11institutetext: US Naval Observatory, Washington DC 20392 USA
11email:  michael.efroimsky @ usno.navy.mil

#  Justification of the two-bulge method in the theory of bodily tides

Michael Efroimsky 11

(Received: 25 April 2012; accepted: … )

###### Abstract

Mathematical modeling of bodily tides can be carried out in various ways. Most
straightforward is the method of complex amplitudes, which is often used in
the planetary science. Another method, employed both in planetary science and
in astrophysics, is based on decomposition of each harmonic of the tide into
two bulges oriented orthogonally to one another. We prove that the two methods
are equivalent. Specifically, we demonstrate that the two-bulge method is not
a separate approximation, but ensues directly from the Fourier expansion of a
linear tidal theory equipped with an arbitrary rheological model involving a
departure from elasticity. To this end, we use the most general mathematical
formalism applicable to linear bodily tides. To express the tidal amendment to
the potential of the perturbed primary, we act on the tide-raising potential
of the perturbing secondary with a convolution operator. This enables us to
interconnect a complex Fourier component of the tidally generated potential of
the perturbed primary with the appropriate complex Fourier component of the
tide-raising potential of the secondary. Then we demonstrate how this
interrelation entails the two-bulge description. While less economical
mathematically, the two-bulge approach has a good illustrative power, and may
be employed on a par with a more concise method of complex amplitudes. At the
same time, there exist situations where the two-bulge method becomes more
practical for technical calculations.

###### Key Words.:

Celestial Mechanics – Stars: binaries: close – Stars: planetary systems –
Stars: rotation – Planets and satellites: general – Planets and satellites:
dynamical evolution and stability

##  1 Introduction and aim

On several occasions, it was suggested by different authors to model bodily
tides with superposition of two symmetrical bulges. One bulge is always aimed
at the secondary, and thus implements the instantaneous reaction of the
primary’s shape and potential to the tide-rising gravitational pull exerted by
the secondary. This portion of the tide is called “adiabatic tide” (Zahn
1966a,b) or “elastic tide” (Ferraz Mello 2012; Krasinsky 2006). The second
bulge is assumed to align orthogonally to the direction to the tide-raising
secondary, and thus is set to implement the entire nonelastic portion of the
primary’s deformation. This, second bulge is called “dissipative tide” (Zahn
1966a,b; Krasinsky 2006) or “creep tide” (Ferraz Mello 2012).

In this note, we demonstrate that the two-bulge method is not a separate
approximation, but ensues directly from the Fourier expansion of a linear
tidal theory equipped with an arbitrary rheological model involving a
departure from elasticity.

##  2 The static linear theory of bodily tides

Let a spherical primary of radius $\,R\,$ be subject to the gravitational pull
by a secondary of mass $M_{sec}\,$, residing at the position
$\,{\mbox{{\boldmath$\@vec{r}$}}}^{\;*}=(r^{*},\,\phi^{*},\,\lambda^{*})\,$,
where $\,r^{*}\geq R\,$. At a surface point
$\mbox{{\boldmath$\@vec{R}$}}=(R,\phi,\lambda)$ of the primary body, the tidal
potential generated by the secondary can be expanded over the Legendre
polynomials $\,P_{\it l}(\cos\gamma)\;$ as

$\displaystyle
W(\mbox{{\boldmath$\@vec{R}$}}\,,\,\mbox{{\boldmath$\@vec{r}$}}^{\leavevmode\nobreak\
*})$ $\displaystyle=$
$\displaystyle\sum_{{\it{l}}=2}^{\infty}\leavevmode\nobreak\
W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\leavevmode\nobreak\
\mbox{{\boldmath$\@vec{r}$}}^{\leavevmode\nobreak\ *})\leavevmode\nobreak\
=\leavevmode\nobreak\ -\leavevmode\nobreak\
\frac{G\;M_{sec}}{r^{\,*}}\leavevmode\nobreak\
\sum_{{\it{l}}=2}^{\infty}\,\left(\,\frac{R}{r^{\,*}}\,\right)^{\textstyle{{}^{\it{l}}}}\,P_{\it{l}}(\cos\gamma)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (1)

Here $\,G\,$ denotes Newton’s gravity constant, while $\gamma\,$ is the
angular separation between the vectors ${\mbox{{\boldmath$\@vec{r}$}}}^{\;*}$
and $\@vec{R}$ pointing from the primary’s centre. The longitudes
$\lambda,\,\lambda^{*}$ are measured from a fixed meridian on the primary
body, the latitudes $\phi,\,\phi^{*}$ being reckoned from the equator. The
index $\,l\,$ is conventionally named as the degree. In (1) the $\,l=0\,$ term
is missing, because it corresponds to the principal, Newtonian part of the
secondary’s potential, and is not a part of the perturbation. Omission of the
$\,l=1\,$ term is a more subtle point related to the fact that we are
describing the motion of the secondary relative to the primary body, and not
relative to an inertial frame (see equations 8 - 11 in Efroimsky & Williams
2009).

Within the linear theory, the $\,{\emph{l}}^{\leavevmode\nobreak\ th}$ term
$\,W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\leavevmode\nobreak\
\mbox{{\boldmath$\@vec{r}$}}^{\leavevmode\nobreak\ *})\,$ of the secondary’s
potential generates a linear alteration of the primary’s shape. This
alteration, in its turn, causes a linear amendment
$\,U_{\it{l}}(\mbox{{\boldmath$\@vec{r}$}})\,$ to the gravitational potential
of the primary, where linear  means: linear in
$\,W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\leavevmode\nobreak\
\mbox{{\boldmath$\@vec{r}$}}^{\leavevmode\nobreak\ *})\,$. The theory of
potential requires that outside the primary body
$\,U_{\it{l}}(\mbox{{\boldmath$\@vec{r}$}})\,$ should scale with the distance
as $\,r^{-(\it{l}+1)}\,$. Hence the said change in the primary’s potential may
be written down as

$\displaystyle
U_{\it{l}}(\mbox{{\boldmath$\@vec{r}$}})\,=\,k_{l}\,\left(\,\frac{R}{r}\,\right)^{{\it
l}+1}\;W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\;\mbox{{\boldmath$\@vec{r}$}}^{\;*})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (2)

where $R\,$ is the mean equatorial radius of the primary,
$\,\mbox{{\boldmath$\@vec{R}$}}\,=\,(R\,,\,\phi\,,\,\lambda)\,$ is a point on
the primary’s surface, while
$\,\mbox{{\boldmath$\@vec{r}$}}\,=\,(r\,,\,\phi\,,\,\lambda)\,$ is an exterior
point right above the surface point $\@vec{R}$ , at a radius $\,r\,\geq\,R\,$.
The numerical factors $\,k_{l}\,$ are the degree-$l$ Love numbers calculated
from the rheology of the primary body.

The overall tidally caused change of the primary’s potential thus amounts to

$\displaystyle U(\mbox{{\boldmath$\@vec{r}$}})\leavevmode\nobreak\
=\leavevmode\nobreak\ \sum_{{\it l}=2}^{\infty}\leavevmode\nobreak\
U_{\it{l}}(\mbox{{\boldmath$\@vec{r}$}})\leavevmode\nobreak\
=\leavevmode\nobreak\ \sum_{{\it l}=2}^{\infty}\leavevmode\nobreak\ k_{\it
l}\;\left(\,\frac{R}{r}\,\right)^{{\it
l}+1}\;W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\;\mbox{{\boldmath$\@vec{r}$}}^{\;*})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ .\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ $ (3)

##  3 Dynamical linear theories of bodily tides

In realistic situations, the disturbing potential is evolving, so equation (1)
assumes the form of

$\displaystyle
W(\mbox{{\boldmath$\@vec{R}$}}\,,\,\mbox{{\boldmath$\@vec{r}$}}^{\leavevmode\nobreak\
*}(t)\,)$ $\displaystyle=$
$\displaystyle\sum_{{\it{l}}=2}^{\infty}\leavevmode\nobreak\
W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\leavevmode\nobreak\
\mbox{{\boldmath$\@vec{r}$}}^{\leavevmode\nobreak\
*}(t)\,)\leavevmode\nobreak\ =\leavevmode\nobreak\ -\leavevmode\nobreak\
\frac{G\;M_{sec}}{r^{\,*}(t)}\leavevmode\nobreak\
\sum_{{\it{l}}=2}^{\infty}\,\left(\,\frac{R}{r^{\,*}(t)}\,\right)^{\textstyle{{}^{\it{l}}}}\,P_{\it{l}}(\cos\gamma(t)\,)\leavevmode\nobreak\
\leavevmode\nobreak\ .\leavevmode\nobreak\ $ (4)

Then one should expect the distortion of the primary, as well as the
corresponding amendment to its potential at an exterior point $\@vec{r}$ , to
become a function of time: $\,U(\mbox{{\boldmath$\@vec{r}$}},\,t)\,$.

###  3.1 Elastic dynamical tides

Had the tides contained only instantaneous, elastic components, the
expressions for the tidal potential would mimic (2 \- 3). At each instant of
time $\,t\,$, the degree-$l$ term of the tide-raising potential $\,W\,$ of the
orbiting secondary would generate instantaneously an appropriate degree-$l$
term of the tidal potential of the primary:

${}^{\textstyle{{}^{(elastic)}}}U_{\it{l}}(\mbox{{\boldmath$\@vec{r}$}},\,t)\,=\,k_{l}\,\left(\,\frac{R}{r}\,\right)^{{\it
l}+1}\;W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\;\mbox{{\boldmath$\@vec{r}$}}^{\;*}(t)\,)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (5)

so the total tidal amendment to the potential of the primary would look:

${}^{\textstyle{{}^{(elastic)}}}U(\mbox{{\boldmath$\@vec{r}$}},\,t)\leavevmode\nobreak\
=\leavevmode\nobreak\ \sum_{{\it l}=2}^{\infty}\leavevmode\nobreak\
\leavevmode\nobreak\
{}^{\textstyle{{}^{(elastic)}}}U_{\it{l}}(\mbox{{\boldmath$\@vec{r}$}},\,t)\leavevmode\nobreak\
=\leavevmode\nobreak\ \sum_{{\it l}=2}^{\infty}\leavevmode\nobreak\
k_{l}\leavevmode\nobreak\ \left(\,\frac{R}{r}\,\right)^{{\it
l}+1}\leavevmode\nobreak\
W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\leavevmode\nobreak\
\mbox{{\boldmath$\@vec{r}$}}^{\leavevmode\nobreak\
*}(t)\,)\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
.\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (6)

Needless to say, in realistic materials the internal friction prevents the
tidal deformation from being instantaneous.

###  3.2 Realistic dynamical tides

To describe deviation from elasticity, we spell the two basic assumptions
whereon a linear dynamical theory of bodily tides is based:

  * [1]

tidal deformation is linear with respect to the stress generated by the tide-
raising potential;

  * [2]

the deformation is not fully elastic: it incorporates both an immediate and
delayed portions (delayed – relative to the tide-raising potential).

Mathematically, assumption [1] means that an infinitesimal increment $\,\Delta
W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\;\mbox{{\boldmath$\@vec{r}$}}^{\;*}(t\,^{\prime})\,)\,$,
whereby the perturbing potential increased at the time $\,t\,^{\prime}\,$ in
the past, results in a proportional present-time increment of the tidally
distorted shape of the primary and, accordingly, in a proportional increment
of the tidal amendment to its potential:

$\displaystyle\Delta U_{\it
l}(\mbox{{\boldmath$\@vec{r}$}},\,t)\leavevmode\nobreak\ =\leavevmode\nobreak\
\left(\frac{R}{r}\right)^{{\it
l}+1}{\it{k}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})\leavevmode\nobreak\
\,\Delta
W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\;\mbox{{\boldmath$\@vec{r}$}}^{\;*}(t\,^{\prime})\,)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (7)

${\it{k}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})\,$ being a function describing
the delayed reaction of the shape. The mathematical form of this function is
defined by the rheology of the body and by its self-gravitation.

