2010 Vol. 10 No. XX, 000–000

11institutetext: North West University (Potchefstroom Campus), School of
Physics (Unit for Space Research), Private Bag X6001, Potchefstroom $2531$,
Republic of South Africa, Email<EMAIL_ADDRESS>
Received 2010 May 14; accepted 2010 June 27

# Bipolar Outflows as a Repulsive Gravitational Phenomenon
– Azimuthally Symmetric Theory of Gravitation (II)

Golden Gadzirayi Nyambuya00footnotetext: ∗Supported by the Republic of South
Africa’s National Research Foundation and the North West University, and
Germany’s DAAD Programme via the University of K$\ddot{\rm{o}}$ln.

###### Abstract

This reading is part in a series on the Azimuthally Symmetric Theory of
Gravitation (ASTG) set-out in Nyambuya ([$2010a$]). This theory is built on
Laplace-Poisson’s well known equation and it has been shown therein (Nyambuya
[$2010a$]) that the ASTG is capable of explaining – from a purely classical
physics standpoint; the precession of the perihelion of solar planets as being
a consequence of the azimuthal symmetry emerging from the spin of the Sun.
This symmetry has and must have an influence on the emergent gravitational
field. We show herein that the emergent equations from the ASTG – under some
critical conditions determined by the spin – do possess repulsive
gravitational fields in the polar regions of the gravitating body in question.
This places the ASTG on an interesting pedal to infer the origins of outflows
as a repulsive gravitational phenomena. Outflows are an ubiquitous phenomena
found in star forming systems and their true origins is a question yet to be
settled. Given the current thinking on their origins, the direction that the
present reading takes is nothing short of an asymptotic break from
conventional wisdom; at the very least, it is a complete paradigm shift as
gravitation is not at all associated; let alone considered to have anything to
do with the out-pour of matter but is thought to be an all-attractive force
that tries only to squash matter together into a single point. Additionally,
we show that the emergent Azimuthally Symmetric Gravitational Field from the
ASTG strongly suggests a solution to the supposed Radiation Problem that is
thought to be faced by massive stars in their process of formation. That is,
at $\sim 8-10\,\mathcal{M}_{\odot}$, radiation from the nascent star is
expected to halt the accretion of matter onto the nascent star. We show that
in-falling material will fall onto the equatorial disk and from there, this
material will be channelled onto the forming star via the equatorial plane
thus accretion of mass continues well past the curtain value of $\sim
8-10\,\mathcal{M}_{\odot}$ albeit via the disk. Along the equatorial plane,
the net force (with the radiation force included) on any material there-on
right up-till the surface of the star, is directed toward the forming star,
hence accretion of mass by the nascent star is un-hampered.
PACS (2010): $97.10.$Bt, $97.10.$Gz, $97.10.$Fy

###### keywords:

stars: formation – stars: mass-loss – stars: winds, outflows – ISM: jets and
outflows.

##  1 Introduction

Champagne like bipolar molecular outflows are an unexpected natural phenomenon
that grace the star formation podium. Bipolar molecular outflows are the most
spectacular physical phenomenon intimately associated with newly formed stars.
Studies of bipolar outflows reveal that they [bipolar outflows] are ubiquitous
toward High Mass Star (HMS) forming regions. These outflows in HMS forming
regions are far more massive and energetic than those found associated with
Low Mass Stars (LMS) forming regions (see e.g. Shepherd & Churchwell
[$1996a$]; Shepherd & Churchwell [$1996b$]; Zhang et al. [$2001$]; Zhang et
al. [$2005$]; Beuther [$2002$]). Obviously, this points to a correlation
between the mass of the star and the outflow itself. Independent studies have
established the existence of such a correlation. The mass outflow rate
$\dot{\mathcal{M}}_{out}$ has been shown to be related to the bolometric
luminosity $\mathcal{L}$ by the relationship:
$\dot{\mathcal{M}}_{out}\propto\mathcal{L}^{0.60}_{star}$, and this is for
stars in the luminosity range:
${0.30}\mathcal{L}_{\odot}\leq\mathcal{L}_{star}\leq{10}^{5}\mathcal{L}_{\odot}$
(we shall use the term luminosity to mean bolometric luminosity). Another
curious property of outflows is that the mass-flow rate,
$\dot{\mathcal{M}}_{out}$, is related to the speed of the molecular outflow
$\dot{\mathcal{M}}_{out}\propto V^{-\gamma}_{out}$ where $\gamma\sim{1.80}$
and $V_{out}$ is the speed of the outflow. How and why outflows come to
exhibit these properties is an interesting field of research that is not part
of the present reading. However, we shall show that these relationships do
emerge from our proposed ASTG Outflow Model. In the present, we simple want to
show that an outflow model emerges from the ASTG model. We set herein the
mathematical foundations for such a model. Once we have a fully-fledged
mathematical model, we shall move on to building a numerical model (i.e.
computer code). Once this computer code is available, an endeavor to answer
the above and other questions surrounding the nature of outflows will be made.

Pertaining to their association with star formation activity, it is believed
that molecular outflows are a necessary part of the star formation process
because their existence may explain the apparent angular momentum imbalance.
It is well known that the amount of initial angular momentum in a typical
star-forming molecular cloud core is several orders of magnitude too large to
account for the observed angular momentum found in formed or forming stars
(see e.g. Larson [$2003b$]). The sacrosanct Law of Conservation of angular
momentum informs us that this angular momentum can not just disappear into the
oblivion of interstellar spacetime. So, the question is where does this
angular momentum go to? It is here that outflows are thought to come to the
rescue as they can act as a possible agent that carries away the excess
angular momentum. This angular moment, if it where to remain as part of the
nascent star, it would, via the strong centrifugal forces, tear the star
apart. This however does not explain, why they exist and how they come to
exist but simply posits them as a vehicle needed to explain the mystery of
“The Missing Angular Momentum Problem” in star forming systems and the
existence of stars in their intact and compact form as stable firery balls of
gas.

In the existing literature, viz the question why and how molecular outflows
exist, there are about four proposed leading models that endeavor to explain
the aforesaid. These four major proposals are:

Wind Driven Outflow Model: In this model, a wide-angle radial wind blows into
the stratified surrounding ambient material, forming a thin swept-up shell
that can be identified as the outflow shell (see Shu et al. [$1991$]; Li & Shu
[$1996$]; Matzner & McKee [$1999$]).

Jet Driven Bow Shocks Model: In this model, a highly collimated jet propagates
into the surrounding ambient material producing a thin outflow shell around
the jet (see Raga et al. [$1993a$]; Masson & Chernin [$1993$]).

Jet Driven Turbulent Outflow Model: In this model, Kelvin-Helmholtz
instabilities along the jet and or environmental boundary leading to the
formation of a turbulent viscous mixing layer, through which the molecular
cloud gas in entrained (see Cantó & Raga [$1991$]; Raga et al. [$1993b$];
Stahler [$1994$]; Lizano & Giovanardi [$1995$]; Cantó et al. [$2003$]).

Circulation Flows Model: In this model, the molecular outflow is not entrained
by an underlying wind jet but is rather formed by in-falling matter that is
deflected away from the protostar in the central torus of high magneto-
hydrodynamic pressure through a quadrupolar circulation pattern around the
protostar and is accelerated above escape speeds by local heating (see Fiege &
Henriksen [$1996a$]; Fiege & Henriksen [$1996a$]).

All these ad hoc models and some that are not mentioned here explain outflows
as a feedback effect. The endeavor of the work presented in this reading is to
make an alternative suggestion albeit a complete, if not a radical departure
from the already existing models briefly mentioned above. Our model flows
naturally from the Laplace-Poison equation, namely from the Azimuthally
Symmetric Theory of Gravitation (ASTG) laid down in Nyambuya ([$2010a$])
(hereafter Paper I). This model is new and has never before appeared in the
literature. Because we are at the stage of setting this model, we see no need
to get into the details of the existing models as this would lead to an
unnecessary digression, confusion, and an un-called for lengthy reading.

Our model is a complete departure from the already existing models because, of
all the agents that could lead to outflows, gravitation is not even considered
to be a possible agent because it is thought of as, or assumed to be, an all-
attractive force. Actually, the idea of a gravitating body such as a star
producing a repulsive gravitational field, is at the very least unthinkable.
Contrary to this, we show here that an azimuthally symmetric gravitational
system does in-principle give rise to a bipolar repulsive gravitational field
and this – in our view, clearly suggests that these regions of repulsive
gravitation, possibly are the actual driving force of the bipolar molecular
outflows. We also see that the ASTG provides a neat solution (possibly and
very strongly so) to the so-called Radiation Problem thought to bedevil and
bewilder the formation of HMSs (see Larson & Starrfield [$1971$]; Kahn
[$1974$]; Bonnell et al. [$1998$]; Bonnell & Bate [$2002$]; Palla & Stahler
[$1993$]) and as-well the observed Ring of Masers (Bartkiewicz et al.
[$2008$], [$2009$]).

