1-m}}{m}R,0\right) ,\nonumber \\ \end{aligned}$$
(B19)
with the cut-off at the radial position along the equatorial plane \(\rho \left( R^{\prime },0\right) \simeq 0\) explicitly taken into account. Similarly, the radial gradient of the gravitational potential is
$$\begin{aligned}&\dfrac{\partial \phi \left( R,0\right) }{\partial R} \nonumber \\&\quad \simeq 2\sqrt{2\pi }G\displaystyle \int _{\epsilon }^{1-\epsilon }dm\left( \dfrac{2-m-2\sqrt{1-m} }{m}\right) ^{2}\nonumber \\&\qquad \times \Delta \left( \dfrac{2-m-2\sqrt{1-m}}{m}R\right) \rho \left( \dfrac{2-m-2\sqrt{1-m}}{m}R,0\right) \nonumber \\&\qquad \times \left( \dfrac{K\left( m\right) }{2\sqrt{1-m}\left( 1-\sqrt{1-m}\right) }\right. \nonumber \\&\qquad \left. -\dfrac{E\left( m\right) }{2\sqrt{1-m}\left( 1-m-\sqrt{1-m}\right) }\right) \nonumber \\&\qquad +2\sqrt{2\pi }G\displaystyle \int _{m_{\min }}^{1-\epsilon }dm\left( \dfrac{2-m+2\sqrt{1-m}}{m}\right) ^{2}\nonumber \\&\qquad \times \Delta \left( \dfrac{2-m+2\sqrt{1-m}}{m}R\right) \rho \left( \dfrac{2-m+2\sqrt{1-m}}{m}R,0\right) \nonumber \\&\qquad \times \left( \dfrac{K\left( m\right) }{2\sqrt{1-m}\left( 1+\sqrt{1-m}\right) }\right. \nonumber \\&\qquad \left. -\dfrac{E\left( m\right) }{2\sqrt{1-m}\left( 1-m+\sqrt{1-m}\right) }\right) . \end{aligned}$$
(B20)
Appendix C: A model for the galactic width
The mass density distribution \(\rho \left( R^{\prime },0\right) \) in the above integral expression of \(\phi \left( R,0\right) \) can be taken from the luminosity profile of the galaxy under consideration. However, a suitable model for the galactic width \(\Delta \left( R^{\prime }\right) \) must be developed to complete the description. This can be done considering the constant density contours defined by the Miyamoto and Nagai solution [8]
$$\begin{aligned} \rho \left( R,Z\right) =\dfrac{MB^{2}}{4\pi }\dfrac{AR^{2}+\left( A+3\sqrt{B^{2}+Z^{2}}\right) \left( A+\sqrt{B^{2}+Z^{2}}\right) ^{2} }{\left[ R^{2}+\left( A+\sqrt{B^{2}+Z^{2}}\right) ^{2}\right] ^{5/2}\left( B^{2}+Z^{2}\right) ^{3/2}}, \end{aligned}$$
(C1)
where A and B are the semi-axes of a spheroid of revolution. An exact solution of the Poisson equation gives the gravitational potential
$$\begin{aligned} \phi \left( R,Z\right) =-\dfrac{GM}{\left[ R^{2}+\left( A+\sqrt{B^{2} +Z^{2}}\right) ^{2}\right] ^{1/2}}, \end{aligned}$$
(C2)
where M is the total mass given by
$$\begin{aligned} M=2\pi \displaystyle \int _{0}^{\infty }\displaystyle \int _{-\infty }^{\infty } \rho \left( R,Z\right) R\,dR\,dZ. \end{aligned}$$
(C3)
Introducing the normalization radial distance \(R_{0}=1\,\)kpc, the Miyamoto and Nagai profile solution becomes
$$\begin{aligned} \varrho \left( r,z\right) =\left( \dfrac{3M}{4\pi R_{0}^{3}\rho _{0}}\right) \dfrac{b^{2}\left[ ar^{2}+\left( a+3\sqrt{b^{2}+z^{2}}\right) \left( a+\sqrt{b^{2}+z^{2}}\right) ^{2}\right] }{3\left[ r^{2}+\left( a+\sqrt{b^{2}+z^{2}}\right) ^{2}\right] ^{5/2}\left( b^{2}+z^{2}\right) ^{3/2}},\nonumber \\ \end{aligned}$$
(C4)
where all distances are normalized by \(R_{0}\) and the density \(\varrho =\rho /\rho _{0}\) is normalized by the value \(\rho _{0}=\rho \left( 0,0\right) \) at the center. One may introduce in the above expression a coefficient