Thus the addition $\,U_{l}\,$ to the primary’s potential gets expressed
through the tide-raising potential $\,W_{l}\,$ by a linear integral operator:

$\displaystyle U_{\it
l}(\mbox{{\boldmath$\@vec{r}$}},\,t)\,=\,\left(\frac{R}{r}\right)^{{\it
l}+1}\int_{t\,^{\prime}=-\infty}^{t\,^{\prime}=t}k_{\it
l}(t-t\,^{\prime})\stackrel{{\scriptstyle\bf\centerdot}}{{W}}_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\,\mbox{{\boldmath$\@vec{r}$}}^{\;*}(t\,^{\prime})\,)\,dt\,^{\prime}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (8)

overdot denoting a time derivative. Integration of (8) by parts renders:

$\displaystyle U_{\it
l}(\mbox{{\boldmath$\@vec{r}$}},\,t)\,=\,\left(\frac{R}{r}\right)^{{\it
l}+1}\left[k_{l}(0)W_{l}(t)\,-\,k_{l}(\infty)W_{l}(-\infty)\right]\,+\,\left(\frac{R}{r}\right)^{{\it
l}+1}\int_{-\infty}^{t}{\bf\dot{\it{k}}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})\,\leavevmode\nobreak\
W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\,\mbox{{\boldmath$\@vec{r}$}}^{\;*}(t\,^{\prime})\,)\,dt\,^{\prime}\leavevmode\nobreak\
\,,\leavevmode\nobreak\ \quad$ (9)

where the relaxed term $\,-\,k_{l}(\infty)W(-\infty)\,$ should be neglected.
Indeed, the current events cannot be influenced by the perturbation
$\,W_{l}(-\infty)\,$ in the infinite past, wherefore $\,k_{l}(\infty)=0\,$. Of
the remaining two terms, the unrelaxed term $\,k_{l}(0)W_{l}(t)\,$ reflects
the elastic part of the deformation, while the integral expresses the delayed
components – viscous and anelastic.

Assumption [2] means that both the unrelaxed term $\,k_{l}(0)W_{l}(t)\,$ and
the delayed term given by the integral should be kept. As explained in
Efroimsky (2012a,b), the relaxed term may be easily incorporated into the
integral, where it should show up multiplied with a Heaviside step function
$\,\Theta(t-t\,^{\prime})\,$. Then our expression for the tidal potential will
acquire the simple form of

$\displaystyle U_{\it
l}(\mbox{{\boldmath$\@vec{r}$}},\,t)\;=\;\left(\frac{R}{r}\right)^{{\it
l}+1}\int_{-\infty}^{t}{\stackrel{{\scriptstyle\bf\centerdot}}{{\it{k}}}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})\leavevmode\nobreak\
W_{\it{l}}(\mbox{{\boldmath$\@vec{R}$}}\,,\;\mbox{{\boldmath$\@vec{r}$}}^{\;*}(t\,^{\prime})\,)\,dt\,^{\prime}\leavevmode\nobreak\
,$ (10)

${\bf{\it{k}}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})$ now incorporating both
the delayed-reaction terms and the elastic term
$\,{k}_{l}(0)\,\Theta(t-t\,^{\prime})\,$. The elastic part will enter the
kernel
$\,{\stackrel{{\scriptstyle\bf\centerdot}}{{\it{k}}}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})\,$
as $\,{k}_{l}(0)\,\delta(t-t\,^{\prime})\,$, with $\,\delta(t-t\,^{\prime})\,$
being the Dirac delta function. Integration of this term will furnish
$\,k_{l}(0)W_{l}(t)\,$, as in (9).

##  4 Fourier components of tidal stresses and strains

###  4.1 Tidal modes and tidal frequencies

The sidereal angle and the spin rate of a tidally-perturbed primary are
normally denoted with $\,\theta\,$ and
$\,\stackrel{{\scriptstyle\bf\centerdot}}{{\theta\,}}\,$, while the node,
pericentre, and mean anomaly of a tide-raising secondary, as seen from the
primary, are denoted with $\,\Omega\,$, $\omega\,$, and ${\cal M}$.

In the Darwin-Kaula theory, the tide-raising potential $W$, the primary’s
deformation, and the tidal amendment $U$ to the primary’s potential are
expanded over the modes

$\displaystyle\omega_{lmpq}\;\equiv\;(l-2p)\;\dot{\omega}\,+\,(l-2p+q)\;\dot{\cal{M}}\,+\,m\;(\dot{\Omega}\,-\,\dot{\theta})\,\approx\,(l-2p+q)\;n\,-\,m\;\dot{\theta}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (11)

$l,\,m,\,p,\,q\,$ being integers, and $\,n\,$ being the mean motion. Dependent
upon the values of the mean motion, spin rate, and the indices, the tidal
modes $\,\omega_{{\it l}mpq}\,$ may be positive or negative or zero.

The actual forcing frequencies of the resulting stresses and strains in the
primary’s material are the absolute values of the tidal modes:

$\displaystyle\chi_{lmpq}\equiv\,|\,\omega_{lmpq}\,|\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (12)

so these frequencies are always positive.

###  4.2 Fourier expansions

In practical calculations, it is extremely convenient to employ complex
stresses and strains, under the convention that the actual, physical
quantities are the real parts of their complex counterparts. This way, the
Fourier series for the stress and strain look:

$\displaystyle\sigma_{\gamma\nu}(t)\,\,=\,\sum_{s=0}^{\infty}\,\sigma_{\gamma\nu}(\chi_{\textstyle{{}_{s}}})\,\cos\left[\,\chi_{\textstyle{{}_{s}}}t+\varphi_{\sigma}(\chi_{\textstyle{{}_{s}}})\,\right]\;=\;\sum_{s=0}^{\infty}\,{\cal{R}}{\it{e}}\left[\,{\bar{\sigma}}_{\gamma\nu}(\chi_{\textstyle{{}_{s}}})\,\;\exp\left({\textstyle{{\,{\it
i}\chi_{\textstyle{{}_{s}}}t}}}\right)\,\;\right]\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (13)
$\displaystyle
u_{\gamma\nu}(t)\,=\,\sum_{s=0}^{\infty}\,u_{\gamma\nu}(\chi_{\textstyle{{}_{s}}})\,\cos\left[\,\chi_{\textstyle{{}_{s}}}t+\varphi_{u}(\chi_{\textstyle{{}_{s}}})\,\right]\;=\;\sum_{s=0}^{\infty}\,{\cal{R}}{\it{e}}\left[\,{\bar{u}}_{\gamma\nu}(\chi_{\textstyle{{}_{s}}})\;\,\exp\left({\textstyle{{\,{\it
i}\chi_{\textstyle{{}_{s}}}t}}}\right)\,\;\right]\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (14)

$\gamma\nu$ being tensor indices, and $s$ being a concise notation for $lmpq$.
The complex amplitudes are

$\displaystyle{\bar{{\sigma}}_{\gamma\nu}}(\chi)={{{\sigma}}_{\gamma\nu}}(\chi)\,\;\exp\left[{{\it
i}\varphi_{\sigma}(\chi)}\right]\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
{\bar{{u}}_{\gamma\nu}}(\chi)={{{u}}_{\gamma\nu}}(\chi)\,\;\exp\left[{{\it
i}\varphi_{u}(\chi)}\right]\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (15)

where the initial phases $\,\varphi_{\sigma}(\chi)\,$ and
$\,\varphi_{u}(\chi)\,$ are chosen so that the real amplitudes
$\,\sigma_{\gamma\nu}(\chi_{\textstyle{{}_{s}}})\,$ and
$\,u_{\gamma\nu}(\chi_{\textstyle{{}_{s}}})\,$ are non-negative.

For a continuous spectrum, the sums get replaced with integrals over
frequency:

$\displaystyle\sigma_{\gamma\nu}(t)\leavevmode\nobreak\ =\leavevmode\nobreak\
{\cal{R}}{\it{e}}\leavevmode\nobreak\
\int_{0}^{\infty}\,\bar{\sigma}_{\gamma\nu}(\chi)\leavevmode\nobreak\
e^{\textstyle{{}^{\,{\it i}\chi t}}}\leavevmode\nobreak\
d\chi\quad\quad\mbox{and}\leavevmode\nobreak\ \quad\leavevmode\nobreak\ \quad
u_{\gamma\nu}(t)\leavevmode\nobreak\ =\leavevmode\nobreak\
{\cal{R}}{\it{e}}\leavevmode\nobreak\
\int_{0}^{\infty}\,\bar{u}_{\gamma\nu}(\chi)\leavevmode\nobreak\
e^{\textstyle{{}^{\,{\it i}\chi t}}}\leavevmode\nobreak\
d\chi\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ .$ (16)

Similarly, the tide-raising potential $W_{l}$ and the potential $U_{l}$ of the
primary get expanded into a sum or integral over the tidal modes:

$\displaystyle W_{l}(t)\leavevmode\nobreak\ =\leavevmode\nobreak\
{\cal{R}}{\it{e}}\leavevmode\nobreak\
\int_{-\infty}^{\infty}\,\bar{W}_{l}(\omega)\leavevmode\nobreak\
e^{\textstyle{{}^{\,{\it i}\omega t}}}\leavevmode\nobreak\
d\omega\quad\quad\mbox{and}\leavevmode\nobreak\ \quad\leavevmode\nobreak\
\quad U_{l}(t)\leavevmode\nobreak\ =\leavevmode\nobreak\
{\cal{R}}{\it{e}}\leavevmode\nobreak\
\int_{-\infty}^{\infty}\,\bar{U}_{l}(\omega)\leavevmode\nobreak\
e^{\textstyle{{}^{\,{\it i}\omega t}}}\leavevmode\nobreak\
d\omega\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (17)

where the complex amplitudes are expressed via the real amplitudes and the
initial phases by

$\displaystyle{\bar{{W}}_{l}}(\omega)={{{W}}_{l}}(\omega)\,\;\exp\left[{{\it
i}\varphi_{\textstyle{{}_{W_{l}}}}(\omega)}\right]\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ ,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
{\bar{{U}}_{l}}(\omega)={{{U}}_{l}}(\omega)\,\;\exp\left[{{\it
i}\varphi_{\textstyle{{}_{U_{l}}}}(\omega)}\right]\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (18)

The phases $\,\varphi_{\textstyle{{}_{W_{l}}}}(\omega)\,$ and
$\,\varphi_{\textstyle{{}_{U_{l}}}}(\omega)\,$ can always be set in such a way
that the real amplitudes $\,W_{l}(\chi)\,$ and $\,U_{l}(\chi)\,$ are non-
negative.

Both in (16) and (17), the actual, physical spectral components are the real
parts of the complex ones. However, there also is an important difference
between (16) and (17). While the stresses and strains are habitually expanded
in (16) over positive frequencies $\,\chi\,$ only, the potentials in (17) are
expanded over the tidal modes $\,\omega\,$, which can be positive or negative
or zero, as demonstrated in the Darwin-Kaula theory of tides.