We need to reiterate this so as to make it clear to our reader, that, the work
presented in this reading is meant to lay down the mathematical foundations of
the outflow model emergent from the ASTG. It is not a comparative study of
this outflow model with those currently in existence. We believe we have to
put thrust on lying down these ideas and only worry about their plausibility,
i.e. whether or not they correspond with experience and only thereafter make a
literature wide comparative study. Given that this model flows naturally from
a well accepted equation (the Poisson-Laplace equation), against the
probability of all unlikelihood, this model should have a bearing with
reality. If it does not have a bearing with reality, then, at the very least,
it needs to be investigated since this solution of the Poisson-Laplace
equation has not been explored anywhere in the literature111In our exhaustive
survey of the accessible literature, we have not come across a treatment of
the Poisson-Laplace equation as is done in the present, hence our proclamation
that this solution of the Poisson-Laplace equation is the first such..

Also, we should say that as we build this model, we are doing this with
expediency, that is, watchful of what experience dictates, at the end of the
day, if our efforts are to bear any fruits, our model must correspond with
reality. This literature wide comparative study is expected to be done once a
mathematical model of our proposed outflow model is in full-swing. This
mathematical model is expected to form part of the future works where only-
after that, it would make sense then to embark on this literature wide
comparative study. How does one compare a baby human-being to a human-embryo?
It does not make sense, does it? Should not the baby be born first and only
thereafter a comparative study be conducted of this baby with those babies
already in existence? We hope the reader concurs with us that this is perhaps
the best way to set into motion a new idea amid a plethora of ideas that
champion a similar if not the same endeavor.

Further, we need to say this; that, as already stated above, the direction
that the present reading takes is nothing short of an asymptotic break from
conventional wisdom; at the very least, it is a complete paradigm shift as
gravitation is not at all associated; let alone considered to have anything to
do with the out-pour of matter but is thought to be an all-attractive force
that tries only to squash matter together into a single point. Because of this
reason, that, the present is “nothing short of an asymptotic break from
conventional wisdom” and that “at the very least, it is a complete paradigm
shift”, we strongly believe that this is enough to warrant the reader’s
attention to this seemingly seminal theoretical discovery.

The synopsis of this reading is as follows. In the subsequent section, we
present the theory to be used in setting up the proposed ASTG Outflow Model.
In §($3$), we revisit the persistent problem of the ASTG model, that of “The
ASTG’s Undetermined Parameter Problem”. Therein, we present what we believe
may be a solution to this problem. As to what really these parameters may be,
this is still an open question subject to debate. In §($4$), we present the
main findings of the present reading, that is. the repulsive bipolar
gravitational field and therein we argue that this field fits the description
of outflows. We present this for both the empty and non-empty space solutions
of the Poisson-Laplace equation. In §($5$), we look at the anatomy of the
outflow model, i.e. the switching on and off outflows, the nature of the
repulsive polar field, the emergent shock rings and the collimation factor of
these outflows. In §($6$), we show that the ASTG model posits what strongly
appears to be a perdurable solution to the so-called Radiation Problem that is
thought to be faced by massive stars during their formation process. Lastly,
in §($7$), we give a general discussion and make conclusion that cane be drawn
from this reading.

Lastly, it is important that we mention here in the penultimate of this
introductory section that this reading is fundamental in nature and because of
this, we shall seek to begin whatever argument we seek to rise, from the soils
of its very basic and fundamental level. This is done so that we are at the
same level of understanding with the reader. With the aforesaid approach, if
at any point we have errored, it would be easy to know and understand where
and how we have errored.

##  2 Theory

Newton’s Law of universal gravitation can be written in a more general and
condensed form as Poisson’s Law, i.e.:

$\vec{\nabla}^{2}\Phi=4\pi G\rho,$ (1)

where $\rho$ is the density of matter and $G=6.667\times
10^{-11}\textrm{kg}^{-1}\textrm{ms}^{-2}$ is Newton’s universal constant of
gravitation and the operator $\vec{\nabla}^{2}$ written for a spherical
coordinate system [see figure (1) for the coordinate setup] is given by:

$\vec{\nabla}^{2}=\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\frac{\partial}{\partial
r}\right)+\frac{1}{r^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}}{\partial\varphi^{2}},$
(2)

where the symbols have their usual meanings. For a spherically symmetric
setting, the solution to Poisson’s equation outside the vacuum space (where
$\rho=0$) of a central gravitating body of mass $\mathcal{M}_{star}$ is given
by the traditional inverse distance Newtonian gravitational potential which is
given by:

$\Phi(r)=-\frac{G\mathcal{M}_{star}}{r},$ (3)

where $r$ is the radial distance from the center of the gravitating body. The
Poisson equation for the case $(\rho=0)$ is known as the Laplace equation. The
Poisson equation is an extension of the Laplace equation. Because of this, we
shall generally refer to the Poisson equation as the Poisson-Laplace equation.
In the case where there is material surrounding this central mass, that is
$\mathcal{M}=\mathcal{M}(r)$, where:

$\mathcal{M}(r)=\int^{r}_{0}\int^{2\pi}_{0}\int^{2\pi}_{0}r^{2}\rho(r,\theta,\varphi)\sin\theta
d\theta d\varphi dr,$ (4)

we must – in (3), make the replacement:
$\mathcal{M}_{star}\longmapsto\mathcal{M}(r)$. As already argued in Paper I,
if the gravitating body in question is spinning, we ought to consider an
Azimuthally Symmetric Gravitational Field (ASGF). Thus, we shall solve the
azimuthally symmetric setting of (1) for both cases of empty and non-empty
space and show from these solutions that Poisson’s equation entails a
repulsive bipolar gravitational field. We shall assume that if one has the
empty space solution, to obtain the non-empty space solutions, one has to make
the replacement: $\mathcal{M}_{star}\longmapsto\mathcal{M}(r)$, just as is
done in Newtonian gravitation. This is a leaf that we shall take from
spherically symmetric Newtonian gravitation into the ASTG model.

Figure (1): This figure shows a generic spherical coordinate system, with the
radial coordinate denoted by $r$, the zenith (the angle from the North Pole;
the co-latitude) denoted by $\theta$, and the azimuth (the angle in the
equatorial plane; the longitude) by $\varphi$.

###  2.1 Empty Space Solutions

As already argued in Paper I, for a scenario or setting that exhibits
azimuthal symmetry such as a spinning gravitating body as the Sun and also the
stars that populate the heavens (where the unexpected and spectacular
champagne like bipolar molecular outflows are the observed); we must have:
$\Phi=\Phi(r,\theta)$. There-in Paper I, the Poisson equation for empty space
has been “solved” for a spinning gravitating system and the solution to it is:

$\Phi(r,\theta)=-\sum^{\infty}_{\ell=0}\left[\lambda_{\ell}c^{2}\left(\frac{G\mathcal{M}_{star}}{rc^{2}}\right)^{\ell+1}P_{\ell}(\cos\theta)\right],$
(5)

where $\lambda_{\ell}$ is an infinite set of dimensionless parameters with
$\lambda_{0}=1$ and the rest of the parameters $\lambda_{\ell}$ for
$(\ell>1)$, generally take values different from unity. There-in Paper I, a
suggestion as to what these parameters may be has been made. In §($3$) we go
further and suggest a form for these parameters. This suggestion, if correct,
puts the ASTG on a pedestal to make predictions without first seeking these
values (i.e. the $\lambda_{\ell}$’s) from observations. We will show that
there lays embedded in (5) a solution that is such that the polar regions of
the gravitating central body will exhibit a repulsive gravitational field. It
is this repulsive gravitational field that we shall propose as the driving
force causing the emergence of outflows. But, we must bare in mind that
outflows are seen in regions in which the central gravitating body is found in
the immensement of ambient circumstellar material, thus we must – for the
azimuthally symmetric case (where the central gravitating body is spinning),
solve the Poisson-Laplace equation for the setting $(\rho\neq 0)$.