$$\begin{aligned} \lambda =\dfrac{4\pi R_{0}^{3}\rho _{0}}{3M}, \end{aligned}$$
(C5)
which gives the ratio between the mass of a sphere of radius \(R_{0}\) and uniform density \(\rho _{0}\), and the total mass M of the spheroidal distribution (\(\lambda \) measures the strength of the central density \(\rho _{0}\)). Hence,
$$\begin{aligned} \lambda ^{-1}=\dfrac{3}{2}\displaystyle \int _{0}^{\infty }\displaystyle \int _{-\infty }^{\infty }\varrho \left( r,z\right) r\,dr\,dz. \end{aligned}$$
(C6)
The normalized potential \(\varphi =\phi /c^{2}\) becomes
$$\begin{aligned} \varphi \left( r,z\right) =-\dfrac{r_{s}}{2\left[ r^{2}+\left( a+\sqrt{b^{2}+z^{2}}\right) ^{2}\right] ^{1/2}}, \end{aligned}$$
(C7)
where
$$\begin{aligned} r_{s}=\dfrac{2GM}{c^{2}R_{0}} \end{aligned}$$
(C8)
is the normalized Schwarzschild radius of the spheroidal mass.
Although the Miyamoto and Nagai density profile extends to infinity, a galactic edge can be defined taking a constant density contour along the profile. According to the thin disk approximation (B14), one can define an edge such that \(\varrho \left( 0,\delta \left( 0\right) \right) \) on the vertical axis decreases by \(\ell \) characteristic widths with respect to the central density \(\varrho \left( 0,0\right) =1\). In this way, the galactic shape is defined by \(\delta \left( r\right) \) along the profile and calculated by a root of the equation
$$\begin{aligned}&\dfrac{b^{2}\left[ ar^{2}+\left( a+3\sqrt{b^{2}+\delta \left( r\right) ^{2}}\right) \left( a+\sqrt{b^{2}+\delta \left( r\right) ^{2}}\right) ^{2}\right] }{3\left[ r^{2}+\left( a+\sqrt{b^{2}+\delta \left( r\right) ^{2}}\right) ^{2}\right] ^{5/2}\left( b^{2}+\delta \left( r\right) ^{2}\right) ^{3/2}}\nonumber \\&\quad =\lambda \exp \left( -\dfrac{\ell ^{2}}{2}\right) , \end{aligned}$$
(C9)
where a is the major semi-axis in the radial direction and b the minor semi-axis in the vertical direction. According to the thin disk approximation it is assumed that \(a>3b\) (in general \(a\gg b\)). Exact analytic solution of this equation for \(\sqrt{b^{2}+\delta \left( r\right) ^{2}}\) is very difficult to achieve. It involves combination of roots 2, 4, 6 and 8 of a sixteenth order polynomial (roots 1, 3, 5 and 7 are negative and the remaining roots are complex). It is possible to obtain an approximate cubic solution for \(\delta \left( r\right) \) in Eq. (C9), but the numerical root-finding procedure can be easily implemented. Numerical calculation of \(\delta \left( r\right) \) as a function of r is straightforward for appropriate ranges of initial values a, b and \(\lambda \exp \left( -\ell ^{2}/2\right) \). The range of possible values of \(\lambda \) is
$$\begin{aligned} 0<\lambda \exp \left( -\dfrac{\ell ^{2}}{2}\right) \le \dfrac{a+3b}{3b\left( a+b\right) ^{3}}, \end{aligned}$$
(C10)
so that \(r=0\) and \(\delta \left( r\right) =0\) at the upper value in the range. Taking \(\delta \left( r\right) =0\), a given value of \(\lambda \) less than the upper limit defines the maximum value of the disk radius \(r=r_{\max }\) on the equatorial plane (\(r_{\max }\rightarrow \infty \) for \(\lambda \rightarrow 0\)):