It is of course a common fact that a real function can be decomposed into a
Fourier series or integral over only positive frequencies. This way,
expansions of the potentials over $\,\chi\equiv|\omega|\,$ may appear to be
sufficient, because a contribution from some negative tidal mode
$\,\omega<0\,$ can be shown to coincide with the contribution from the
appropriate positive mode $\,|\omega|>0\,$. Therefore, (17) may be rewritten
simply as

$\displaystyle W_{l}(t)\leavevmode\nobreak\ =\leavevmode\nobreak\
{\cal{R}}{\it{e}}\leavevmode\nobreak\
\int_{0}^{\infty}\,\bar{W}_{l}(\chi)\leavevmode\nobreak\
e^{\textstyle{{}^{\,{\it i}\chi t}}}\leavevmode\nobreak\
d\chi\quad\quad\mbox{and}\leavevmode\nobreak\ \quad\leavevmode\nobreak\ \quad
U_{l}(t)\leavevmode\nobreak\ =\leavevmode\nobreak\
{\cal{R}}{\it{e}}\leavevmode\nobreak\
\int_{0}^{\infty}\,\bar{U}_{l}(\chi)\leavevmode\nobreak\
e^{\textstyle{{}^{\,{\it i}\chi t}}}\leavevmode\nobreak\
d\chi\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (19)

where $\,\bar{W}_{l}(\chi)\,=\,2\,\bar{W}_{l}(\omega)\,$ and
$\,\bar{U}_{l}(\chi)\,=\,2\,\bar{U}_{l}(\omega)\,$.

Surprisingly, the theory of tides is a rare exception from the rule, in that
this theory does distinguish between the contribution from a negative tidal
mode and that from a positive mode of the same absolute value. Fortunately,
the difference shows up only at the stage when one calculates tidal forces or
torques (Efroimsky 2012a,b). As in the current paper we discuss potentials
only, we shall ignore this subtlety and shall employ (19) instead of (17).

###  4.3 Dynamical analogues to the Love number

Insertion of (19) into (10) entails

$\displaystyle\bar{U}_{\textstyle{{}_{l}}}(\chi)\;=\;\left(\frac{R}{r}\right)^{l+1}\bar{k}_{\textstyle{{}_{l}}}(\chi)\;\,\bar{W}_{\textstyle{{}_{l}}}(\chi)$
(20)

or, in a more detailed form:

$\displaystyle|\,\bar{U}_{\textstyle{{}_{l}}}(\chi)\,|\leavevmode\nobreak\
e^{\,\textstyle{{}^{i\,\chi\,t\,+\,i\,\varphi_{\textstyle{{}_{{}_{U_{l}}}}}(\chi)}}}\leavevmode\nobreak\
=\leavevmode\nobreak\ \left(\frac{R}{r}\right)^{l+1}\leavevmode\nobreak\
|\,\bar{k}_{\textstyle{{}_{l}}}(\chi)\,|\;\leavevmode\nobreak\
e^{\,\textstyle{{}^{i\,\chi\,t\,-\,i\,\epsilon_{l}(\chi)}}}\;|\,\bar{W}_{\textstyle{{}_{l}}}(\chi)\,|\;\leavevmode\nobreak\
e^{\,\textstyle{{}^{i\,\chi\,t\,+\,i\,\varphi_{\textstyle{{}_{{}_{W_{l}}}}}(\chi)}}}\;\;\;,$
(21)

where the complex function

$\displaystyle\bar{k}_{\textstyle{{}_{l}}}(\chi)\,=\,|\bar{k}_{\textstyle{{}_{l}}}(\chi)|\leavevmode\nobreak\
e^{\textstyle{{}^{\,-\,i\,\epsilon_{l}(\chi)\,}}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,$ (22)

is a Fourier component of the kernel
$\,{\bf\dot{\it{k}}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})\,$ of the integral
operator (10). The kernels
$\,{\bf\dot{\it{k}}}_{\textstyle{{}_{l}}}(t-t\,^{\prime})\,$ are named Love
functions, a term suggested by Churkin (1998). The functions
$\,\bar{k}_{\textstyle{{}_{l}}}(\chi)\,$ are called complex Love numbers, in
understanding however that these “numbers” change with frequency. An approach
similar to (22) was taken by Mathis and Le Poncin Lafitte (2009). These
authors introduced a complex impedance as the ratio between the complex Love
number and the static Love number. Their equation (107) is equivalent to our
(22), up to a convention on the sign of the argument.

We see from (21) that at each frequency $\,\chi\,$, the negative argument
$\,\epsilon_{l}(\chi)\leavevmode\nobreak\ $ is a measure of lagging of the
spectral component $\,U_{l}(\chi)\,$, relative to appropriate spectral
component $\,W_{l}(\chi)\leavevmode\nobreak\ $:

$\displaystyle\varphi_{\textstyle{{}_{U_{l}}}}(\chi)\leavevmode\nobreak\
=\leavevmode\nobreak\
\varphi_{\textstyle{{}_{W_{l}}}}(\chi)\leavevmode\nobreak\
-\leavevmode\nobreak\ \epsilon_{l}(\chi)\leavevmode\nobreak\
\leavevmode\nobreak\ .$ (23)

##  5 Decomposition of a tidal mode into an in-phase part and an in-quadrature
(lagging by $\bf{90^{o}}$) part

In the expression (21), we may set the initial phase of
$\,\bar{W}_{l}(\chi)\,$ to be zero, and reckon the phase of
$\,\bar{U}_{l}(\chi)\,$ from $\,\bar{W}_{l}(\chi)\,$. This will enable us to
single out, in $\,\bar{U}_{l}(\chi)\,$, a part which is in phase with
$\,\bar{W}_{l}(\chi)\,$, and also to see what part of $\,\bar{U}_{l}(\chi)\,$
is out of phase with $\,\bar{W}_{l}(\chi)\,$.

According to (23), nullification of the phase of $\,\bar{W}_{l}(\chi)\,$ makes
the phase of $\,\bar{U}_{l}(\chi)\,$ equal to the negative argument of
$\,\bar{k}_{l}(\chi)\,$. So (21) will assume the form of

$\displaystyle U_{l{\textstyle{{}_{\,0}}}}(\chi)\leavevmode\nobreak\
e^{\,\textstyle{{}^{i\,\chi\,t\,-\,i\,\epsilon_{l}(\chi)}}}\leavevmode\nobreak\
=\leavevmode\nobreak\ \left(\frac{R}{r}\right)^{l+1}\leavevmode\nobreak\
k_{l{\textstyle{{}_{\,0}}}}(\chi)\;\leavevmode\nobreak\
e^{\,\textstyle{{}^{\,-\,i\,\epsilon_{l}(\chi)}}}\;W_{l{\textstyle{{}_{\,0}}}}(\chi)\;\leavevmode\nobreak\
e^{\,\textstyle{{}^{i\,\chi\,t}}}\;\;\;,$ (24)

where we introduced simplified notations
$U_{l{\textstyle{{}_{\,0}}}}(\chi)\,$, $\,W_{l{\textstyle{{}_{\,0}}}}(\chi)\,$
and $\,k_{l{\textstyle{{}_{\,0}}}}\,$ for real amplitudes:

$\displaystyle U_{l{\textstyle{{}_{\,0}}}}(\chi)\leavevmode\nobreak\
\equiv\leavevmode\nobreak\
|\,\bar{U}_{l}(\chi)\,|\,\quad,\quad\quad\,W_{l{\textstyle{{}_{\,0}}}}(\chi)\leavevmode\nobreak\
\equiv\leavevmode\nobreak\
|\,\bar{W}_{l}(\chi)\,|\,\quad,\quad\quad\,k_{l{\textstyle{{}_{\,0}}}}(\chi)\leavevmode\nobreak\
\equiv\leavevmode\nobreak\ |\,\bar{k}_{l}(\chi)\,|\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ .$ (25)

By means of the Euler formula, (24) can be trivially expanded as

$\displaystyle U_{l{\textstyle{{}_{\,0}}}}(\chi)\leavevmode\nobreak\
\left[\,\cos\left(\chi t-\epsilon_{l}(\chi)\,\right)+i\sin\left(\chi
t-\epsilon_{l}(\chi)\,\right)\,\right]\leavevmode\nobreak\
\quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\
\quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\
\quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\
\quad\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle=\leavevmode\nobreak\
\left(\frac{R}{r}\right)^{2l+1}k_{l{\textstyle{{}_{\,0}}}}(\chi)\leavevmode\nobreak\
\left[\,\cos\epsilon_{l}(\chi)-i\sin\epsilon_{l}(\chi)\,\right]\leavevmode\nobreak\
W_{l{\textstyle{{}_{\,0}}}}(\chi)\leavevmode\nobreak\ \left[\,\cos(\chi
t)+i\sin(\chi t)\,\right]\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ .$ (26)

The actual, physical tide-raising potential $W_{l}(\chi)$ is the real part of
the complex $\bar{W}_{l}(\chi)$. So it is rendered by
$\,{W_{l}}_{0}(\chi)\cos(\chi t)\,$, where we set the initial phase nil, as
agreed above. Similarly, the actual, physical tidal potential $U_{l}(\chi)$ is
the real part of the complex $U_{l}(\chi)$, and it reads as:

$\displaystyle{U_{l}}_{0}(\chi)\leavevmode\nobreak\
\cos\epsilon_{l}(\chi)\leavevmode\nobreak\ \cos(\chi
t)-{U_{l}}_{0}(\chi)\leavevmode\nobreak\
\sin\epsilon_{l}(\chi)\leavevmode\nobreak\ \sin(\chi t)\leavevmode\nobreak\
\quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\
\quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\
\quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\ \quad\leavevmode\nobreak\
\quad\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle=\,\left(\frac{R}{r}\right)^{2l+1}{k_{l}}_{0}(\chi)\leavevmode\nobreak\
\cos\epsilon_{l}(\chi)\leavevmode\nobreak\
{W_{l}}_{0}(\chi)\leavevmode\nobreak\ \cos(\chi
t)\,+\,\left(\frac{R}{r}\right)^{2l+1}{k_{l}}_{0}(\chi)\leavevmode\nobreak\
\sin\epsilon_{l}(\chi)\leavevmode\nobreak\
{W_{l}}_{0}(\chi)\leavevmode\nobreak\ \sin(\chi t)\leavevmode\nobreak\
\,.\leavevmode\nobreak\ \leavevmode\nobreak\ $ (27)

In this expression for the actual real $\,U_{l}\,$, we see not one but two
terms. One is the elastic part of the tide, a part that is in phase with the
real $\,W_{l}\,$. This is the term proportional to $\,\cos(\chi
t)\leavevmode\nobreak\ $:

${}^{\textstyle{{}^{(in\leavevmode\nobreak\
phase)}}}{U_{l}}\leavevmode\nobreak\ =\leavevmode\nobreak\
{U_{l}}_{0}(\chi)\leavevmode\nobreak\
\cos\epsilon_{l}(\chi)\leavevmode\nobreak\ \cos(\chi
t)=\left(\frac{R}{r}\right)^{2l+1}{k_{l}}_{0}(\chi)\leavevmode\nobreak\
\cos\epsilon_{l}(\chi)\leavevmode\nobreak\
{W_{l}}_{0}(\chi)\leavevmode\nobreak\ \cos(\chi t)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (28)

We see that the “response factor” is equal to
${k_{l}}_{0}(\chi)\cos\epsilon_{l}(\chi)\leavevmode\nobreak\ $, where
${k_{l}}_{0}(\chi)$ is the real amplitude of the complex Love number (call
this amplitude dynamical Love number).