###  2.2 Non-Empty Space Solutions

Clearly, in the event that $(\rho\neq 0)$ for the azimuthally symmetric case,
we must have $\rho=\rho(r,\theta)$. In Paper I, an argument has been advanced
in support of this claim that: $\Phi(r,\theta)\Rightarrow\rho(r,\theta)$.
Taking this as given, the question we wish to answer is; what form does
$\Phi(r,\theta)$ take for a given mass distribution $\rho(r,\theta)$? or the
reverse, what form does $\rho(r,\theta)$ take for a given $\Phi(r,\theta)$? It
is reasonable and most logical to assume that the gravitational field is what
influences the distribution of mass and not the other way round. Taking this
as the case, then, we must have $\rho(r,\theta)=\rho(\Phi)$, i.e. the
distribution of the matter in any mass distribution must be a function of the
gravitational field. We find that the form for $\rho(r,\theta)$ that meets the
requirement: $\rho(r,\theta)=\rho(\Phi)$, and most importantly the requirement
that to obtain the non-empty space solution from the empty space solution one
simply makes the replacement: $\mathcal{M}_{star}\longmapsto\mathcal{M}(r)$,
is:

$\rho(r,\theta)=-\frac{1}{4\pi G}\left[\frac{2}{r}\frac{\partial}{\partial
r}-\frac{1}{r^{2}}\right]\frac{\partial\Phi(r,\theta)}{\partial\theta}.$ (6)

How did we arrive at this? We have to answer this question. To make life very
easy for us to arrive at the answer, we shall write Poisson’s equation in
rectangular coordinates, i.e.:

$\left(\sum_{j=1}^{3}\frac{\partial^{2}}{\partial
x^{2}_{j}}\right)\Phi(x,y,z)=4\pi G\rho(x,y,z),$ (7)

where $x_{1}=x,x_{2}=y,x_{3}=z$. Now suppose we had a function $F(x,y,z)$ such
that:

$\left(\sum_{j=1}^{3}\frac{\partial}{\partial x_{j}}\right)^{2}F(x,y,z)=0.$
(8)

This equation can be written as:

$\left(\sum_{j=1}^{3}\frac{\partial^{2}}{\partial
x^{2}_{j}}\right)F(x,y,z)=-\left(\sum_{j}^{3}\sum_{i\neq
j}^{3}\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}\right)F(x,y,z).$ (9)

Now, if and only if the gravitational potential did satisfy (7), then,
comparison of (7) with (9) requires the identification: $\Phi(x,y,z,)\equiv
F(x,y,z)$, and as-well the identification:

$\rho(x,y,z)=-\frac{1}{4\pi G}\left(\sum_{j}^{3}\sum_{i\neq
j}^{3}\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}\right)\Phi(x,y,z).$
(10)

What this means is that the non-linear terms of (7) come about because of the
presence of matter. Now, if we transform to spherical coordinates, it is now
understood as to why and how we came to the choice of $\rho$ given in (6). At
the end of the day, what this means is that we can choose whatever form for
$\Phi$, the density $\rho$ will have to conform and prefigure to this setting
of the gravitational field via (10). Only and only after accepting (10), do we
have the mathematical legitimacy to choose to maintain the form (5) which we
found for the case of empty space such that in the place of
$\mathcal{M}_{star}$ we now can put $\mathcal{M}(r)$, hence thus in the case
where a central gravitating condensation of mass is in the immensement of
ambient circumstellar material, we must have:

$\Phi(r,\theta)=-\sum^{\infty}_{\ell=0}\lambda_{\ell}c^{2}\left(\frac{G\mathcal{M}(r)}{rc^{2}}\right)^{\ell+1}P_{\ell}(\cos\theta),$
(11)

where $\mathcal{M}(r)$ is given in (4). We believe this answers the question
“What form does $\rho(r,\theta)$ take for a given $\Phi(r,\theta)$?” and at
the same-time we have justified (6) viz how we have come to it. Importantly,
it should be noted that the observed radial density profile is maintained by
the choice (10), i.e.
$\rho(r)=\int^{r}_{0}\int^{2\pi}_{0}r^{2}\rho(r,\theta)\sin\theta d\theta
dr\propto r^{-\alpha_{\rho}}$. Also important to state clearly is that, all
the above implies that the gravitational field is what influences the
distribution of matter – this, in our view, resonates both with logic and
intuition. We shall demonstrated the assertion that:
$\rho(r)=\int^{r}_{0}\int^{2\pi}_{0}r^{2}\rho(r,\theta)\sin\theta d\theta
dr\propto r^{-\alpha_{\rho}}$. We know that:

$\int^{r}_{0}\int^{2\pi}_{0}r^{2}\rho(r,\theta)\sin\theta d\theta
dr=\int^{r}_{0}r^{2}\left(\int^{2\pi}_{0}\rho(r,\theta)\sin\theta
d\theta\right)dr=4\pi\int^{r}_{0}r^{2}\rho(r)dr$ (12)

this means:

$\rho(r)=\frac{1}{4\pi}\int^{2\pi}_{0}\rho(r,\theta)\sin\theta d\theta.$ (13)

Our claim is that if $\rho(r,\theta)$ is given by (6) such that
$\Phi(r,\theta)$ is given by (11), where $\mathcal{M}(r)$ in (11) is such that
$\mathcal{M}(r)\propto r^{\alpha}$ for some constant $\alpha$, then:

$\rho(r)=\frac{1}{4\pi}\int^{2\pi}_{0}\rho(r,\theta)\sin\theta d\theta\propto
r^{\alpha_{\rho}},$ (14)

where $\alpha_{\rho}$ is some constant. We know that:

$\frac{1}{4\pi}\int^{2\pi}_{0}\rho(r,\theta)\sin\theta
d\theta=-\frac{1}{16\pi^{2}G}\int^{2\pi}_{0}\left(\frac{2}{r}\frac{\partial}{\partial
r}-\frac{1}{r^{2}}\right)\frac{\partial\Phi(r,\theta)}{\partial\theta}\sin\theta
d\theta.$ (15)

We have substituted $\rho(r,\theta)$ in (6) into the above. This simplifies
to:

$\frac{1}{4\pi}\int^{2\pi}_{0}\rho(r,\theta)\sin\theta
d\theta=-\frac{1}{16\pi^{2}G}\left(\frac{2}{r}\frac{\partial}{\partial
r}-\frac{1}{r^{2}}\right)\int^{2\pi}_{0}\frac{\partial\Phi(r,\theta)}{\partial\theta}\sin\theta
d\theta.$ (16)

From (11), we know that:

$\frac{\partial\Phi(r,\theta)}{\partial\theta}=\sum^{\infty}_{\ell=0}\lambda_{\ell}c^{2}\left(\frac{G\mathcal{M}(r)}{rc^{2}}\right)^{\ell+1}\sin\theta\frac{\partial
P_{\ell}(\cos\theta)}{\partial(\cos\theta)}$ (17)

and this implies:

$\frac{1}{4\pi}\int^{2\pi}_{0}\rho(r,\theta)\sin\theta
d\theta=-\frac{c^{2}}{16\pi^{2}G}\left(\frac{2}{r}\frac{\partial}{\partial
r}-\frac{1}{r^{2}}\right)\sum^{\infty}_{\ell=0}\lambda_{\ell}\int^{2\pi}_{0}\left(\frac{G\mathcal{M}(r)}{rc^{2}}\right)^{\ell+1}\sin^{2}\theta\frac{\partial
P_{\ell}(\cos\theta)}{\partial(\cos\theta)}d\theta$ (18)

this simplifies to:

$\frac{1}{4\pi}\int^{2\pi}_{0}\rho(r,\theta)\sin\theta
d\theta=-\frac{c^{2}}{16\pi^{2}G}\left(\frac{2}{r}\frac{\partial}{\partial
r}-\frac{1}{r^{2}}\right)\sum^{\infty}_{\ell=0}\lambda_{\ell}\left(\frac{G\mathcal{M}(r)}{rc^{2}}\right)^{\ell+1}\overbrace{\int^{2\pi}_{0}\sin^{2}\theta\frac{dP_{\ell}(\cos\theta)}{d(\cos\theta)}d\theta}^{\textrm{Let}\,\,\textrm{this}\,\,\textrm{be:}\,\,\textrm{I}_{\ell}\textrm{(}\theta\textrm{)}}$
(19)

where $I_{\ell}(\theta)$ is as defined above. It should not be difficult to
see that $I_{0}(\theta)=0$, $I_{1}(\theta)=1$ and that $I_{\ell}(\theta)\equiv
0$ for all $\ell\geq 2$. From this, it follows that:

$\rho(r)=\frac{1}{4\pi}\int^{2\pi}_{0}\rho(r,\theta)\sin\theta
d\theta=\left(\frac{\lambda_{1}c^{2}}{16\pi^{2}G}\right)\left(\frac{2}{r}\frac{\partial}{\partial
r}-\frac{1}{r^{2}}\right)\left(\frac{G\mathcal{M}(r)}{rc^{2}}\right)^{2},$
(20)

Now, if $\mathcal{M}(r)\propto r^{\alpha}$ this means
$\mathcal{M}(r)=kr^{\alpha}$ for some adjustable constant $k$. Plugging this
into the above, one obtains:

$\rho(r)=\left(\frac{\left(2\alpha-1\right)\lambda_{1}c^{2}}{16\pi^{2}G}\right)\left(\frac{Gk}{c^{2}}\right)^{2}r^{2\alpha-4}.$
(21)

This222Under the prescribed conditions $\mathcal{M}(r)\propto r^{\alpha}$
leads to $\rho(r)\propto r^{2\alpha-4}$. While
$\mathcal{M}(r)=\int^{r}_{0}\int^{2\pi}_{0}\rho(r,\theta)\sin\theta d\theta
dr$, the basic definition $\mathcal{M}(r)=4\pi r^{3}\rho(r)/3$ must hold too,
since $\mathcal{M}(r)$ is the amount of mass enclosed in volume sphere of
radius $r$ and $\rho(r)$, is the mass-density of material in this volume
sphere. These two definitions must lead to identical formulas. If this is to
be so – then; one is lead to the conclusion that $\alpha=1$, and this means
$\mathcal{M}(r)\propto r$ and $\rho(r)\propto r^{-2}$. In the face of
observations, the later result is very interesting since MCs seem to favor
this density profile. verifies our claim in (14). As already said, all the
above implies that the gravitational field is what influences the distribution
of matter. Co-joining this result with the result $(0\leq\alpha_{\rho}<3)$ in
Nyambuya ([$2010c$]) (hereafter Paper III), it follows that
$(0.5\leq\alpha<2)$. Further, a deduction to be made from the above result is
that the spin does control the mass distribution via the term $\lambda_{1}$.

##  3  The Undetermined Constants $\lambda_{\ell}$

Again, as already stated in Paper I, one of the draw backs of the ASTG is that
it is heavily dependent on observations for the values of $\lambda_{\ell}$
have to be determined from observations. Without knowledge of the
$\lambda_{\ell}^{\prime}s$, one is unable to produce the hard numbers required
to make any numerical quantifications. Clearly, a theory incapable of making
any numerical quantifications is – in the physical realm, useless. To avert
this, already in Paper I and as-well in Nyambuya ([$2010b$]) (hereafter Paper
II) an effort to solve this problem has been made. In Paper I, a reasonable
suggestion was made to the effect that:

$\lambda_{\ell}=\left(\frac{(-1)^{\ell+1}}{\left(\ell^{\ell}\right)!\left(\ell^{\ell}\right)}\right)\lambda_{1}.$
(22)

This suggestion meets the intuitive requirements stated there-in Paper I. If
these $\lambda$’s are to be given by (22), then, there is just one unknown
parameter and this parameter is $\lambda_{1}$. The question is what does this
depend on? We strongly feel/believe that $\lambda_{1}$ is dependent on the
spin angular frequency and the radius of the gravitating body in question and
our reasons are as follows.

The ASTG will be shown shortly to be able to explain outflows as a
gravitational phenomenon. Pertaining to their association with star formation
activity, it is believed that molecular outflows are a necessary part of the
star formation process because their existence may explain the apparent
angular momentum imbalance. It is well known that the amount of initial
angular momentum in a typical star-forming cloud core is several orders of
magnitude too large to account for the observed angular momentum found in
formed or forming stars (see e.g. Larson [$2003b$]). The sacrosanct Law of
Conservation of angular momentum informs us that this angular momentum can not
just disappear into the oblivion of interstellar spacetime. So, the question
is where does this angular momentum go to? It is here that outflows are
thought to come to the rescue as they can act as a possible agent that carries
away the excess angular momentum. Whether or not this assertion is true or may
have a bearing with reality, no one really knows.

This angular momentum, if it where to remain as part of the nascent star, it
would, via the strong centrifugal forces (the centrifugal acceleration is
given by: $a_{c}=\omega^{2}_{star}\mathcal{R}_{star}$), tear the star apart.
This however does not explain, why they [outflows] exist and how they come to
exist but simple posits them as a vehicle needed to explain the mystery of
“The Missing Angular Momentum Problem” in star forming systems and the
existence of stars in their intact and compact form as firery balls of gas.

In Paper II, guided more by intuition than anything else, it was drawn from
the tacit thesis “that outflows possibly save the star from the detrimental
centrifugal forces”, the suggestion that
$\lambda_{1}\propto(a_{c})^{\zeta_{0}}$ where $\zeta_{0}$ is a pure constant
that must be universal, that is, it must be the same for all spinning
gravitating systems. This suggestion, if correct leads us to:

$\lambda_{\ell}=\left(\frac{(-1)^{\ell+1}}{\left(\ell^{\ell}\right)!\left(\ell^{\ell}\right)}\right)\left(\frac{a_{c}}{a_{*}}\right)^{\zeta_{0}}.$
(23)

Knowing the solar values of $\lambda_{1}$ and as-well the value of
$\zeta_{0}$, one is lead to:
$a_{*}=\omega^{2}_{\odot}\mathcal{R}_{\odot}(\lambda_{1}^{\odot})^{-\frac{1}{\zeta_{0}}}$.
As will be demonstrated soon, the term $\lambda_{1}$ controls outflows. Given
that $\lambda_{1}$ controls outflows and that outflows possibly aid the star
in shedding off excess spin angular momentum, the best choice333We speak of
“choice” here as though the decision is ours on what this parameter must be.
No, the decision was long made by Nature, ours is to find out what choice
Nature has made. That said, we should say that, this “choice” is made with
expediency – i.e., this choice which is based on intuition, is to be measured
against experience. for this parameter is one that leads to these outflows
responding to the spin of the star and as well the centrifugal forces
generated by this spin in such a way that the star is able to shed off this
excess spin angular momentum. So, what led to this proposal
$\lambda_{1}\propto(a_{c})^{\zeta_{0}}$ is the aforesaid. Now, we shall revise
this suggestion by advancing what we believe is a far much better argument.

If outflows are there to save the nascent star from the ruthlessness of the
centrifugal forces, then, it is logical to imagine that at the moment the
centrifugal forces are about to rip the star apart, outflows will switch-on,
thus shedding off this excess spin angular momentum. The centrifugal forces
have their maximum toll on the equatorial surface of the star hence if the
centrifugal forces are to rip the nascent star apart, this would start at the
equator of the nascent star. The centrifugal force on the surface of the star
acting on a particle of mass $m$ is
$F_{c}=m\omega^{2}_{star}\mathcal{R}_{star}=ma_{c}$ and the gravitational
force on the same particle is
$F_{g}=G\mathcal{M}m/\mathcal{R}_{star}^{2}=mg_{star}$. Now lets define the
quotient $\mathcal{Q}=F_{c}/F_{g}=a_{c}/g_{star}$. If the particle where to
stay put on the surface of the star, then we will have
$F_{c}-F_{g}<0\Rightarrow\mathcal{Q}<1$; and if the particle where to fly off
the surface, we will have $F_{c}-F_{g}>0\Rightarrow\mathcal{Q}>1$. The
critical condition before the star begins to be torn apart is
$F_{c}-F_{g}=0\Rightarrow\mathcal{Q}=1$. All the above can be summarized as:

$\mathcal{Q}:=\left\\{\begin{array}[]{l l l}<1&&\textrm{No\,\,
Outflow\,\,Activity}\\\ =1&&\textrm{Critical\,\, Condition}\\\ >1&&\
textrm{Outflow\,\,Activity}\end{array}\right..$ (24)

Lets call this quotient, the Outflow Control Quotient (OCQ). Clearly, the OCQ
determines the necessary conditions for outflows to switch on. Given this, and
as-well the thinking that $\lambda_{1}$ controls outflows, the suggestion is
clear that $\lambda_{1}\propto\mathcal{Q}^{\zeta_{0}}$. If this is correct,
then:

$\lambda_{1}=\zeta\mathcal{Q}^{\zeta_{0}}.$ (25)

We shall take this as our proposal for $\lambda_{1}$ and this means we must
determine $(\zeta,\zeta_{0})$. From the above, it follows that:

$\frac{\lambda_{1}^{\oplus}}{\lambda_{1}^{\odot}}=\left(\frac{\mathcal{Q}_{\oplus}}{\mathcal{Q}_{\odot}}\right)^{\zeta_{0}},$
(26)

where $\mathcal{Q}_{\oplus}=a_{c}^{\oplus}/g_{\oplus}$ and $a_{c}^{\oplus}$ is
the centripetal acceleration generated by the Earth’s spin at the equator and
$g_{\oplus}$ is the gravitational field strength at the Earth equator.
Likewise, $\mathcal{Q}_{\odot}=a_{c}^{\odot}/g_{\odot}$, is the solar outflow
quotient where $a_{c}^{\odot}$ is the centripetal acceleration generated by
the Sun’s spin at the solar equator and $g_{\odot}$ is the gravitational field
strength at the solar equator. Given that: $(\omega_{\oplus}=7.27\times
10^{-5}\,\textrm{Hz}$ and $\omega_{\odot}=2.04\times 10^{-5}\,\textrm{Hz})$,
$(\mathcal{R}_{\oplus}=6.40\times 10^{6}\,\textrm{m}$ and
$\mathcal{R}_{\odot}=6.96\times 10^{8}\,\textrm{m})$ and
$(g_{\oplus}=9.80\,\textrm{ms}^{-2}$ and $g_{\odot}=27.9g_{\oplus})$. From
this data, it follows that:

$\frac{\mathcal{Q}_{\oplus}}{\mathcal{Q}_{\odot}}=169.$ (27)

Now, in Paper II, we did show that depending on how one interprets the flyby
equation, one obtains two values of $\lambda_{1}^{\oplus}$, i.e.
$\lambda_{1}^{\oplus}=(2.00\pm 0.80)\times 10^{3}$ and
$\lambda_{1}^{\oplus}=(1.50\pm 0.70)\times 10^{4}$. If the spin of the Earth
is significantly variable during the course of its orbit around the Sun, we
will have $\lambda_{1}^{\oplus}=(2.00\pm 0.80)\times 10^{3}$ and if the spin
is not significantly variable, then, $\lambda_{1}^{\oplus}=(1.50\pm
0.70)\times 10^{4}$. If $\lambda_{1}^{\oplus}=(2.00\pm 0.80)\times 10^{3}$,
then:

$\frac{\lambda_{1}^{\oplus}}{\lambda_{1}^{\odot}}=\frac{15000\pm
7000}{21.00\pm 4.00}=800\pm 500,$ (28)

and from this it follows that $800\pm 500=169.19^{\zeta_{0}}$, hence
$\zeta_{0}=1.30\pm 0.10$. If $\lambda_{1}^{\oplus}=(1.50\pm 0.70)\times
10^{4}$, then:

$\frac{\lambda_{1}^{\oplus}}{\lambda_{1}^{\odot}}=\frac{2000\pm 800}{21.00\pm
4.00}=100\pm 60,$ (29)

and from this it follows that $100\pm 60=169.19^{\zeta_{0}}$ hence
$\zeta_{0}=0.90\pm 0.10$.

If $\zeta_{\oplus}$ and $\zeta_{\odot}$ are the $\zeta$-values for the Earth
and the Sun respectively, then, for $\lambda_{1}^{\oplus}=15000\pm 7000$, we
will have $\zeta_{\oplus}=(3.40\pm 2.70)\times 10^{7}$ and
$\zeta_{\odot}=(8.00\pm 4.00)\times 10^{10}$; and for
$\lambda_{1}^{\oplus}=2000\pm 800$, we will have $\zeta_{\oplus}=(3.40\pm
2.70)\times 10^{7}$ and $\zeta_{\odot}=(8.00\pm 4.00)\times 10^{10}$. Table
(I) is a self explanatory summary of all the above calculations. The mean
values of $\zeta$ for the case $\lambda_{1}^{\oplus}=(2.00\pm 0.80)\times
10^{3}$ is $\zeta=(8.00\pm 1.00)\times 10^{5}$ and for the case
$\lambda_{1}^{\oplus}=(1.50\pm 0.70)\times 10^{4}$ is $\zeta=(4.00\pm
2.00)\times 10^{7}$. These mean values have been obtained by taking the values
of $\zeta_{\oplus}$ and $\zeta_{\odot}$ where they intersect in their error
margins.

Table (I): : The $(\zeta_{0},\zeta)$ Values for the Two Different Values of $\lambda_{1}^{\oplus}$. $\lambda_{1}^{\oplus}$ | $\lambda_{1}^{\odot}$ | $\zeta_{0}$ | $\zeta_{\odot}$ | $\zeta_{\oplus}$
---|---|---|---|---
($10^{3}$) |  |  | ($10^{5}$) | ($10^{5}$)
$\,\,\,2.00\pm 0.80$ | $21.00\pm 4.00$ | $0.90\pm 0.10$ | $13.00\pm 6.00$ | $5.00\pm 3.00$
$15.00\pm 7.00$ | $21.00\pm 4.00$ | $1.30\pm 0.10$ | $500\pm 400$ | $400\pm 200$

As argued in Paper II, the value $\lambda_{1}^{\oplus}=(2.00\pm 0.80)\times
10^{3}$ has been obtained from the assumption that the spin of the Earth
varies widely during its course on its orbit around the Sun. This is not
supported by observations thus we are not persuaded to take-up/recommend this
value of $\lambda_{1}^{\oplus}=(2.00\pm 0.80)\times 10^{3}$. Also, as argued
in Paper II, the value $\lambda_{1}^{\oplus}=(1.50\pm 0.70)\times 10^{4}$ is
obtained from the assumption that the spin of the Earth does not vary widely
during its course on its orbit. Thus, we shall adopt the values of
$(\zeta_{0},\zeta)$ that conform with $\lambda_{1}^{\oplus}=(1.50\pm
0.70)\times 10^{4}$ and $\lambda_{1}^{\odot}=21.0\pm 0.40$, hence:

$\lambda_{1}=(4.00\pm 2.00)\times
10^{7}\left(\frac{a_{c}}{g_{star}}\right)^{1.30\pm 0.10}.$ (30)

Obviously, the greatest criticism against this result is that it is obtained
from just two data points. To obtain something more reliable, one needs more
data points. This is something that a future study must handle, at present, we
simple want to set-up the mathematical model from the little available data
and when data becomes available, amendments are made accordingly. While we
have used the minimal possible data points, one thing that can be deduced from
this data is that this result obtained points to a correlation as proposed in
(25) – otherwise, if there was no correlation as proposed, the values of
$(\zeta_{0},\zeta)$ obtained the two values of $\lambda_{1}$ do not vary
widely as is expected if the proposed relationship (25) did not hold at all.

##  4 Outflows as a Gravitational Phenomenon

We shall look into the empty and non-empty space solution of the solution of
the Poisson-Laplace equation and show that both these solutions exhibit a
repulsive bipolar gravitational field and that this repulsive gravitational
field is controlled by the parameter $\lambda_{1}$.

###  4.1 Non-Empty Space Solutions

Now, if one accepts what has been presented thus far – as will be shown in
this section; it follows that outflows may-well be a gravitational phenomena.
First, from the previous section, it follows that we must take the ASTG only
up to second order, i.e.:

$\Phi=-\frac{G\mathcal{M}(r)}{r}\left[1+\frac{\lambda_{1}G\mathcal{M}(r)\cos\theta}{rc^{2}}+\lambda_{2}\left(\frac{G\mathcal{M}(r)}{rc^{2}}\right)^{2}\left(\frac{3\cos^{2}\theta-1}{2}\right)\right].$
(31)

We know that the gravitational field intensity:
$\vec{\textbf{g}}(r,\theta)=-\mbox{\boldmath$\nabla$}\Phi(r,\theta)=g_{r}(r,\theta)\hat{\textbf{r}}+g_{\theta}(r,\theta)\hat{\mbox{\boldmath$\theta$}}$,
this means:

$g_{r}=g_{N}\left[\overbrace{1+\frac{2\lambda_{1}G\mathcal{M}(r)\cos\theta}{rc^{2}}}^{\textbf{Term
I}}+\overbrace{3\lambda_{2}\left(\frac{G\mathcal{M}(r)}{rc^{2}}\right)^{2}\left(\frac{3\cos^{2}\theta-1}{2}\right)}^{\textbf{Term
II}}\right],$ (32)

where: $g_{N}=-G\mathcal{M}(r)/r^{2}$, is the Newtonian gravitational field
intensity and:

$g_{\theta}=g_{N}r^{2}\sin\theta\left[\frac{\lambda_{1}G\mathcal{M}(r)}{rc^{2}}+9\lambda_{2}\left(\frac{G\mathcal{M}(r)}{rc^{2}}\right)^{2}\cos\theta\right].$
(33)

For gravitation to be exclusively attractive (as is expected), we must have:
[$g_{r}(r,\theta)>0$] and [$g_{\theta}(r,\theta)~{}>~{}0$]. From (32) and
(33), it is clear that regions of exclusively repulsive gravitation will exist
and these will occur in the region where: [$g_{r}(r,\theta)<0$] and
[$g_{\theta}(r,\theta)~{}<~{}0$]. This region where gravity is exclusively
repulsive is the region where it is not attractive, it is the negated region
of the region of attractive gravitation: [i.e.
$\left\\{g_{r}(r,\theta)>0\right\\}$ and
$\left\\{g_{\theta}(r,\theta)~{}>~{}0\right\\}$]. Let us start by treating the
case: [$g_{r}(r,\theta)~{}<~{}0$]. From (32), if: [$g_{r}(r,\theta)<0$], then:
(Term I $<0$) and (Term II $<0$), as well. The condition: (Term I $<0$),
implies:

$r<-\lambda_{1}\left(\frac{2G\mathcal{M}(r)}{c^{2}}\right)\cos\theta=\lambda_{1}\left(\frac{2G\mathcal{M}(r)}{c^{2}}\right)\cos\theta,$
(34)

(NB: $\cos\theta\equiv-\cos\theta$) and if one where to take $r$ such that it
only takes positive values, then, (34) must be written in the equivalent form:

$r<\lambda_{1}\left(\frac{2G\mathcal{M}(r)}{c^{2}}\right)\left|\cos\theta\right|,$
(35)

where the brackets $\left|[]\right|$ represents the absolute value. We have to
explain this, i.e. why we concealed the negative sign in (34) and inserted the
absolute value operator in (35). From (34), it is seen that this inequality
includes negative values of $r$ and to avoid any confusion as to what these
negative values of $r$ really mean, this needs to be explained for failure to
do so or failure by the reader to understand this means they certainly will be
unable to agree with the outflow “picture” laid down herein. This explanation
is important in order to understand the morphology of the outflow and as-well
the ASGF.

For a moment, imagine a flat Euclidean plane and on this plane let O, A and P
be distinct and separate points on this plane with O and A being fixed and P
is a variable point. In polar coordinates, as in the present case, a point P
is characterized by two numbers: the distance $(r\geq{0})$ to the fixed pole
or origin O, and the angle $\theta$ the line OP makes with the fixed reference
line OA. The angle $\theta$ is only defined up to a multiple of
${360}\hbox{${}^{\circ}$}$ (or ${2}\pi\,\textrm{rad}$, in radians). This is
the conventional definition. Sometimes it is convenient as in the present case
to relax the condition $(r\geq 0)$ and allow $r$ to be assigned a negative
value such that the point $(r,\theta)$ and
$(-r,\theta+{180}\hbox{${}^{\circ}$})$ represent the same-point, hence thus
when ever we have $(-r,\theta)$ this must be replaced by
$(r,\theta-{180}\hbox{${}^{\circ}$})$. It is easier for us to always think of
$r$ as always being positive. To achieve this, given the fact that
$(-r,\theta)\equiv(r,\theta-{180}\hbox{${}^{\circ}$})$, we must write (34) as
has been done in (35), hence (35) finds justification. This explanation can be
found in any good mathematics textbook that deals extensively with polar
coordinates. Hereafter, whenever a similar scenario arises where negative
values of $r$ emerge, we will automatically and without notification assume
that $(-r,\theta)$ is $(r,\theta-{180}\hbox{${}^{\circ}$})$ and this will come
with the introduction of the absolute value sign as has been done in (35).

Now, proceeding from where we left. As has already been explained at the
beginning of this section, we have to substitute the Mass Distribution
Function (MDF) $\mathcal{M}(r)$ into (35) and having done so we would have to
make $r$ the subject. It has been argued in equation $24$ of Paper III, that
for a MC that exhibits a density profile: $\rho(r)\propto r^{-\alpha_{\rho}}$,
where $\alpha_{\rho}$ is the density index, that the MDF is given by:

$\mathcal{M}(r)=\overbrace{\mathcal{M}_{csl}\left(\frac{r^{3-\alpha_{\rho}}-\mathcal{R}_{star}^{3-\alpha_{\rho}}}{\mathcal{R}^{3-\alpha_{\rho}}_{core}-\mathcal{R}_{star}^{3-\alpha_{\rho}}}\right)}^{\textrm{\tiny\bf{Circumstellar}\,{Mass}\,{Inside}\,{Region}\,{of
\,\,Radius}\,{r}}}+\overbrace{\mathcal{M}_{star}}^{\textbf{ {\tiny
Nascent\,Star's\,Mass}}}\,\,\,\,\,\,\,\,\,\textrm{for}\,\,\,\,\,\,r\geq\mathcal{R}_{star},$
(36)

where $\mathcal{M}_{csl}$ is total mass of the circumstellar material at any
given time, $\mathcal{R}_{star}$ is the radius of the nascent star at any
given time, $\mathcal{R}_{core}$ is the radius at any given time of the
gravitationally bound core from which the star is forming.

Now, substituting the MDF (given above) into (35) and thereafter making $r$
the subject of the formula would lead to a horribly complicated inequality
that would require the use of the Newton-Ralphson approach to solve. Since
ours in the present is but a qualitative analysis, we can make some very
realistic simplifying assumptions that can make our life much easier. If the
spatial extent of the star is small compared to that of the core i.e.:
$(\mathcal{R}_{star}\ll\mathcal{R}_{core}\Rightarrow\mathcal{R}_{core}^{3-\alpha_{\rho}}-\mathcal{R}_{star}^{3-\alpha_{\rho}}\simeq\mathcal{R}_{core}^{3-\alpha_{\rho}})$
and the mass of the star is small compared to the mass of the core i.e.:
$(\mathcal{M}_{star}\ll\mathcal{M}_{core}\Rightarrow\mathcal{M}_{csl}\simeq\mathcal{M}_{core})$,
then, the MDF simplifies to:

$\mathcal{M}(r)\simeq\mathcal{M}_{core}\left(\frac{r}{\mathcal{R}_{core}}\right)^{3-\alpha_{\rho}}.$
(37)

Inserting this into (35) and thereafter performing some basic algebraic
computations that see $r$ as the subject of the formula, one is lead to:

$r<\left[\lambda_{1}\left(\frac{2G\mathcal{M}_{core}}{c^{2}\mathcal{R}_{core}}\right)\mathcal{R}_{core}\right]\left|\cos\theta\right|^{\frac{1}{2-\alpha_{\rho}}}.$
(38)

Now, if we set:

$\epsilon_{1}^{core}=\left(\left[\lambda_{1}\left(\frac{2G\mathcal{M}_{core}}{c^{2}\mathcal{R}_{core}}\right)\right]^{\frac{1}{2-\alpha_{\rho}}}\right)\left(\frac{\mathcal{R}_{core}}{\mathcal{R}_{star}}\right),$
(39)

then (38) reduces to:

$r<\epsilon_{1}^{core}\mathcal{R}_{star}\left|\cos\theta\right|^{\frac{1}{2-\alpha_{\rho}}}=l_{max}\left|\cos\theta\right|^{\frac{1}{2-\alpha_{\rho}}},$
(40)

where: $l_{max}=\epsilon_{1}^{core}\mathcal{R}_{star}$. On the xy-plane as
shown in figure (2), the equation:
$r=l_{max}\left|\cos\theta\right|^{\frac{1}{2-\alpha_{\rho}}}$, describes two
lobs. For the purposes of this reading, let the volume of revolution of the
lob be called a loboid, and the loboid above the x-axis shall be called the
upper loboid, and likewise the loboid below the x-axis shall be called the
lower loboid.

Now, the condition: (Term II $<0$), implies: $[\theta~{}<~{}\cos^{-1}(\pm
1/\sqrt{3})]$, which means: $(-54.7<\theta<54.7)$. Now, for the azimuthal
component to be repulsive, we must have: $[g_{\theta}(r,\theta)>0]$, we will
have from (33), the condition:

$r>-\left(\frac{9\lambda_{2}}{2\lambda_{1}^{2}}\right)\left(\frac{2\lambda_{1}G\mathcal{M}(r)}{c^{2}}\right)\cos\theta.$
(41)

Now going through the same procedure as above, (41) can be written as:

$r>l_{min}\left|\cos\theta\right|^{\frac{1}{2-\alpha_{\rho}}},$ (42)

where:

$l_{min}=\left(\left|\frac{9\lambda_{2}}{2\lambda_{1}^{2}}\right|^{\frac{1}{2-\alpha_{\rho}}}\right)l_{max}.$
(43)

Thus, coalescing the results, invariably, one is led to conclude that the
region of repulsive gravitation is:

$\left[l_{min}<r<l_{max}\right]\,\textit{\&}\,\left[\cos^{-1}\left(-\frac{1}{\sqrt{3}}\right)<\theta<\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)\right].$
(44)

In the region described above, the gravitational field is both radially and
azimuthally repulsive, that is, there is complete gravitational repulsion in
this region. Pictorially, a summary of the emergent picture of the repulsive
gravitational field in shown in figure (2). This picture – in our view, fits
the description of outflows, the limiting factors are the sizes of $l_{max}$
and $l_{min}$, these values all depend on the one parameter $\lambda_{1}$,
hence thus, this parameter is the crucial parameter which determines the
properties of outflows. Shortly, we will discuss this picture but before this,
it is necessary that we go through the empty space solutions as-well.