$$\begin{aligned} \dfrac{ar_{\max }^{2}+\left( a+3b\right) \left( a+b\right) ^{2}}{3b\left[ r_{\max }^{2}+\left( a+b\right) ^{2}\right] ^{5/2}}=\lambda \exp \left( -\dfrac{\ell ^{2}}{2}\right) . \end{aligned}$$
(C11)
Assuming \(b\ll a\), this equation gives an estimate for the maximum radius of the galactic disk as a function of the relevant geometrical and physical parameters
$$\begin{aligned} \left( r_{\max }^{2}+a^{2}\right) ^{3/2}\sim \dfrac{a^{2}}{3b}\left( \dfrac{3M}{4\pi R_{0}^{3}\rho _{0}}\right) \exp \left( \dfrac{\ell ^{2}}{2}\right) . \end{aligned}$$
(C12)
This expression indicates that \(\ell \) can be identified as a range parameter. One would expect that a region described by the width \(\delta \left( r\right) \) calculated using equation (C9) contains approximately \(39.3\%\), \(86.5\%\) and \(98.9\%\) of the total galactic mass for \(\ell =1\), 2 and 3, respectively. A region defined by \(\ell =4\) contains practically all the total mass M (large values of \(\ell \) are required for very large disks containing a dim dust distribution at large distances).
Finally, note that \(\lambda \) cannot be specified independently, since its value is defined by the integral (C6), which, in the thin disk approximation, becomes
$$\begin{aligned} \lambda ^{-1}=3\sqrt{\dfrac{\pi }{2}}\displaystyle \int _{0}^{r_{\max }} \delta \left( r\right) \varrho \left( r,0\right) r\,dr\,. \end{aligned}$$
(C13)
This constraint actually establishes the relation between a, b and the arbitrary normalization distance \(R_{0}\). The value of \(\lambda \) can be determined by an iteration procedure as follows: \(\varvec{(1)}\) Take \(\lambda _{1}<1\) and determine \(\delta \left( r\right) \) solving equation (C9) for given values of \(\ell \), a and b; \(\varvec{(2)}\) Calculate \(r_{\max }\) solving equation (C11 ) for the same values of \(\ell \), a, b and \(\lambda _{1}\); \(\varvec{(3)} \) Calculate a new value \(\lambda _{2}\) by the integral (C13); \(\varvec{(4)}\) Compare the two values of \(\lambda _{2}\) and \(\lambda _{1}\) and stop the iteration if the error \(\left| \left( \lambda _{2}-\lambda _{1}\right) /\lambda _{1}\right| \) is less than the required precision (say \(10^{-3}\)); \(\varvec{(5)}\) Define the new value \(\lambda _{1}=\left( \lambda _{1}+\lambda _{2}\right) /2\) by the bisection method and repeat the iteration procedure until the required precision. As a result of this iteration, \(\lambda \) is determined as a function \(\lambda \left( \ell ,a,b\right) \) of \(\ell \), a and b. A simple numerical procedure can be established so that \(\lambda \left( \ell ,a,b\right) \) is readily determined. In this way, one may consider the galactic width \(\delta \left( r\right) =\delta \left( \ell ,a,b,r\right) \) as given by a solution of (an algebraic–integral equation for \(\delta \left( r\right) \))