The second component is the in-quadrature term, which is proportional to
$\sin(\chi t)\leavevmode\nobreak\ $:

${}^{\textstyle{{}^{(in\leavevmode\nobreak\
quadrature)}}}{U_{l}}\leavevmode\nobreak\ =\leavevmode\nobreak\
-\leavevmode\nobreak\ {U_{l}}_{0}(\chi)\leavevmode\nobreak\
\sin\epsilon_{l}(\chi)\leavevmode\nobreak\ \sin(\chi
t)=\left(\frac{R}{r}\right)^{2l+1}{k_{l}}_{0}(\chi)\leavevmode\nobreak\
\sin\epsilon_{l}(\chi)\leavevmode\nobreak\
{W_{l}}_{0}(\chi)\leavevmode\nobreak\ \sin(\chi t)\leavevmode\nobreak\
\,.\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (29)

Naturally, the expression of this term via $W_{l}$ contains
${k_{l}}_{0}(\chi)\sin\epsilon_{l}(\chi)\leavevmode\nobreak\ $.

Therefore, as soon as we express $U_{l}$ via $W_{l}$ by an integral operator
permitting delayed action (Eqn. 9), we automatically arrive at the two
components of the tidal $U_{l}$, at each frequency involved, – the adiabatic
component and the dissipative component (using the terms of Zahn 1966) or the
elastic one and the creep one (as Ferraz-Mello 2012 named them).111 Also see
the work by Krasinsky (2006), who employed the terms elastic and dissipative.

Similar decomposition will take place for the harmonic modes of the primary’s
surface elevation, except that the Love number $\,h_{l}\,$ will be involved
instead of $\,k_{l}\,$. For the principal, $\,\\{lmpq\\}=\\{2200\\}\,$, tidal
mode, this situation is illustrated, in a very exaggerated manner, by Figure
1.

Figure 1: Decomposition of the semidiurnal tide into the elastic and
dissipative components. The primary is spinning at the rate
$\,\dot{\theta}\,$, with $\,\theta\,$ being its sidereal angle. The secondary
is orbiting the primary with the mean motion $\,n\,$. The ellipse shaded green
depicts the principal, semidiurnal mode of the tides exerted on the primary by
the secondary. This is the mode numbered with $\,\\{lmpq\\}=\\{2200\\}\,$. The
semimajor axis of the green ellipse deviates from the direction towards the
perturber by a geometric lag angle. For the semidiurnal tide, this angle is
equal to $\,\frac{\textstyle 1}{\textstyle 2}\,\epsilon_{2}(\omega_{2200})\,$,
where $\,\epsilon_{2}\,$ is the phase lag as a function of the principal tidal
mode $\,\omega_{2200}\,$. It is shown here, in an exaggerated manner, how the
principal tidal mode gets decomposed into two components. One is the elastic
bulge depicted by the ellipse aligned with the perturber. Another is the
dissipative bulge depicted with the ellipse aligned orthogonally.

It should finally be mentioned that in the case of stars and giant planets it
is, technically, difficult to solve the emerging hydrodynamical problem
analytically. Therefore, in practical calculations of fluid equilibrium tides
in such bodies, the two-bulge method is the only known elegant option of
getting an acceptable analytical solution. One first solves the hydrodynamical
equations governing the adiabatic component (26). The adiabatic adjustment of
the structure is deduced from the hydrostatic balance. Then, since the
velocity field is preserved along isobars, one obtains the radial component of
the adiabatic velocity field induced by the tide. The subsequent calculation
of the horizontal component is based on the velocity field being divergence-
free. The knowledge of the adiabatic velocity field makes it possible to
determine the viscous force. This force drives the dissipative component of
the velocity field. The process leads to mass redistribution inside the body,
and thus to perturbation of the density and gravitational potential (27). The
described, two-bulge approach is available because in the fluid equilibrium
tide the dissipative component is much weaker than the adiabatic component.
The method is implemented, e.g., in the work by Remus et al. (2012a), which
furthers the original two-bulge approach offered by Zahn (1966).

##  6 Conclusions

In this short note, we pointed out a simple rule linking the two methods (or,
possibly better to say, two languages), in which bodily tides have been
described by different authors. The language of complex amplitudes is more
economical and is conventional to those who studied the theory of vibrations,
in physics or engineering. The language of two bulges turns out to be
equivalent to that of complex amplitudes. Although less economical
mathematically, the two-bulges language has some illustrative power, and may
be employed (like, e.g., in Remus et al. 2012a,b) on a par with the more
concise method of complex amplitudes.

Importantly, the existence of two mutually orthogonal bulges at each tidal
frequency is not a separate approximation (as was presumed by some of the
devotees of the two-bulge method), but is a consequence of the linearity
assumption implemented by the integral operator (10). As soon as we say that
the tidal deformation is linear but not fully elastic – this deformation can
be decomposed, at each tidal frequency, into an in-phase and an in-quadrature
part.

##  7 Acknowledgements

It is my pleasure to thank Françoise Remus for her useful advises and for her
help in preparing the figure. I also acknowledge with gratitude the
stimulating conversations on the topic of this work, which I had on various
occasions with Sylvio Ferraz Mello, Valéry Lainey, Valeri Makarov, Stephane
Mathis, and Jean-Paul Zahn.

##

## References

  * (1) Churkin, V. A. 1998. “The Love numbers for the models of inelastic Earth.” Preprint No 121. Institute of Applied Astronomy. St.Petersburg, Russia. /in Russian/
  * (2) Efroimsky, M., and Williams, J. G. 2009. “Tidal torques. A critical review of some techniques.” Celestial mechanics and Dynamical Astronomy, Vol. 104, pp. 257 - 289. doi: 10.1007/s10569-009-9204-7 arXiv:0803.3299
  * (3) Efroimsky, M. 2012a. “Bodily tides near spin-orbit resonances.” Celestial mechanics and Dynamical Astronomy, Vol. 112, pp. 283 - 330. doi: 10.1007/s10569-011-9397-4 . Extended version available at arXiv:1105.6086
  * (4) Efroimsky, M. 2012b. “Tidal dissipation compared to seismic dissipation: in small bodies, earths, and superearths.” the Astrophysical Journal, Vol. 746, No 2, article id 150. doi: 10.1088/0004-637X/746/2/150 arXiv:1105.3936
  * (5) Ferraz-Mello, S. 2012. “Tidal synchronisation of close-in satellites and exoplanets. A rheophysical approach.” Submitted to: _Celestial Mechanics and Dynamical Astronomy_ . arXiv:1204.3957
  * (6) Mathis, S., and Le Poncin-Lafitte, C. 2009. “Tidal dynamics of extended bodies in planetary systems and multiple stars.” Astronomy & Astrophysics, Vol. 497, pp. 889 – 910
  * (7) Krasinsky, G. A. 2006. “Numerical theory of rotation of the deformable Earth with the two-layer fluid core. Part 1: Mathematical model.” _Celestial Mechanics and Dynamical Astronomy_ , Vol. 96, pp. 169 - 217. $\left[\right.$In this paper, see formulae (96 - 97) and the subsequent sections 4.2 – 4.3.$\left.\right]$
  * (8) Remus, F.; Mathis, S.; and Zahn, J.-P. 2012a. “The Equilibrium Tide in Stars and Giant Planets. I - The Coplanar Case.” Astronomy & Astrophysics. In press.
  * (9) Remus, F.; Mathis, S.; Zahn, J.-P.; and Lainey, V. 2012b. “Anelastic tidal dissipation in multi-layer planets.” Astronomy & Astrophysics. In press.
  * (10) Zahn, J.-P. 1966a. “Les marées dans une étoile double serrée.” Annales d’Astrophysique, Vol. 29, pp. 313 - 330
  * (11) Zahn, J.-P. 1966b. “Les marées dans une étoile double serrée (suite).” Annales d’Astrophysique, Vol. 29, pp. 489 - 506


11institutetext: National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100012, China

# Kinematics and flux evolution of superluminal components in QSO B1308+326

S.J. Qian 11

(Compiled by using A&A-latex)

###### Abstract

Context. Search for Doppler-boosting effect in flux evolution of superluminal
components in blazars has been an important subject, which can help to clarify
their kinematic and emission properties.

Aims. The kinematics and flux evolution observed at 15 GHz for the three
superluminal components (knot-c, -i and -k) in QSO B1308+326 (z=0.997) were
investigated in detail.

Methods. It is shown that the precessing jet nozzle model previously proposed
by Qian et al. (1991, 2014, 2017, 2022a, 2022b) can be used to fully simulate
their kinematics on pc-scales with a nozzle precession period of 16.9 yr.

Results. With the acceleration/deceleration in their motion found in the
model-simulation of their kinematics we can derive their bulk Lorentz factor
and Doppler factor as function of time and predict their Doppler-boosting
effect.

Conclusions. Interstingly, the flux evolution of the three superluminal
components can be well intrepreted in terms of their Doppler-boosting effect.
The full explanation of both their kinematic behavior and flux evolution
validates our precessing nozzle model and confirms that superluminal
components are physical entities moving relativistically toward us at small
viewing angles.

###### Key Words.:

galaxies: active – galaxies: black holes – galaxies: jets – quasars:
individual B1308+326

##  1 Introduction

B1308+326 is a high redshift quasar (z=0.997). In historical records it was
observed as being optically variable with a long-term variability amplitude of
$M_{B}\sim$5.6 mag and highly polarization and was classified as one of the
most variable BL Lac objects (Angel & Stockman An80 (1980)). It is a
$\gamma$-ray source detected by the ”Fermi Gamma-ray Observatory” (Ackermann
et al. Ack11 (2011), Acero et al. Ac15 (2015)). So B1308+326 radiates across
the entire electromagnetic spectrum from radio-mm-NIR-optical through X-ray to
$\gamma$-ray bands. Very strong variability has been observed in all these
wavebands with various timescales (hours/days to years).
B1308+326 is a low-synchrotron-peaked high polarization quasar, showing a
radio core-jet structure on pc-scales with superluminal components ejected
from its core. In the previous works (Qian et al. Qi17 (2017), Britzen et al.
Br17 (2017)) the kinematic behaviors of the five superlumianl components
(knot-c, -h, -i, -j and -k) observed at 15 GHz in B1308+326 were well model-
simulated in terms of our precessing nozzle scenario with a precession period
of 16.9$\pm$0.85 years, revealing their motion along the precessing common
trajectory in the innermost jet regions corresponding to their precession
phases.
The interesting results (including the modeled distribution of the precessing
common trajectories, the model-fits of the observed trajectories, the periodic
swing of their ejection position angle and the relation between the position
angle and their initial viewing angle) are re-plotted in Figures 1 and 2.
Although these results are ”very nice !!” as an anonymous referee commented,
they are not consummate, because the bulk Lorentz factor and Doppler factor
derived for the superluminal knots in our model-simulations were not further
investigated to find out the association of their flux evolution with the
Doppler-boosting effect. Therefore, one would ask the questions: What about
the flux-density evolution of the superluminal knots ? Would the evolving
line-of-sight of the feature trajectories and their bulk Lorentz factor
produce measurable changes in the flux densities? Would the used precession
model be able to predict (at least part of) the light curves of the different
componnets ?
In this paper we shall discuss the flux evolution combined with the model-
fitting of the kinematic behavior for three superluminal components (knot-c,
-i and -k), demonstrating that their Doppler-boosting effect plays an
important role for understanding their kinematic, dynamic and emission
properties. That is, the inclusion of flux evolution can help to accurately
interpret their VLBI-kinematics, correctly deriving their bulk Lorentz factor
and Doppler factor as function of time in the model-simulations and showing
that their flux evolution can be well explained in terms of the Doppler-
boosting effect.