Figure (2): This figure illustrates the emergent picture from the azimuthally
symmetric considerations of the Poisson equation. While fanning out matter in
the region of repulsive gravitation, the rotating star is surrounded by an
equatorial disk; once the outflow switches-on, this disk is the only channel
via which the mass of the star feeds. The disk is not affected by radiation in
the sense that some of its material close to the nascent star will be swept
away by the radiation field, no! The force of gravity along this disk is
purely radial and is directed toward the nascent.

###  4.2 Empty Space Solutions

As will be demonstrated in this section, the picture imaging from the empty
space solution is not different from that of the non-empty space solution.
However, there is an important difference between these two pictures and this
difference need to be stated. If our spinning gravitating body is not giving
off material like the Sun, then the region of repulsive gravitation will occur
inside the this body. We shall consider the star to be a point mass i.e., all
of its mass is concentrated at the star’s center of mass.

As before, from (32) and (33), it is clear that regions of repulsive
gravitation will exist and these will occur where [$g_{r}(r,\theta)<0$] and or
[$g_{\theta}(r,\theta)~{}<~{}0$]. We shall as before start by treating the
case [$g_{r}(r,\theta)~{}<~{}0$]. From (32), if [$g_{r}(r,\theta)<0$], then
(Term I $<0$) and (Term II $<0$) as well. The condition (Term I $<0$) implies:

$r<-\lambda_{1}\left(\frac{2G\mathcal{M}}{c^{2}}\right)\cos\theta,$ (45)

where in the present case $\mathcal{M}(r)$ must be replaced by
$\mathcal{M}_{star}$ and this can be written in the equivalent form:

$r<\lambda_{1}\left(\frac{2G\mathcal{M}_{star}}{c^{2}}\right)\left|\cos\theta\right|.$
(46)

Now, if we set:

$\epsilon_{1}^{star}=\lambda_{1}\left(\frac{\mathcal{R}_{star}^{s}}{\mathcal{R}_{star}}\right),$
(47)

where $\mathcal{R}_{star}^{s}=2G\mathcal{M}_{star}/c^{2}$ is the Schwarzchild
radius of the star, then (46) reduces to:

$r<\epsilon_{1}^{star}\mathcal{R}_{star}|\cos\theta|=l_{max}|\cos\theta|.$
(48)

Now, the condition (Term II $<0$), as before, implies:
$[\theta~{}<~{}\cos^{-1}(\pm 1/\sqrt{3})]$ , which means:
$(-54.7<\theta<54.7)$. Again as before, for the azimuthal component to be
repulsive, we must have: $[g_{\theta}(r,\theta)>0]$, we will have from (33),
that:

$r>\left(\frac{9\lambda_{2}}{2\lambda_{1}^{2}}\right)\left(\frac{2\lambda_{1}G\mathcal{M}_{star}}{c^{2}}\right)\left|\cos\theta\right|,$
(49)

and we need not explain anymore why the above can be written as:

$r>l_{min}|\cos\theta|,$ (50)

where this time:

$l_{min}=\left|\frac{9\lambda_{2}}{2\lambda_{1}^{2}}\right|l_{max}.$ (51)

Coalescing the results, invariably, one is led to conclude that the region of
repulsive gravitation is:

$\left[l_{min}<r<l_{max}\right]\,\textit{\&}\,\left[\cos^{-1}\left(-\frac{1}{\sqrt{3}}\right)<\theta<\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)\right].$
(52)

As in the case of the non-empty space, in the region described above, the
gravitational field is both radially and azimuthally repulsive, hence there is
complete gravitational repulsion in this region. The emergent picture is no
different from that of the case of non-empty space. The important difference
is that the region of gravitational repulsion is confined in the interior of
the star if $(\epsilon_{1}<1)$, it is not visible outside. If
$(\epsilon_{1}<1)$, there will exist no repulsive bipolar gravitational field
that is visible to beyond the surface of the spinning star. In the interior of
the star, the solutions obtained for the case of non-empty space is what must
apply.

##  5 ASGF of a Spinning Core with an Embedded Spinning Star

Central to the ASTG is that the material under consideration possesses some
finite spin angular momentum. In the case of a nascent star embedded inside a
gravitationally bound core, we are going to have the star’s spin angular
frequency being different to that of the circumstellar material; because, in
the early stages when the nascent star is forming, the spin angular frequency
of the circumstellar material and the star will, on the average, be the same
since it is expected that circumstellar material and the star will co-rotate;
but, because of the increasing mass and spin angular momentum of the nascent
star due to the accretion of material, at some-point, the star must break-off
from this co-rotational motion and spin independently of the circumstellar
material, thus in the end, the star will have a different spin angular
frequency to that of the circumstellar material. The different spin angular
momentum of the nascent star and the circumstellar material will come along
with different $\lambda$-values. Assuming the circumstellar material is co-
rotating with itself, it must have its own $\lambda$-value, let us call this
$\lambda_{\ell}^{csl}$ and that for the star be $\lambda_{\ell}^{star}$.

If there is a way of calculating the ASGF of the star at point $(r,\theta)$
and that of the circumstellar material at that same point $(r,\theta)$, then
one will be able to calculate the resultant ASGF at any point $(r,\theta)$
because the gravitational field is here a scalar. Let $\Phi_{star}$ be the
Azimuthally Symmetric Gravitational Potential (ASGP) of the star and that of
the circumstellar material be $\Phi_{csl}$. Knowing $\Phi_{star}$ and
$\Phi_{csl}$, clearly the resultant ASGP $\Phi_{eff}$ at any point
$(r,\theta)$ is $\Phi_{eff}=\Phi_{star}+\Phi_{csl}$, hence one will be able to
obtain the resultant ASGF. The ASGF of the star is not difficult to obtain, we
already know that it must be given by:

$\Phi_{star}=-\frac{G\mathcal{M}_{star}}{r}\left[1+\frac{\lambda_{1}^{star}G\mathcal{M}_{star}\cos\theta}{rc^{2}}+\lambda_{2}^{star}\left(\frac{G\mathcal{M}_{star}}{rc^{2}}\right)^{2}\left(\frac{3\cos^{2}\theta-1}{2}\right)\right].$
(53)

Now, we have to obtain the ASGF of a spinning core. The gravitational
potential (31) is the potential of star that is co-rotating with the
circumstellar material. If we remove the central star from this gravitational
potential what remains is the gravitational potential of a spinning core.
Removing the central star from this potential means set
$\mathcal{M}_{star}=0$, hence, the gravitational potential of a spinning core
must be:

$\Phi_{csl}=-\frac{G\mathcal{M}_{csl}(r)}{r}\left[1+\frac{\lambda_{1}^{csl}G\mathcal{M}_{csl}(r)\cos\theta}{rc^{2}}+\lambda_{2}^{csl}\left(\frac{G\mathcal{M}_{csl}(r)}{rc^{2}}\right)^{2}\frac{3\cos^{2}\theta-1}{2}\right],$
(54)

where $\lambda_{\ell}^{csl}$ is the $\lambda_{\ell}$-value for the spinning
circumstellar material and:

$\mathcal{M}_{csl}(r)=\mathcal{M}_{csl}\left(\frac{r^{3-\alpha_{\rho}}-\mathcal{R}_{cav}^{3-\alpha_{\rho}}(t)}{\mathcal{R}^{3-\alpha_{\rho}}_{core}(t)-\mathcal{R}_{cav}^{3-\alpha_{\rho}}(t)}\right)\,\,\textrm{for}\,\,r\geq\mathcal{R}_{cav}(t),$
(55)

is the circumstellar material enclosed in radius $r$. Now, as argued already:
$\Phi_{eff}=\Phi_{star}+\Phi_{csl}$, thus adding these two potentials (i.e. 53
& 54), one obtains:

$\Phi_{eff}(r,\theta)=-\sum^{\infty}_{\ell=0}c^{2}\left(\frac{G\left\\{\lambda_{\ell}^{star}\mathcal{M}_{star}^{\ell+1}+\lambda_{\ell}^{csl}\mathcal{M}_{csl}^{\ell+1}(r)\right\\}^{\frac{1}{{}^{\ell+1}}}}{rc^{2}}\right)^{\ell+1}P_{\ell}(\cos\theta).$
(56)