$$\begin{aligned}&\dfrac{b^{2}\left[ ar^{2}+\left( a+3\sqrt{b^{2}+\delta \left( r\right) ^{2}}\right) \left( a+\sqrt{b^{2}+\delta \left( r\right) ^{2}}\right) ^{2}\right] }{3\left[ r^{2}+\left( a+\sqrt{b^{2}+\delta \left( r\right) ^{2}}\right) ^{2}\right] ^{5/2}\left( b^{2}+\delta \left( r\right) ^{2}\right) ^{3/2}}\nonumber \\&\quad =\lambda \left( \ell ,a,b\right) \exp \left( -\dfrac{\ell ^{2}}{2}\right) , \end{aligned}$$
(C14)
and the maximum radius \(r_{\max }=r_{\max }\left( \ell ,a,b\right) \) of the disk by
$$\begin{aligned} \dfrac{ar_{\max }^{2}+\left( a+3b\right) \left( a+b\right) ^{2}}{3b\left[ r_{\max }^{2}+\left( a+b\right) ^{2}\right] ^{5/2}}=\lambda \left( \ell ,a,b\right) \exp \left( -\dfrac{\ell ^{2}}{2}\right) . \end{aligned}$$
(C15)
Recall that the minimum value of \(m_{\min }=m_{\min }\left( \ell ,a,b,r\right) \), which sets the radial extend in the disk region of the integrals for \(\phi \left( R,0\right) \) and \(\partial \phi \left( R,0\right) /\partial R\) defined in Appendix B, is given by
$$\begin{aligned} m_{\min }=\dfrac{4rr_{\max }\left( \ell ,a,b\right) }{\left[ r+r_{\max }\left( \ell ,a,b\right) \right] ^{2}}<1. \end{aligned}$$
(C16)
Introducing this lower (distant) limit, the normalized form the integral expressions for the gravitational potential and its radial gradient become:
$$\begin{aligned}&\varphi \left( r,0\right) \nonumber \\&\quad \simeq -\sqrt{\dfrac{2}{\pi }}\left( \dfrac{3}{2}\lambda \left( \ell ,a,b\right) r_{s}\right) r\displaystyle \int _{\epsilon }^{1-\epsilon }dm\dfrac{K\left( m\right) }{2\sqrt{m\left( 1-m\right) }}\nonumber \\&\qquad \times \left( \dfrac{2-m-2\sqrt{1-m}}{m}\right) ^{3/2}\nonumber \\&\qquad \times \delta \left( \dfrac{2-m-2\sqrt{1-m}}{m}r\right) \varrho \left( \dfrac{2-m-2\sqrt{1-m}}{m}r,0\right) \nonumber \\&\qquad -\sqrt{\dfrac{2}{\pi }}\left( \dfrac{3}{2}\lambda \left( \ell ,a,b\right) r_{s}\right) r\displaystyle \int _{m_{\min }}^{1-\epsilon }dm\dfrac{K\left( m\right) }{2\sqrt{m\left( 1-m\right) }}\nonumber \\&\qquad \times \left( \dfrac{2-m+2\sqrt{1-m}}{m}\right) ^{3/2}\nonumber \\&\qquad \times \delta \left( \dfrac{2-m+2\sqrt{1-m}}{m}r\right) \varrho \left( \dfrac{2-m+2\sqrt{1-m}}{m}r,0\right) , \end{aligned}$$
(C17)
and
$$\begin{aligned}&\dfrac{\partial \varphi \left( r,0\right) }{\partial r} \nonumber \\&\quad \simeq \dfrac{1}{\sqrt{2\pi }}\left( \dfrac{3}{2}\lambda \left( \ell ,a,b\right) r_{s}\right) \displaystyle \int _{\epsilon }^{1-\epsilon }dm\nonumber \\&\qquad \times \left( \dfrac{2-m-2\sqrt{1-m}}{m}\right) ^{2}\nonumber \\&\qquad \times \delta \left( \dfrac{2-m-2\sqrt{1-m}}{m}r\right) \varrho \left( \dfrac{2-m-2\sqrt{1-m}}{m}r,0\right) \nonumber \\&\qquad \times \left( \dfrac{K\left( m\right) }{2\sqrt{1-m}\left( 1-\sqrt{1-m}\right) }\right. \nonumber \\&\qquad \left. -\dfrac{E\left( m\right) }{2\sqrt{1-m}\left( 1-m-\sqrt{1-m}\right) }\right) \nonumber \\&\qquad +\dfrac{1}{\sqrt{2\pi }}\left( \dfrac{3}{2}\lambda \left( \ell ,a,b\right) r_{s}\right) \displaystyle \int _{m_{\min }}^{1-\epsilon }dm\left( \dfrac{2-m+2\sqrt{1-m}}{m}\right) ^{2}\nonumber \\&\qquad \times \delta \left( \dfrac{2-m+2\sqrt{1-m}}{m}r\right) \varrho \left( \dfrac{2-m+2\sqrt{1-m}}{m}r,0\right) \nonumber \\&\qquad \times \left( \dfrac{K\left( m\right) }{2\sqrt{1-m}\left( 1+\sqrt{1-m}\right) }\right. \nonumber \\&\qquad \left. -\dfrac{E\left( m\right) }{2\sqrt{1-m}\left( 1-m+\sqrt{1-m}\right) }\right) . \end{aligned}$$