Figure 1: Left panel: the distribution of precessing common trajectories
modeled for superluminal components in B1308+326 at precession phases
$\phi$=0.5, 1.5, 2.5, 3.5, 4.5 and 5.5 rad. Right panel: model fits of the
trajectories for knot-c, -h, -i, -j and -k in the innermost jet regions.
Within $X_{n}$$\sim$0.3–0.5 mas all the trajectories can be well fitted in
terms of the precessing common trajectory pattern.

Figure 2: Left panel: the periodic swing of ejection position angle PA(t)
produced by the precession of the jet-nozzle for knot-c, -d, -h, -i, -j and
-k. Right panel: relation $\theta(PA)$ between the ejection position angle and
the viewing angle for these components.

##  2 Recapitulation of the precessing nozzle scenario

Figure 3: Geometry of the precession nozzle model (not to scale), adopted from
Qian (Qi17 (2017)). Parameters $\epsilon$ and $\psi$ define the (X,Z)-plane
relative to the coordinate system ($X_{n},Y_{n},Z_{n}$) with Z-axis taken as
the axis of precession. The $Y_{p}$ axis indicates the direction toward the
observer; the position of a superluminal component is defined by parameters
(A, $\Phi$): $A(Z)$ is the amplitude function and $\Phi$ is the azimuthal
angle (or phase); the plane ($X_{p}$,$Z_{p}$) represents the plane of the sky;
$\theta$ denotes the angle between the knot’s velocity vector (${\bf{S}}$) and
the direction toward the observer (${\bf{V}}$). $\eta$ is the initial half
opening angle of the precessing jet cone.

In order to perform the model-fitting of the kinematics of B1308+326 we will
use the precessing nozzle scenario previously proposed in Qian et al. (Qi17
(2017)). Firstly, we recapitulate the formalism of the model. The geometry of
this model is shown in Figure 3, where three coordinate systems are used.
The coordinate system (${X_{p}},{Y_{p}},{Z_{p}}$) has the $Y_{p}$-axis
directed toward the observer, i.e. the plane ($X_{p}$, $Z_{p}$) is defined as
the sky plane. In this plane the $Z_{n}$-axis is defined as the direction
toward the north pole and the $X_{n}$-axis as opposite to the direction of
right ascension. The observed position angle of VLBI knots is measured in
clockwise from the $Z_{n}$-axis. We define a third coordinate system
$(X,Y,Z)$: the $X$-axis coincides with the axis $X_{p}$ and the $Z$-axis is
situated in the $Y_{p}$-$Z_{p}$ plane forming an angle $\epsilon$ with the
$Y_{p}$-axis. The $Z$-axis is defined as the jet-axis around which the
precessing nozzle rotates. $\psi$ denotes the angle between the
$X(X_{p})$-axis and the $X_{n}$-axis. The precession cone has an initial half
opening angle of $\eta$.
We assume that the superluminal knots move along curved trajectories (as shown
in Figure 3), defined by the amplitude function A(Z) and phase $\Phi$, which
changes successively due to the nozzle precession. In Figure 3, ${\bf{S}}$
denotes the direction of the spatial velocity and ${\bf{V}}$ denotes the
direction toward the observer (parallel to the direction $Y_{p}$). $\theta$
denotes the viewing angle of the knot’s motion.
Thus, the trajectory of a knot can be described in the $(X,Y,Z)$ system as
follows.

$X({Z},\Phi)=A({Z}){\cos{\Phi}},$ (1) $Y({Z},\Phi)=A({Z}){\sin\Phi}.$ (2)

The projection of the spatial trajectory on the sky plane is defined by

${X_{n}({Z},{\Phi})}={X_{p}({Z,\Phi})}{\cos{\psi}}-{Z_{p}({Z,\Phi})}{\sin{\psi}},$
(3)
${Z_{n}({Z},{\Phi})}={X_{p}({Z,\Phi})}{\sin{\psi}}+{Z_{p}({Z,\Phi})}{\cos{\psi}},$
(4)

where $\psi$ is the angle between the $X(X_{p})$-axis and the $X_{n}$-axis,

${X_{p}(Z,\Phi)}=X(Z,\Phi),$ (5)
${Z_{p}(Z,\Phi)}={Z}{\sin{\epsilon}}-{Y({Z},\Phi)}{\cos{\epsilon}}.$ (6)

We give the formulas for calculating the viewing angle $\theta$, Doppler
factor $\delta$, apparent transverse velocity $v_{app}$ and elapsed time $T$
after ejection as follows.

  * •

Viewing angle $\theta$

${\theta}=\arccos[\cos{\Delta}(\cos{\epsilon}+\sin{\epsilon}\tan{{\Delta}_{p}})].$
(7)

Where

${\Delta}=\arctan\left[\left(\frac{dX}{dZ}\right)^{2}+\left(\frac{dY}{dZ}\right)^{2}\right]^{1/2},$
(8)

$\Delta$ is the angle between the spatial velocity vector and the $Z$-axis,
and

${{\Delta}_{p}}=\arctan\left(\frac{dY}{dZ}\right)$ (9)

is the projection of $\Delta$ on the $(Y,Z)$-plane.

  * •

Apparent transverse velocity $v_{app}$ and Doppler factor $\delta$

${v_{app}}=c{{\beta}_{app}}={\frac{c{\beta}{\sin{\theta}}}{1-{\beta}{\cos{\theta}}}},$
(10)

and

${\delta}={\frac{1}{{\Gamma}(1-{\beta}{\cos{\theta}})}},$ (11)

where $\beta$=$\frac{v}{c}$, $v$ is the spatial velocity of the knot and
$\Gamma$=$(1-{\beta}^{2})^{-1/2}$ is the bulk Lorentz factor.

  * •

Elapsed time $T$, at which the knot reaches the axial distance $Z$:

${T}={\int_{0}^{Z}}{\frac{1+z}{{\Gamma}{\delta}{v}{\cos{{\Delta}_{s}}}}}{d{Z}},$
(12)

where $z$ is the redshift of B1308+326,

${\Delta}_{s}=\arccos\left[\left(\frac{dX}{dZ}\right)^{2}+\left(\frac{dY}{dZ}\right)^{2}+1\right]^{-1/2},$
(13)

where ${\Delta}_{s}$ is the instantaneous angle between the velocity vector
and the $Z$-axis.

All coordinates and the amplitude $A(Z)$ are measured in units of
milliarcsecond (mas). $\theta$ and v are instantaneous quantities at an
elapsed time $T$.

###  2.1 Precessing common trajectory pattern

As defined above, each of the superluminal components moves along a curved
(collimated) track described by the amplitude function A(Z) and a constant
phase $\Phi$, while for the successive knots the phase changes due to
precession. That is, the superluminal components move along the precessing
common trajectory. We choose the following pattern for describing the
precessing common trajectory.
Its amplitude $A(Z)$ is taken as a function of $Z$:
when $Z$${\leq}b$,

${A(Z)}={A_{0}}{\frac{2b}{\pi}}{\sin\left(\frac{{\pi}{Z}}{2b}\right)},$ (14)

and when $Z>b$,

${A(Z)}={A_{0}}{\frac{2b}{\pi}}.$ (15)

Parameter $b$ may be regarded as a ’collimation parameter’ to describe the
shape of the jet collimation. The phase $\Phi$ is defined by parameter $\phi$
for a specific trajectory:

${\Phi}={\phi}+{{\Phi}_{0}},$ (16)

${\Phi}_{0}$ is an arbitrary constant and $\phi$ is defined as the precessing
phase.

Figure 4: The assumed amplitude function $A(Z)$ and phase $\Phi$ for
describing the trajectory of a knot. Here $\Phi$=$\Phi_{0}$+$\phi$=4.783 rad,
corresponding to the precession phase $\phi$=1.0 rad and ${\Phi}_{0}$=3.783
rad. See text.

Since ${\frac{d\Phi}{dZ}}$=0, we have

${\frac{dX}{dZ}}={\frac{dA}{dZ}}{\cos{\Phi}},$ (17)
${\frac{dY}{dZ}}={\frac{dA}{dZ}}{\sin{\Phi}}.$ (18)

Thus, from Eqs. (8), (9) and (13) we have

${\Delta}={\arctan\left(\frac{dA}{dZ}\right)},$ (19)
${{\Delta}_{p}}={\arctan\left(\frac{dA}{dZ}\sin{\Phi}\right)},$ (20)
${{{\Delta}_{s}}}=\arccos\left[\left(1+\left(\frac{dA}{dZ}\right)^{2}\right)^{-1/2}\right].$
(21)

Substituting $\Delta$, ${\Delta}_{p}$ and ${\Delta}_{s}$ into Eqs. (7) and
(10)–(12), we can calculate the viewing angle $\theta$, apparent velocity
${\beta}_{app}$, Doppler factor $\delta$ and elapsed time $T$.
We should point out that the assumed pattern of the precessing common
trajectory (see Figure 4 below) closely represents the field structure
configurations observed in the jets of radio galaxies and blazars. For
example, the giant radio galaxy M87 which has a powerful optical-radio jet and
a supermassive black hole of $\sim$6$\times$$10^{9}{M_{\odot}}$, is the best
possible target for studying the initial jet formation/collimation process
(Biretta et al. Bi02 (2002)). Nakamura & Asada (Na13 (2013)) (also, Asada &
Nakamura As12 (2012); Doeleman et al. Do12 (2012)) have found that its
innermost jet emission components follow an extrapolated parabolic streamline,
so that the jet has a single power-law structure with a nearly five orders of
magnitude in the distance starting from the vicinity of the supermassive black
hole, less than ten Schwarzschild radii. They have also proposed a
magnetohydrodynamic nozzle model to interpret the property of the bulk jet
acceleration and assumed that the MHD nozzle consists of a hollow parabolic
tube. Most recently, Lu et al. (Lu23 (2023)) (also cf. Kim et al. Kim23
(2023)) present the parabolic jet profile extending to the jet-axis distance
of $\sim$70$\mu$as (or $\sim$10$R_{s}$, $R_{s}$ – Schwarzschild radius).
Moreover, general relativistic MHD simulations (e.g., McKinney et al. Mc12
(2012)) reveal that the magnetic field structures (or configurations) near the
horizon of a rotating black hole closely correspond to a parabolic
configuration which is consistent with the analytic results given by Beskin &
Zheltoukhov (Bes13 (2013)) for a field geometry: a radial field near the
horizon and a vertical field far from the black hole. In these configurations,
the distribution of the magnetic field and the field angular velocity profile
near the horizon can be described in detail (Punsly Pu01 (2001); McKinney et
al. Mc12 (2012); Beskin & Zheltoukhov Bes13 (2013)). In addition, the assumed
pattern is also quite similar to the fork-structure observed in the prominent
blazar OJ287 by Tateyama (Ta13 (2013)).
In fact, in the previous works we have already adopted such a kind of common
precessing trajectory pattern to study the kinematics of the superluminal
components in a few blazars, e.g., 3C279 (Qian et al. Qi19 (2019); Qian Qi12
(2012), Qi13 (2013),), 3C454.3 (Qian et al. Qi14 (2014), Qi21 (2021)), OJ287
(Qian 2018b ), 3C345 (Qian et al. Qi91 (1991), Qian 2022a , 2022b ), and also
in the QSOs PG1302-132 (Qian et al. 2018a ) and NRAO 150 (Qian Qi16 (2016)).
In this paper we will adopt the concordant cosmology model with
${\Omega}_{\Lambda}$=0.73, ${\Omega}_{m}$=0.27 and Hubble constant $H_{0}$=71
km ${{\rm{s}}^{-1}}$ ${{\rm{Mpc}}^{-1}}$ (Spergel et al. Sp03 (2003)). For the
redshift z=0.997 of B1308+326, we have its luminosity distance $D_{L}$=6.61
Gpc and the angular diameter distance $D_{A}$=1.66 Gpc (Hogg Ho99 (1999); Pen
Pe99 (1999)). The angular scale is 1 mas=8.04 pc and the proper motion of 1
mas/yr is equivalent to an apparent velocity of 52.34c.

##  3 Selection of model parameters

In our precessing jet nozzle model, the jet nozzle precesses around a fixed
jet axis and the knots are ejected from the nozzle, moving along their
individual trajectories (of a common pattern, ballistic or helical, Qian Qi16
(2016)) with different bulk Lorentz factors. The precession of the nozzle
leads to the rotation of the ejection direction of the knots or the periodic
position angle swing. The combination of a sequence of isolated knots (and
associated magnetized plasma flows) ejected from this nozzle will construct
the structure of the whole jet and its evolution seen on VLBI-maps (e.g.,
Tateyama & Kingham Ta04 (2004); Qian et al. Qi17 (2017); Tateyama Ta09 (2009),
Ta13 (2013); Qian Qi14 (2014)).
In order to model-fit the kinematics of the superluminal knots in terms of our
precessing jet-nozzle model, the model parameters, defining the jet-axis
direction ($\epsilon$, $\psi$), the pattern of the precessing common track
($A_{0}$, b), precession period $T_{p}$ and phase ($\Phi_{0}$, $\phi$) should
be approriately selected. We shall adopt the same values as used in the
previous work (Qian et al. Qi17 (2017)) as follows: 111Here for the
superluminal knots, we shall use the changes in parameters $\epsilon$ and
${\psi}$ (instead of changes in amplitude A(Z)) to describe the transition
from the common precessing tracks in the innermost jet regions to their own
individual trajectories in the outer jet regions.

$\epsilon$ = $1.5^{\circ}$
$\psi$ = –$42.8^{\circ}$
$A_{0}$ = 1.375$\times{10^{-2}}$ mas
$b$ = 50 mas
${\Phi}_{0}$ = 3.783 rad
$T_{p}$ = 16.9 yr

The ejection epoch $t_{0}$ for the knots can be calculated from their
precession phase $\phi$:

${t_{0}}=1995.54+{\frac{T_{p}}{2\pi}}(6.0-{\phi}).$ (22)

The kinematic parameters including the bulk Lorentz factor, viewing angle,
apparent velocity and Doppler factor as function of time will be derived
through the model-fitting process.
In order to model-fit the light curves of the superluminal knots the observed
flux density ${S_{obs}}(\nu,t)$ can be calculated as:

${{S_{obs}}(\nu,t)}={{S_{int}}(\nu,t)}{[\delta(t)]^{3+\alpha(\nu,t)}}$ (23)

${S_{int}}(\nu,t)$ – intrinsic flux density; $\delta(t)$ – Doppler factor;
$\alpha(\nu,t)$ – spectral index ($S_{\nu}$ $\propto$ ${\nu^{-\alpha}}$). In
most cases ${S_{int}}(\nu,t)$=${S_{int}}(\nu)$ and
$\alpha(\nu,t)$=$\alpha(\nu)$ are assumed.

Figure 5: Knot-c: Model fitting of the entire kinematics (1995–2014). Left
panel: the modeled traveled distance Z(t) along the Z-axis. Right panel: the
modeled curves for parameters $\epsilon(t)$ and $\psi(t)$. After 1996.40
$\psi$ increased quickly, implying a rotation of the (X,Z)-plane relative to
the coordinate system ($X_{n},Y_{n},Z_{n}$) and knot-c started to move along
its own individual track. Figure 6: Knot-c: the model fit of the entire
trajectory extending to core distance $r_{n}$$\sim$8.5 mas (during 1996–2014).
The curvature within $X_{n}{\sim}$1.7 mas is well fitted.

Figure 7: Knot-c: Model fitting of the entire kinematics (1995–2014). Left
panel: the model-fit of the core separation $r_{n}(t)$. Right panel: the
model-fits of coordinates $X_{n}(t)$ and $Z_{n}(t)$. All the kinematic
features are well fitted in terms of the precessing nozzle model extending to
core distance $r_{n}{\sim}$8.5 mas (during 1996–2014).

Figure 8: Knot-c: Model fitting of the entire kinematics (1995–2014). Left
panel: the model-derived apparent velocity $\beta_{app}(t)$ and viewing angle
$\theta(t)$. Right panel: the model-derived bulk Lorentz factor $\Gamma(t)$
and Doppler factor $\delta(t)$. The prominent feature is
$\delta(t)$$<$$\Gamma(t)$ during the peaking stage.

##  4 Knot-c: Model-fitting results

The model fitting results for knot-c will be presented in two parts: (1) for
entire kinematics (1995–2014) in Fig.5–8 and (2) for the inner jet region
(1995–2001.5, $r_{n}{\leq}2.5$ mas) in Fig.9-13.

###  4.1 Knot-c: model-fitting of entire kinematics (1995-2014)

Its ejection epoch $t_{0}$=1995.54, corresponding to a precession phase
$\phi$=6.0 rad.
In Fig.5 the traveled distance Z(t) along Z-axis (left panel) and the
parameters $\epsilon(t)$ and $\psi(t)$ (right panel) are presented. During the
time-interval $\sim$1996–2000 parameter $\psi$ changed quickly, implying a
rotation of the XY-plane 222XY-plane is defined as the reference-plane for
calculating the precession phase of knots. relative to the coordinate system
($X_{n},Z_{n}$).
The model-fits of the entire trajectory $Z_{n}(X_{n})$, core separation
$r_{n}(t)$, coordinates $X_{n}(t)$ and $Z_{n}(t)$ are shown in Fig.6 and
Fig.7, respectively. Within $r_{n}(t)\sim$8.6 mas (or till 2014.09) all these
kinematic features were well fitted.
The model-derived apparent velocity $\beta_{app}(t)$ and viewing angle
$\theta(t)$ (left panel) and bulk Lorentz factor $\Gamma(t)$ and Doppler
factor $\delta(t)$ (right panel) are presented in Fig.8. Both show a peak
structure during $\sim$1995.5–1997.0, coincident with the radio burst (see
Fig.13, below).

Figure 9: Knot-c: model fitting of the inner kinematics (1995.5–2001.5). Left
panel: the modeled traveled distance Z(t) along the Z-axis. Right panel: the
modeled curves for parameters $\epsilon(t)$ and $\psi(t)$. Within core
separation $r_{n}\simeq$0.46 mas (or before 1996.40, corresponding to traveled
distance Z$\leq$13.8 mas=106 pc) $\epsilon$=$1.5^{\circ}$ and
$\psi$=–$42.8^{\circ}$, knot-c moved along the precessing common trajectory.
Beyond $r_{n}$=0.46 mas, parameter $\psi$ increased quickly and knot-c started
to move along its own individual track. Figure 10: Knot-c: Model fitting of
the inner trajectory ($X_{n}{\leq}$2 mas, 1995.5–2001.5). Within core
separation $r_{n}$=0.46 mas (or coordinate $X_{n}{\leq}$0.43 mas) knot-c moved
along the precessing common trajectory. Beyond that knot-c started to move
along its own individual trajectory. The red and green curves are calculated
for the precession phases $\phi{\pm}$0.31 rad ($\phi$=6.0 rad), demonstrating
that the observational data-points are within the area confined by the two
curves and showing that the precession period was derived correctly with an
uncertainty of ${\sim}\pm$5% of the precession period.

Figure 11: Knot-c: Model fitting of the kinematics (1995.5–2001.5). Left
Panel: the model fit of the core separation $r_{n}(t)$. Right panel: the model
fits of coordinates $X_{n}(t)$ and $Z_{n}(t)$. All the kinematic features are
well fitted extending to $r_{n}{\sim}$ 2.5 mas. Before 1996.40 knot-c moved
along the precessing common traejctory, while after that it started to move
along its own individual track.

Figure 12: Knot-c: model fitting of the inner kinematics (1995.5–2001.5). Left
panel: the model-derived apparent velocity $\beta_{app}(t)$ and viewing angle
$\theta(t)$. Right panel: the model-derived bulk Lorentz factor $\Gamma(t)$
and Doppler factor $\delta(t)$. At 1996.15 $\delta$=$\delta_{max}$=30.2,
$\Gamma$=29.7, $\beta$=29.6, while at 1996.02 $\Gamma$=$\Gamma_{max}$=38.0 and
$\beta_{app}$=$\beta_{a,max}$=37.0. $\theta(t)$ varied in the range of
[$1.92^{\circ}$, $5.32^{\circ}$] during 1995.5–2001.0. It is worthy of note
that both $\Gamma(t)$ and $\delta(t)$ have a bump structure and during the
peaking stage $\delta(t)$$<$$\Gamma(t)$.

###  4.2 Knot-c: Model-fitting of inner kinematics (1995.5-2001.5)

In order to investigate the flux evolution associated with its Doppler-
boosting effect, the results of detailed studies on its kinematics in the
inner jet regions (1995.5-2001.5) are presented in Figs.9–13.
The traveled distance Z(t) of knot-c along the jet-axis is shown in Fig.9
(left panel). The curves of parameters $\epsilon(t)$ and $\psi(t)$ are
presented in the right panel. Before 1996.40 (Z$\leq$13.8 mas=106 pc)
$\epsilon$=$1.5^{\circ}$ and $\psi$=–$42.8^{\circ}$ knot-c moved along the
precessing common trajectory. After 1996.40 parameter $\psi$ quickly increased
to $\sim$–$20^{\circ}$ and knot-c started to move along its own individual
trajectory, deviating from the precessing common track. Such kind of
transition from the common track pattern to the individual paths could involve
some complex physical conditions (e.g., evolution of the kinetic and magnetic
energy of the jet associated with the change in its current distribution,
interaction between the jet and its surrounding environments, hydrodynamic and
magnetohydrodynamic instabilities (e.g., Hardee Hard87 (1987), Har11 (2011);
Falke et al. Fa96 (1996), Nakakura & Meier Na04 (2004), etc.).
The model-fit of the trajectory $Z_{n}(X_{n})$ during 1995.5–2001.5 is shown
in Figure 10. Within coordinate $X_{n}$$\sim$0.46 mas knot-c moved along the
precessing common trajectory and beyond that knot-c started to move along its
own individual track, deviating from the precessing common trajectory. In the
figure the red and green curves are calculated for precession phases
$\phi{\pm}$0.31 rad ($\phi$=6.0 rad for ejection at 1995.54), indicating that
the observational data-points are within the range defined by the two curves
and the precession period is determined with an uncertainty of $\sim$$\pm$5%
of the precession period (or $\sim$$\pm$0.85 yr.).
The model-fits of core separation $r_{n}(t)$ (left panel), coordinates
$X_{n}(t)$ and $Z_{n}(t)$ are shown in Figure 11. Within $r_{n}{\sim}$0.46 mas
knot-c moved along the precessing common trajectory and beyond that it started
to move along its own individual track, deviatitng from the common track.
$r_{n}(t)$, $X_{n}(t)$ and $Z_{n}(t)$ are all well fitted during 1995.5–2001.5
(in the range of $r_{n}$ extending to 2.5 mas).
The model-derived apparent velocity ${\beta_{app}}(t)$, viewing angle
$\theta(t)$, bulk Lorentz factor $\Gamma(t)$ and Doppler factor $\delta(t)$
are shown in Figure 12. ${\beta_{app}}(t)$, $\Gamma(t)$ and $\delta(t)$ have a
bump stracture: at 1996.15 $\delta$=$\delta_{max}$=30.2, $\Gamma$=29.7 and
$\beta_{app}$=29.6. At 1996.02 $\Gamma$=$\Gamma_{max}$=38.0 and
$\beta_{app}$=${\beta_{app,max}}$=37.0.333In comparison, an average velocity
$\beta_{app}$=22.9 was given in Britzen et al. Br17 (2017). $\theta(t)$ varied
in the range of [$1.92^{\circ}$, $5.32^{\circ}$] during 1995.5–2001.0. It
should be noted that during the peaking stage $\delta(t)$$<$$\Gamma(t)$.