(C18)
Appendix D: Mass density estimate from surface-brightness profile
This appendix gives a brief presentation of the expressions used by observational astronomers to estimate the galactic luminosity and mass density, and the limitations of these simple definitions. In general, the mass density distribution along the equatorial plane of the galaxy can be evaluated from the measured surface-brightness profile, which can be approximated by the Sérsic profile for the flux density (in W/m\(^{2}\))
$$\begin{aligned} F_{s}\left( R\right) =F_{0}\exp \left[ -b_{s}\left( \dfrac{R}{R_{\text {eff}}}\right) ^{1/s}\right] , \end{aligned}$$
(D1)
where s is the Sérsic index and \(R_{\text {eff}}\) is the half-luminosity radius, i.e., the radius that encloses half of the light, and \(b_{s} ^{-s}R_{\text {eff}}\) is the radius at which the central flux density \(F_{0}\) drops by \(e^{-1}\). The coefficient \(b_{s}\) satisfies the relation
$$\begin{aligned} 2\Gamma \left( 2s,b_{s}\right) =\Gamma \left( 2s\right) , \end{aligned}$$
(D2)
where \(\Gamma \left( 2s,b_{s}\right) \) and \(\Gamma \left( 2s\right) \) denote the incomplete and complete gamma functions, respectively. The coefficient \(b_{s}\) is given approximately by
$$\begin{aligned} b_{s}\simeq 2s-\dfrac{1}{3}. \end{aligned}$$
(D3)
The partial radiant flux (in W) is given by
$$\begin{aligned} L_{s}\left( R\right)= & {} 2\pi \displaystyle \int _{0}^{R}F_{s}\left( R^{\prime }\right) R^{\prime }\,dR^{\prime }\nonumber \\= & {} 2\pi s\left\{ \Gamma \left( 2s\right) -\Gamma \left[ 2s,b_{s}\left( \dfrac{R}{R_{\text {eff}}}\right) ^{1/s}\right] \right\} \nonumber \\&\quad \times b_{s}\left( s\right) ^{-2s}R_{\text {eff}}^{2}F_{0}, \end{aligned}$$
(D4)
so that the total radiant flux is (\(R\rightarrow \infty \))
$$\begin{aligned} L_{s}=2\pi s\Gamma \left( 2s\right) b_{s}\left( s\right) ^{-2s} R_{\text {eff}}^{2}F_{0}, \end{aligned}$$
(D5)
and the absolute magnitude is
$$\begin{aligned} M_{s}=-\dfrac{5}{2}\log _{10}\dfrac{L_{s}}{L_{10}}, \end{aligned}$$
(D6)
where \(L_{10}=4\pi \left( 10\text {\, pc}\right) ^{2}F_{10} =3.0128\times 10^{28}\) W is the standard radiant flux (luminosity) given in terms of the standard flux density \(F_{10}=2.518021002\times 10^{-8} \) W/m\(^{2}\) (according to IAU resolution B2, 2015). The distance independent apparent magnitude is
$$\begin{aligned} \mu _{s}\left( R\right) =\mu _{0}+\dfrac{5}{2\ln 10}\left( \dfrac{R}{b_{s}^{-s}R_{\text {eff}}}\right) ^{1/s}. \end{aligned}$$
(D7)
The brightness profile is usually expressed in terms of the apparent magnitude in \({\text {arcsec}}^{2}\)
$$\begin{aligned} \mu \left( \alpha \right) =\mu _{0}+\dfrac{5}{2\ln 10}\left( \dfrac{\alpha }{\alpha _{\text {eff}}}\right) ^{1/s}, \end{aligned}$$
(D8)
where \(\alpha _{\text {eff}}=\left( 180\times 3600/\pi \right) b_{s} ^{-s}R_{\text {eff}}/d\) is the effective half-angle in \({\text {arcsec}}\) and d is the distance to the galaxy. The apparent magnitude of the galaxy seen from Earth is