###  4.3 Knot c: flux evolution and Doppler-boosting effect

Figure 13: Knot-c. Left panel: the model-fit of the 15 GHz light curve with
the modeled peak flux density of 2.37 Jy (at 1996.15) and the intrinsic flux
density of 15.6 $\mu$Jy. Right panel: the light curve normalized by the
modeled peak flux density is very well fitted by the Doppler-boosting profile
$[\delta(t)/\delta_{max}]^{3+\alpha}$ with an assumed value $\alpha$=0.5. The
data-points at 1997.26, 1997.66 and 1997.01 deviating from the Doppler-
boosting profile obviously could be due to variations in its intrinsic flux
density.

The model fit of the 15 GHz light curve is shown in Figure 13. The modeled
peak flux density is 2.37 Jy (at 1996.15) and its intrinsic flux density
15.6$\mu$Jy. The light curve normalized by the modeled peak flux density is
well fitted by the Doppler-boosting profile
$[\delta(t)/\delta_{max}]^{3+\alpha}$ (with an assumed value $\alpha$=0.5).
The flux fluctuations on shorter timescales (e.g. at 1997.26) which obviously
deviate from the Doppler-boosting profile might be induced by the intrinsic
flux variations of knot-c itself.

Figure 14: Knot-i. Left panel: the modeled traveled distance Z(t) along the
jet-axis. Right panel: the modeled curves for parameters $\epsilon(t)$ and
$\psi(t)$. Before 2012.0 (or Z(t) $\leq$12.5 mas=96.1 pc)
$\epsilon$=$1.5^{\circ}$ and $\psi$=–$42.8^{\circ}$, knot-i moved along the
precessing common trajectory. Beyond that distance its motion started to
transfer to follow its own individual track, deviating from the precessing
common trajectory.  Figure 15: Knot-i: the model fit of the observed
trajectory $Z_{n}(X_{n})$. Within coordinate $X_{n}{\simeq}$0.39 mas
(Z$\leq$12.5 mas=96.1 pc), knot-i moved along the precessing common
trajectory. Beyond that distance it started to follow its own individual
track. The red and green curves are calculated for precession phases
$\phi{\pm}$0.31 rad ($\phi$=1.0 rad), demonstrating that the observational
data-points are within the area defined by the two curves and the precession
period is determined correctly with an uncertainty of $\sim$${\pm}$5% of the
precession period (or ${\sim}$$\pm$0.85 yr.).

Figure 16: Knot-i. Left panel: the model fit of the core distance $r_{n}(t)$.
Right panel: the model fits of the coordinates $X_{n}(t)$ and $Z_{n}(t)$. All
the kinematic features are well fitted extending to $r_{n}\simeq$1.2 mas
(during 2010.5–2014.5).

Figure 17: Knot-i. Left panel: the model-derived apparent velocity
$\beta_{app}(t)$ and viewing angle $\theta(t)$. Right panel: the model-derived
bulk Lorentz factor $\Gamma(t)$ and Doppler factor $\delta(t)$.
$\beta_{app}(t)$, $\Gamma(t)$ and $\delta(t)$ all have a bump structure during
2009.5–2012.0, which was associated with a major burst (see Figure 18 below).
At 2010.90 $\delta$= $\delta_{max}$=24.02 and $\Gamma$=$\Gamma_{max}$=13.80.
At 2010.64 $\beta_{app}$=$\beta_{app,max}$=16.96. $\theta(t)$ varied in the
range of [$2.27^{\circ}$, $4.18^{\circ}$] during 2009.5–2014.0.

Figure 18: Knot-i. Left panel: the 15GHz light curve is well fitted with its
modeled peak flux density of 1.20 Jy at 2010.90 and intrinsic flux density of
17.7$\mu$ Jy. Right panel: the light curve normalized by the modeled peak flux
density is fitted very well by the Doppler-boosting profile
$[\delta(t)/\delta_{max}]^{3+\alpha}$.

##  5 Knot i: Model-fitting results

The model-fitting results for knot-i are shown in Figures 14–18. Its ejection
time $t_{0}$=2009.0 and the corresponding precession phase $\phi$=1.0 rad.

###  5.1 Knot-i: model-fit of kinematics

The modeled traveled distance Z(t) along the jet-axis and the modeled curves
for parameters $\epsilon(t)$ and $\psi(t)$ are shown in Figure 14. Before
2012.00 (Z$\leq$12.5 mas=96.1 pc) $\epsilon$=$1.5^{\circ}$ and
$\psi$=–$42.8^{\circ}$ knot-i moved along the precessing common trajectory,
while after that $\epsilon$ increased knot-i started to move along its own
individual track, deviating from the common precessing track ($\psi$ started
to increase quickly after 2013.38).
The model fit of its projected trajectory $Z_{n}(X_{n})$ is shown in Figure
15\. Within core separation $r_{n}$$\sim$0.49 mas knot-i moved along the
common precessing track. Beyond that knot-i started to move along its own
individual track, deviating from the common track. The red and green curves in
the figure are calculated for precession phase $\phi{\pm}$0.31 rad ($\phi$=1.0
mas), indicating that the observational data-points are within the area
defined by the two curves and the precession period is determined with an
uncertainty of $\sim$${\pm}$5%.
Model fits of core separation $r_{n}(t)$ (left panel), coordinates $X_{n}(t)$
and $Z_{n}(t)$ are shown in Figure 16. Before 2012.00 ($r_{n}$$\leq$0.49 mas,
$X_{n}$$\leq$0.39 mas) knot-c moved along the precessing common track, while
after that knot-c stared to move along its own individual trajectory,
deviating from the common precessing track. It can be seen that $r_{n}(t)$,
$X_{n}(t)$ and $Z_{n}(t)$ are all well model-fitted during 2010.5–2014.0.
The model-derived apparent velocity $\beta_{app}(t)$, viewing angle
$\theta(t)$, bulk Lorentz factor $\Gamma(t)$ and Doppler factor $\delta(t)$
are shown in figure 17. $\beta_{app}$, $\Gamma$ and $\delta$ all have a bump
structure coincident with the radio burst (see Figure 18). At 2010.90
$\delta$=$\delta_{max}$=24.02, $\Gamma$=$\Gamma_{max}$=18.0. At 2010.64
$\beta_{app}$=$\beta_{app,max}$=16.94. $\theta(t)$ varied in the range of
[$2.27^{\circ}$ (2009.0), $4.18^{\circ}$ (2014.0)].

###  5.2 Knot-i: flux evolution and Doppler-boosting effect

The model fit of the 15GHz light curve is shown in Fig.18 (left panel) with
its modeled peak flux density 1.20 Jy (at 2010.90) and the intrinsic flux
density 17.7$\mu$Jy. The light curve normalized by the modeled peak flux
density was well fitted by its Doppler-boosting profile
$[\delta(t)/\delta_{max}]^{3+\alpha}$ (right panel).

##  6 Knot k: Model-fitting results

The model fitting results of the kinematics and light curve for knot-k are
shown in Figures 19–23. Its ejection time $t_{0}$=2010.88 and the
corresponding precession phase $\phi$=0.30 rad.

###  6.1 Model-fitting of kinematics

The modeled traveled distance Z(t) along the jet-axis and the modeled curves
for parameters $\epsilon(t)$ and $\psi(t)$ are shown in Figure 19. Before
2012.97 (or Z$\leq$7.2 mas=55.4 pc) $\epsilon$=$1.5^{\circ}$ and
$\psi$=–$42.8^{\circ}$, knot-k moved along the precessing common track and
after that $\epsilon$ decreased slightly knot-k started to move along its own
individual track, slightly deviating from the precessing common trajectory.
The model-fit of the projected trajectory is shown in Figure 20. Within
$X_{n}$=0.24 mas (or before 2012.97) knot-k moved along the precessing common
track. Beyond that knot-k started to move along its own individual track,
slightly deviating from the common precessing track. The red and green curves
in the Figure are calculated for precession phases $\phi{\pm}$0.31 rad
($\phi$=0.30 rad), indicating that the observational data-points are within
the area defined by the two curves and the precession period is determined
with an uncertainty of $\sim$$\pm$5% of the period ($\sim$$\pm$0.85 years).
The model-fits of the core separation $r_{n}(t)$, coordinates $X_{n}(t)$ and
$Z_{n}(t)$ are shown in Figure 21. Within $r_{n}$=0.26 mas (or
$X_{n}$$\leq$0.24 mas; before 2012.97) knot-k moved along the precessing
common track, while beyond that it started to move along its own individual
track, deviating from the common track.
The model-derived apparent velocity $\beta_{app}(t)$, viewing angle
$\theta(t)$, bulk Lorentz factor $\Gamma(t)$ and Doppler factor $\delta(t)$
are shown in Figure 22. $\beta_{app}$, $\Gamma$ and $\delta$ all have a bump
structure: at 2013.00 $\delta$=$\delta_{max}$=24.50, $\Gamma$=16.06,
$\beta_{app}$=13.62 and $\theta$=$1.99^{\circ}$. But $\Gamma_{max}$=16.50
(during 2012.81–2012.97) and $\beta_{app,max}$=14.4 (at 2012.81). Viewing
angle $\theta(t)$ varied in the range of [$2.07^{\circ}$, $1.53^{\circ}$]
during 2011.4–2014.0.

###  6.2 Knot k: flux evolution and Doppler-boosting effect

The model-fit of the 15 GHz light curve is shown in Fig.22 (left panel) with
its modeled peak flux density 1.26 Jy (at 2013.00) and intrinsic flux density
$S_{int}$=17.4 $\mu$Jy. The light curve normalized by the modeled peak flux
density was well fitted by its Doppler-boosting profile
$[\delta(t)/\delta_{max}]^{3+\alpha}$ with a presumed value $\alpha$=0.5
(right panel). The data-point at 2014.07, obviously deviating from the
profile, might be due to the increasing of its intrinsic flux density.

Figure 19: Knot-k. Left panel: the model-derived traveled distance Z(t). Right
panel: the modeled curves of parameter $\epsilon(t)$ and $\psi(t)$. Before
2012.97 (corresponding to traveled distance Z$\leq$7.2 mas=55.4 pc)
$\epsilon$=$1.5^{\circ}$ and $\psi$=–$42.8^{\circ}$, knot-k moved along the
precessing common trajectory. After that epoch $\epsilon$ slightly decreased
with time and knot-k started to move along its own individual trajectory,
slighly deviating from the precessing common track. Figure 20: Knot-k: the
model fit of the observed trajectory $Z_{n}(X_{n})$. Within $X_{n}{\leq}$0.24
mas (Z$\leq$7.2 mas=55.4 pc) knot-k moved along the preceessing common track.
After that knot-k started to move along its own individual trajectory,
deviating from the precessing common track. The red and green curves were
calculated for the precession phases $\phi{\pm}$0.31 rad ($\phi$=0.30 rad),
indicating that the observational data-points are within the area defined by
the two curves and the precessing period is derived correctly with an
uncertainty of $\sim$$\pm$5% of the period (or ${\sim}$$\pm$0.85 yr.).

Figure 21: Knot-k. Left panel: the model fit of the core separation
$r_{n}(t)$. Right panel: the model fits of the coordinates $X_{n}(t)$ and
$Z_{n}(t)$. Before 2012.97 (or $r_{n}{\leq}0.26$ mas, $X_{n}{\leq}$0.24 mas)
$\epsilon$=$1.5^{\circ}$ and $\psi$=–$42.8^{\circ}$, knot-k moved along the
preceessing common track. After that knot-k started to move along its own
individual trajectory, deviating from the precessing common track.

Figure 22: Knot-k. Left panel: the model-drived apparent velocity
$\beta_{app}(t)$ and viewing angle $\theta(t)$. Right panel: the model-drived
bulk Lorentz factor $\Gamma(t)$ and Doppler factor $\delta(t)$. At 2013.00
$\delta$= $\delta_{max}$=24.50, $\Gamma$=16.1 and $\beta_{app}$=13.6, while
$\Gamma$=$\Gamma_{max}$=16.5 during 2012.81–2012.97 and
$\beta_{app}$=$\beta_{app,max}$=14.4 at 2012.81. $\theta(t)$ varied in the
range of [$2.07^{\circ}$, $1.53^{\circ}$] during 2011.4–2014.0.

Figure 23: Knot-k. Left panel: the model fit of the 15 GHz light curve with
its modeled peak flux density of 1.26 Jy (at 2013.0) and intrinsic flux
density of 17.4 $\mu$Jy. Right panel: the light curve normalized by the
modeled peak flux density is well fitted by the Doppler-boosting profile
$[\delta(t)/\delta_{max}]^{3+\alpha}$ with an assumed $\alpha$=0.5. The data-
point at 2014.07 deviating from the profile largely might be due to a
variation in its intrinsic flux density.

##  7 Summary and conclusion

Based on our precessing jet-nozzle secnario (Qian et al. Qi17 (2017), Qi21
(2021)) we have successfully model-fitted the kinematics observed at 15 GHz on
pc-scales for the three superluminal components (knot-c, -i and -k) in QSO
B1308+326 and interpreted their light curves. We briefly summarize the results
as follows:
(1) Superluminal components were ejected from the precessing jet-nozzle with a
precession period of $\sim$16.9$\pm$0.85 years.
(2) The superluminal knots moved along the precessing common tracks
corresponding to their precession phases (or ejection times) in the innermost
jet regions (core separation $r_{n}\leq$0.30–0.5 mas), while in the outer jet-
regions they started to deviate from the precessing common tracks and moved
along their own individual trajectories.
(2) the periodic position angle swing observed for the superluminal components
can be well explained in terms of our precessing nozzle scenario.
(3) The observed kinematic features (trajectory $Z_{n}(X_{n})$, core
separation ($r_{n}(t)$), coordinates $X_{n}(t)$ and $Z_{n}(t)$ and apparent
velocity $\beta_{app}(t)$) were consistently well model-fitted.
(4) The bulk Lorentz factor $\Gamma(t)$, viewing angle $\theta(t)$ and Doppler
factor $\delta(t)$ for the superluminal components were properly derived.
(5) The 15 GHz light curves of the superluminal components can be well
interpreted in terms of their Doppler-boosting effect.
Our precessing jet-nozzle scenario may be described by such a physically
feasible conception:
In the nucleus of B1308+328 there is a energy-engine consisting of a rotating
(Kerr) black-hole and a tilted magnetized accretion-disk around the hole. Due
to the electromagnetic effects induced from the spin of the black hole and the
rotation of the disk with its magnetosphere, a mini-jet (or beam) is formed
with a nozzle steadily ejecting magnetized plasma and superluminal plasmoids
along helical tracks around the axis of the disk. Moreover, due to the frame-
dragging effect the gravitational torque of the rotating black hole will cause
a global precession of the accretion disk with its mini-jet. Thus the
precession of the mini-jet would naturally form the precessing common
trajectory suggested in our scenario, producing the observed periodic position
angle swing of superlumnal components and regular distribution of their inner
trajectories. Obviously, in our scenario, the observed jet (as usually
defined) is originated from the precession of the single mini-jet which ejects
magnetized plasma and superlumnal plasmoids in a long period.
Our precessing nozzle scenario can be understood in the framework of available
relativistic magnetohydrodynamic theories for the
formation/collimation/acceleration of relativistic jets in blazars (cf. a
detailed discussion in Qian et al. Qi17 (2017)). The structure of the jets
observed in the radio galaxy M87 (parabolic hollow jet nozzle structure;
Nakamura & Asada Na13 (2013)) and in OJ287 (fork-like jet structure; Tateyama
Ta13 (2013)) may be regarded as observational evidence.
Theoretically, magnetic nozzles can be probably formed in disk-driven jets in
the magnetospheres of rotating black-hole/accretion-disk systems. These
magnetic nozzles may locate near the classical fast magnetosonic point where
the magnetohydrodynamic flow remains Poynting flux dominated. In some self-
similar axisymmetric MHD flow models, beyond the classic fast magnetosonic
point, the jet will be accelerated untill approaching the modified fast
magnetosonic point (Blandford & Znajek Bl77 (1977); Li et al. Li92 (1992);
Vlahakis & Königl Vl04 (2004)). This extended acceleration is due to the
dominance of Poynting flux at the classic magnetosonic point, thus having
ample electromagnetic energy to be transformed plasma kinetic energy
(Komissarov et al. Kom07 (2007), Komissarov Kom09 (2009), Millas et al. Mi14
(2014)).
. In the case of relativistic jets, the classical fast magnetosonic point is
located in the force-free region of the magnetosphere where the magnetic
energy dominates the plasma kinetic energy. The magnetic field lines anchored
into the innermost disk and the magnetic nozzle would rotate rigidly with the
disk (MacDonald & Throne Mac82 (1982)).
Additionally, in this extended acceleration region beyond the magnetic nozzle
(or beyond the classical fast magnetosonic point), the inertia of the plasma
becomes strong and the electromagnetic fields are neither degenerate nor force
free, and the plasma would flow along its own streamlines, not following the
local field lines and forming its own trajectory pattern. This can explain why
the superluminal knots observed in B1308+326 move along the common precessing
track.444Also in other QSOs and blazars, (e.g., in PG 1302-102, NRAO 150,
OJ287, 3C345, 3C454.3 and 3C279.
Most recently, the black-hole/accretion-disk system in the giant radio galaxy
M87 (having a hole-mass of $\sim$6.5$\times{10^{9}}$$M_{\odot}$) has been
observed by using the world mm-VLBI-network (Lu et al. Lu23 (2023)). Both the
mm-jet emanating from its centeral hole and the associated circumdisk
structure have been mapped. The morphological strcuture and kinematic
properties are well consistent with the whole picture expected by relativistic
MHD theories for jet formation in black-hole/accretion disk systems.

## References

  * (1) Acero, F., Ackermann, M., Ajello, M., et al. 2015, ApJS, 218, 23
  * (2) Ackermann, M., Ajello, M., Allafort, A., et al. 2011, ApJ, 743, 171
  * (3) Angel, J.R.P., & Stockman, H.S. 1980, Ann. Rev. Astron. Astrophys., 8, 321
  * (4) Asada, K., & Nakamura, M. 2012, ApJL, 745, L28
  * (5) Beskin, V.S., & Zheltoukhov, A.A. 2013, Astronomy Letters, 39, 215
  * (6) Biretta, J.A., Junor, W., & Livio, M. 2002, New Astronomy Review (NAR), 46,239
  * (7) Blandford, R.D., & Znajek, R.L. 1977, MNRAS, 179, 433
  * (8) Britzen, S., Qian, S.J., Steffen, W., et al. 2017, A&A, 602, A29
  * (9) Doeleman, S.S., Fish, V.L., Schenck, D.E., et al. 2012, Science, 338, 355
  * (10) Falcke H., Wilson A.S., Simpson C., Bower G.A., 1996, ApJ 470, L31
  * (11) Hardee, P. E. 1987, ApJ, 318, 78
  * (12) Hardee P.E., 2011, in Jets at all Scales, Proceedings of the International Astronomy Union, IAU Symposium 275, 41–49
  * (13) Hogg, D. W. 1999, astro-ph/9905116
  * (14) Kim J.Y., Savolainen T., Voitsik P., et al. 2023, arXiv e-prints, arXiv:2304.09816 [astro-ph.GA]
  * (15) Komissarov, S.S., Barkov, M.V., Vlahakis, N., & Königl, A. 2007, MNRAS, 380, 51
  * (16) Komissarov, S.S. 2009, Journal of the Korean Physical Society (JKPS), 54, 2503
  * (17) Li, Z.Y., Chieuh, Z.H., & Begelman, M.C. 1992, ApJ, 394, 459
  * (18) Lister, M.L., Cohen, M.H., Homan, D.C., et al. 2009, AJ, 138, 3718
  * (19) Lu R.S., Asada K., Krichbaum t.P., et al., 2023, Nature 616 , 686-690
  * (20) Macdonald, D., & Thorne, K.S. 1982, MNRAS, 198, 345
  * (21) McKinney, J.C., Tchekhovskoy, A., & Blandford, R.D. 2012, MNRAS, 423, 2083
  * (22) Milas, D., Katsoulakos, G., Lingri, D., et al. 2014, in High Energy Phenomena in Relativistic Outflows (International Journal of Modern Physics Conference series vol.28, id 1460200)
  * (23) Nakamura, M. & Asada, K. 2013, ApJ, 775, 118
  * (24) Nakamura M. & Meier D.L., 2004, ApJ 617, 123
  * (25) Pen, Ue-Li. 1999, ApJS, 120, 49
  * (26) Punsly, B. 2001, Black Hole Gravitohydromagnetics (New York: Springer)
  * (27) Qian, S. J., Witzel, A., Krichbaum, T., et al. 1991, Acta Astron. Sin., 32, 369 (english translation: in Chin. Astro. Astrophys., 16, 137 (1992))
  * (28) Qian, S.J. 2012, RAA, 12, 46
  * (29) Qian, S.J. 2013, RAA, 13, 783
  * (30) Qian, S.J., Britzen, S., Witzel, A., et al. 2014, RAA, 14, 249
  * (31) Qian, S. J. 2016, RAA, 16, 20
  * (32) Qian, S.J., Britzen S., Witzel, A. et al. 2017, A&A, 604, A90
  * (33) Qian, S.J., Britzen S., Witzel, A. et al. 2018a, A&A, 615, A123
  * (34) Qian S.J., 2018b, arXiv e-prints, arXiv:1811.1154 [astro-ph.GA]
  * (35) Qian, S., Britzen S., Krichbaum T.P., Witzel A., 2019, A&A, 621, A11
  * (36) Qian S.J., Britzen S., Krichbaum T.P., Witzel A., 2021, A&A, 653, A7
  * (37) Qian S.J., 2022a, arXiv e-prints, arXiv:2202.01915 [astro-ph.GA]
  * (38) Qian S.J., 2022b, arXiv e-prints, arXiv:2206.14995 [astro-ph.GA]
  * (39) Spergel, D. N., Verde, L., Peiris, H. V., et al. 2003, ApJS, 148, 175
  * (40) Tateyama, C. E., & Kingham, K. A. 2004, ApJ, 608, 149
  * (41) Tateyama, C.E. 2009, ApJ, 705, 877
  * (42) Tateyama, C.E. 2013, ApJS, 205, 15
  * (43) Vlahakis, N., & Königl, A. 2004, ApJ, 605, 656