$$\begin{aligned} m_{d}=M_{s}+5\log _{10}\left( \dfrac{d}{10\text {\, pc}}\right) . \end{aligned}$$
(D9)
The de Vaucoulers’s profile for elliptical galaxies corresponds to \(s=4\), while the simple exponential profile corresponds to \(s=1\). Larger s gives light profiles concentrated in the central part, and at the same time higher surface brightness at large radial distances. These two profiles cannot be applied singly to represent the mass density \(\rho (r,0)\) because they have singular derivatives at the origin. Nevertheless, it is here proposed to construct a piecewise continuous profile defined by the juxtaposition of two Sérsic profiles:
$$\begin{aligned}&F_{s}\left( R\right) \nonumber \\&\quad = \left\{ \begin{array} [c]{lcc} F_{0}\exp \left[ -b\left( s_{1}\right) \left( \dfrac{R}{R_{1}}\right) ^{1/s_{1}}\right] &{} \qquad &{} 0\le R\le R_{0}\\ F_{0}\exp \left[ -b\left( s_{1}\right) \left( \dfrac{R_{0}}{R_{1}}\right) ^{1/s_{1}}\left( 1-\dfrac{s_{2}}{s_{1}}+\dfrac{s_{2}}{s_{1}}\left( \dfrac{R}{R_{0}}\right) ^{1/s_{2}}\right) \right] &{} \qquad &{} R>R_{0} \end{array} \right. \nonumber \\ \end{aligned}$$
(D10)
This piecewise profile has continuous amplitude and derivative at \(R=R_{0}\) (not to be confounded with the normalization radius introduced in Sect. 4), and finite values at the origin if \(0\le s_{1}<1\) (the derivative vanishes at the origin if \(0\le s_{1}<1\)). The profile is flat for \(s_{1}\gtrsim 0\) and peaked for \(s_{1}\lesssim 1\) (\(s_{1}=1/2\) corresponds to a Gaussian profile near the origin). The piecewise profile for the apparent magnitude in \({\text {arcsec}}^{2}\) becomes:
$$\begin{aligned}&\mu \left( \alpha \right) \nonumber \\&\quad = \mu _{0}+\left\{ \begin{array} [c]{lcc} \dfrac{5}{2\ln 10}\left( \dfrac{\alpha }{\alpha _{1}}\right) ^{1/s_{1}} &{} \qquad &{} 0\le \alpha \le \alpha _{0}\\ \dfrac{5}{2\ln 10}\left( \dfrac{\alpha _{0}}{\alpha _{1}}\right) ^{1/s_{1} }\left( 1-\dfrac{s_{2}}{s_{1}}+\dfrac{s_{2}}{s_{1}}\left( \dfrac{\alpha }{\alpha _{0}}\right) ^{1/s_{2}}\right) &{} \qquad &{} \alpha >\alpha _{0} \end{array} \right. \end{aligned}$$
(D11)
As an example, the surface brightness \(\mu _{B}\) of the dwarf galaxy NGC 1560 analyzed in Sect. 7 can be adjusted to the observed values listed by Broeils [32] by taking \(\mu _{0}=22.28\), \(\alpha _{0}=61.46\) \({\text {arcsec}}\), \(s_{1}=0.435\), \(\alpha _{1}=99.05\) \({\text {arcsec}}\) and \(s_{2}=1.144\) in the above expression. This approximation indicates a transition from a nearly Gaussian to a nearly exponential profile, and can be used to estimate the basic galactic parameters. The projected distances along the equatorial plane are calculated in terms of the angles of observation by \(r=d\left( \pi /\left( 180\times 3600\right) \right) \alpha \). The calculated absolute magnitude is \(M_{s}=-16.5\), the total luminosity is \(L_{s}=3.06\times 10^{8}L_{\odot }\) and, assuming a distance to the galaxy \(d=3.0\text {\, Mpc}\), the apparent magnitude is \(m_{d}=10.9\) (\(L_{\odot }=3.828\times 10^{26}\) W is the solar luminosity).
The normalized mass density profile can be obtained directly from the adjusted luminosity profile by taking
$$\begin{aligned}&\varrho \left( r,0\right) =10^{-\frac{2}{5}\left[ \mu \left( r\right) -\mu _{0}\right] } \nonumber \\&\quad =\left\{ \begin{array} [c]{lcc} \exp \left[ -\left( \dfrac{r}{r_{1}}\right) ^{1/s_{1}}\right] &{} &{} 0\le r\le r_{0}\\ \exp \left[ -\left( \dfrac{r_{0}}{r_{1}}\right) ^{1/s_{1}}\left( 1-\dfrac{s_{2}}{s_{1}}+\dfrac{s_{2}}{s_{1}}\left( \dfrac{r}{r_{0}}\right) ^{1/s_{2}}\right) \right] &{} &{} r>r_{0} \end{array} \right. \nonumber \\ \end{aligned}$$
(D12)
where \(r_{0}=d\left( \pi /\left( 180\times 3600\right) \right) \alpha _{0}\) and \(r_{1}=d\left( \pi /\left( 180\times 3600\right) \right) \alpha _{1}\). Assuming uniform mass-to-light ratio Y, one can take \(Y=1\) without modifying the mass density profile. Although satisfactory for macroscopic estimates, the piecewise continuous approximation of the mass density is, in general, not satisfactory for detailed rotation velocity calculations since the Sérsic index in the above representation makes a discontinuous transition from the constant value \(s_{1}\) in the bulge region, \(0\le r\le r_{0}\), to the constant value \(s_{2}\) in the disk region, \(r>r_{0}\). This limitation can be circumvented adopting a piecewise continuous representation for the Sérsic index instead
$$\begin{aligned} s\left( \alpha \right) =\left\{ \begin{array} [c]{lcc} s_{b}\left( \alpha \right) &{} \qquad &{} 0\le \alpha \le \alpha _{0}\\ s_{d}\left( \alpha \right) &{} \qquad &{} \alpha _{0}<\alpha \le \alpha _{e}\\ s_{e} &{} \qquad &{} \alpha >\alpha _{e} \end{array} \right. \end{aligned}$$
(D13)
where \(s_{b}\left( \alpha \right) \) and \(s_{d}\left( \alpha \right) \) are power series defined separately in the bulge and in the disk regions:
$$\begin{aligned} \begin{array} [c]{ccc} s_{b}\left( \alpha \right) =\displaystyle \sum _{i=0}^{n}b_{i}\alpha ^{i}&\qquad \text {and}&\qquad s_{d}\left( \alpha \right) =\displaystyle \sum _{j=0}^{m}d_{j}\alpha ^{j}. \end{array} \end{aligned}$$
(D14)
The coefficients \(b_{i}\) and \(d_{j}\) can be chosen so that \(s_{b}\left( 0\right) =s_{0}\) and \(s_{d}\left( \alpha _{e}\right) =s_{e}\), where \(s_{0}\) and \(s_{e}\) are the initial and final values of the Sérsic index in the range of measured values. Furthermore, the values and first derivatives of the index must be continuous at the transition point \(\alpha _{0}\) between the bulge and the disk:
$$\begin{aligned} \begin{array} [c]{ccc} s_{b}\left( \alpha _{0}\right) =s_{d}\left( \alpha _{0}\right)&\qquad \text {and}&\qquad \left. \dfrac{ds_{b}}{d\alpha }\right| _{\alpha _{0}}=\left. \dfrac{ds_{d}}{d\alpha }\right| _{\alpha _{0}}. \end{array} \end{aligned}$$
(D15)
Finally, it is required that the radial derivative of the index vanishes at the endpoint \(\alpha _{e}\) (this requirement is not essential, but it is convenient to have a flat index in extending the index beyond the last measurement point)
$$\begin{aligned} \left. \dfrac{ds_{d}}{d\alpha }\right| _{\alpha _{e}}=0. \end{aligned}$$
(D16)
The final result can be written as follows:
$$\begin{aligned} \begin{array} [c]{rcl} s_{b}\left( \alpha \right) &{} = &{} s_{0}+b_{1}\alpha +\displaystyle \sum _{i=2}^{n}b_{i}\alpha ^{i},\\ s_{d}\left( \alpha \right) &{} = &{} s_{e}+d_{2}\left( \alpha _{e}-\alpha \right) ^{2}+\displaystyle \sum _{j=3}^{m}d_{j}\left( \alpha _{e}-\alpha \right) ^{j}, \end{array} \end{aligned}$$
(D17)
where
$$\begin{aligned} \begin{array} [c]{rcl} b_{1} &{} = &{} \dfrac{1}{\alpha _{e}+\alpha _{0}}\left[ 2\left( s_{e} -s_{0}\right) -\displaystyle \sum _{i=2}^{n}\left[ i\left( \alpha _{e} -\alpha _{0}\right) +2\alpha _{0}\right] \alpha _{0}^{i-1}b_{i} \right. \\ &{}&{}\left. -\displaystyle \sum _{j=3}^{m}\left( j-2\right) \left( \alpha _{e}-\alpha _{0}\right) ^{j}d_{j}\right] ,\\ d_{2} &{} = &{} -\dfrac{1}{\alpha _{e}^{2}-\alpha _{0}^{2}}\left[ s_{e} -s_{0}+\displaystyle \sum _{i=2}^{n}\left( i-1\right) \alpha _{0}^{i} b_{i}\right. \\ &{}&{}\left. +\displaystyle \sum _{j=3}^{m}\left[ \alpha _{e}+\left( j-1\right) \alpha _{0}\right] \left( \alpha _{e}-\alpha _{0}\right) ^{j-1}d_{j}\right] . \end{array} \end{aligned}$$
(D18)
According to Eq. (D8), the Sérsic index is given in terms of the apparent magnitude \(\mu \left( \alpha \right) \) at the angular position \(\alpha \) in \({\text {arcsec}}\) by
$$\begin{aligned} s\left( \alpha \right) =\dfrac{\ln \left( \alpha /\alpha _{\text {eff}}\right) }{\ln \left[ \dfrac{2\ln 10}{5}\left( \mu \left( \alpha \right) -\mu _{0}\right) \right] }. \end{aligned}$$
(D19)
This provides a relation between the index \(s_{e}\) at the last measured angular position \(\alpha _{e}\), and the effective half-angle
$$\begin{aligned} s_{e}=\dfrac{\ln \left( \alpha _{e}/\alpha _{\text {eff}}\right) }{\ln \left[ \dfrac{2\ln 10}{5}\left( \mu \left( \alpha _{e}\right) -\mu _{0}\right) \right] }. \end{aligned}$$
(D20)
Furthermore,
$$\begin{aligned} \mu \left( \alpha _{\text {eff}}\right) =\mu _{0}+\dfrac{5}{2\ln 10}. \end{aligned}$$
(D21)
In this way, the values of \(\mu _{0}\), \(\alpha _{\text {eff}}\), \(\alpha _{0}\), \(s_{0}\) and the power series coefficients \(b_{i}\) and \(d_{j}\) can be determined using a least squares method to fit \(s\left( \alpha \right) \) to the measured brightness profile. It is convenient to specify the initial value \(\mu _{0}\), based on measurements, adjusting \(\alpha _{\text {eff}}\), \(\alpha _{0}\) and the power series coefficients independently. The relation (D21) can be used to verify the consistency of the adjusted results. In terms of the radial position r one can simply replace \(\alpha \) by \(\left( 180\times 3600/\pi \right) r/d\).