This is the ASGP of a star that spins independently from its core. For
convenience, we can write:
$\mathcal{M}^{eff}_{\ell}(r)=\left\\{\lambda_{\ell}^{star}\mathcal{M}_{star}^{\ell+1}+\lambda_{\ell}^{csl}\mathcal{M}_{csl}^{\ell+1}(r)\right\\}^{\frac{1}{{}^{\ell+1}}}$,
and call this the effective gravitational mass for the $\ell^{th}$
gravitational-pole. By $\ell^{th}$ gravitational-pole, it shall be understood
to mean the $\ell^{th}$-term in the gravitational potential term. This means
the above can be written in the clearer and simpler form:

$\Phi_{eff}(r,\theta)=-\sum^{\infty}_{\ell=0}c^{2}\left(\frac{G\mathcal{M}^{eff}_{\ell}(r)}{rc^{2}}\right)^{\ell+1}P_{\ell}(\cos\theta).$
(57)

To second order approximation, this potential is given by:

$\Phi_{eff}=-\left(\frac{G\mathcal{M}_{0}^{eff}(r)}{r}\right)\left[1+\gamma_{1}\lambda_{1}^{star}\left(\frac{G\mathcal{M}_{1}^{eff}(r)\cos\theta}{rc^{2}}\right)+\gamma_{2}\lambda_{2}^{star}\left(\frac{G\mathcal{M}_{2}^{eff}(r)}{rc^{2}}\right)^{2}\left(\frac{3\cos^{2}\theta-1}{2}\right)\right],$
(58)

where: $\gamma_{\ell}=\mathcal{M}_{\ell}^{eff}(r)/\mathcal{M}_{0}^{eff}(r)$.
We shall assume this ASGP for a star that spins independently from its core.

##  6  Outflow Power

Clearly, we do have from the ASTG regions of repulsive gravitation whose shape
is similar to that seen in outflows. If these outflows are really powered by
gravity, the question is: does the gravitational field have that much energy
to drive these and if so, where does this energy come from? To answer this
question, one will need to know the dominant radial component of the
gravitational force since outflows dominantly operate along the radial
direction. Clearly, one of the new extra poles in the gravitational field must
be the cause of the outflows since without them, there are no outflow. For our
investigations, the correct gravitational potential to use is (57) and of
interest in this potential is the gravitational potential of the star. This
invariably means we are looking at (53). So doing, one sees that the first
order term (involving $\lambda_{1}$) is an all-repulsive term as already
argued while the second order them (involving $\lambda_{2}$) is repulsive and
attractive, it depends on the region under consideration.

Now, to ask what powers outflows amounts to asking: “What is their energy
source?”. If this energy source is the gravitational field, then, we know that
the energy stored in the gravitational field whose potential is described by
$\Phi(r,\theta)$, is given by:

$\mathcal{E}^{star}_{gpe}(r)=\int^{\mathcal{M}_{star}}_{0}\int^{\Phi(\infty,2\pi)}_{\Phi(r,0)}d\Phi(r,\theta)d\mathcal{M},$
(59)

and plugging into the above the ASGP, and thereafter performing the
integration, one is led to:

$\mathcal{E}^{star}_{gpe}(r)=-\frac{G\mathcal{M}_{star}^{2}}{2r}\left[1+\frac{\lambda_{1}G\mathcal{M}_{star}}{rc^{2}}+\lambda_{2}\left(\frac{G\mathcal{M}_{star}}{rc^{2}}\right)^{2}\right],$
(60)

and using the fact that
$\mathcal{L}_{star}=\mathcal{L}_{\odot}\left(\mathcal{M}_{star}/\mathcal{M}_{\odot}\right)^{3}$,
one is further led to:

$\mathcal{E}^{star}_{gpe}(r)=-\frac{G\mathcal{M}_{\odot}^{2}}{2r}\left(\frac{\mathcal{L}_{star}}{\mathcal{L}_{\odot}}\right)^{\frac{2}{3}}\left[1+\frac{\lambda_{1}G\mathcal{M}_{\odot}}{rc^{2}}\left(\frac{\mathcal{L}_{star}}{\mathcal{L}_{\odot}}\right)^{\frac{1}{3}}\right].$
(61)

If $\mathcal{M}_{out}$ is the mass of the outflow at position $r$ and
$V_{out}$ is the speed of this outflow at this position and
$\mathcal{K}_{out}$ is the kinetic energy, we know that:

$\left<\frac{d\mathcal{M}_{out}(r)}{dt}\right>=\frac{1}{V_{out}^{2}}\frac{d\left[\mathcal{M}_{out}(r)V_{out}^{2}\right]}{dt}=\frac{2}{V_{out}^{2}}\frac{d\mathcal{K}_{out}}{dt},$
(62)

where the bracket $\left<[]\right>$ tells us that we are looking at the
average. Now if the gravitational energy $\mathcal{E}^{star}_{gpe}(r)$ is
equal to the kinetic energy of the outflow, then, from the above and, coupled
with the said, one is led to:

$\left<\frac{d\mathcal{M}_{out}(r)}{dt}\right>=-\frac{\tau_{G}G\mathcal{M}_{\odot}^{2}V_{out}^{-2}}{r}\left(\frac{\mathcal{L}_{star}}{\mathcal{L}_{\odot}}\right)^{\frac{2}{3}}\left[1+\frac{2\lambda_{1}G\mathcal{M}_{\odot}}{rc^{2}}\left(\frac{\mathcal{L}_{star}}{\mathcal{L}_{\odot}}\right)^{\frac{1}{3}}\right],$
(63)

where $\tau_{G}=\dot{G}/G$ and $\dot{G}$ is the time derivative of the
Newton’s gravitational constant. In the derivation of the above, we have
considered only first order terms and we have assumed that the gravitational
constant is not a constant. Evidence that the gravitational constant maybe
changing exists e.g. see Pitjeva ([$2005$]) and references therein. The ASTG
also points to a variation of the gravitational constant and the details of
this are being worked out444We are at an advanced stage of preparation of this
work and it will soon be archived on viXra.org: check Golden Gadzirayi
Nyambuya’s profile. Title of the Paper: A Foundational Basis for Variable-G
and Variable-c Theories. and we give in the subsequent paragraphs how this
comes about.

As it stands, the Poisson equation ($\vec{\nabla}^{2}\Phi=4\pi G\rho$) for a
time varying $\Phi$ & $\rho$, is not in conformity with the Relativity
Principle. According to our current understanding of physics and Nature, the
seemingly sacrosanct Relativity Principle is a symmetry that every Law of
Physics must fulfill. The Relativity Principle states that Laws of Physics
must be independent of the observer’s state of motion and as-well of the
coordinate system used to formulate them. If the Poisson equation is to be a
Law of Nature, then, it must successfully fulfill the Relativity Principle.
This means we must extend the Poisson equation to meet this requirement and
the most natural and readily available such is:

$\vec{\nabla}^{2}\Phi-\frac{1}{c^{2}}\frac{\partial^{2}\Phi}{\partial
t^{2}}=4\pi G\rho,$ (64)

where $t$ is the time coordinate. This equation satisfies the Relativity
Principle simply because it directly emerges from Einstein’s equation of the
General Theory of Relativity (GTR). We know Einstein’s GTR, specifically the
Law of Gravitation relating matter to the curvature of spacetime, does satisfy
the Relativity Principle; hence (64) too, satisfies the Relativity Principle.
This equation (i.e. 64) is what we are working out, we shall show that it
leads to a time variable $G$. So, as will be shown in the near future, the
time variable $G$ in (63) is not without a basis.

Now, from (63), one sees that: $\dot{\mathcal{M}}_{out}(r)\propto
V_{out}^{-2}(r)\mathcal{L}^{2/3}_{star}$. Given as stated in the introduction
that observations find: $\dot{\mathcal{M}}_{out}(r)\propto
V_{out}^{-1.8}\mathcal{L}^{0.6}_{star}$, which is close to what we have
deduced here; this points to the fact that the thesis leading to our
deduction: $\dot{\mathcal{M}}_{out}(r)\propto
V_{out}^{-2}(r)\mathcal{L}^{2/3}_{star}$, may very well be on the right path
of discovery. This clearly points to the need to look into these matters
deeper than has been done here.

From the above, clearly – a meticulous study of outflows should be able to
measure the time variation in the gravitational constant $G$ and this hinges
on the corrects of the ASTG. This would require higher resolution observations
to measure the mass outflow rate [$\dot{\mathcal{M}}_{out}(r)$] at position
$r$ from the star and as well the speed of the outflow at that point and
knowing the mass or luminosity of the central driving source, a graph of,
e.g.: $r\dot{\mathcal{M}}_{out}(r)$ vs
$V_{out}^{-2}(r)\mathcal{L}^{2/3}_{star}$, should in accordance with the ideas
above, produce a straight line graph whose slope is
$\tau_{G}G\mathcal{M}_{\odot}^{2}$. This kind of work, if it where possible,
it would help in making an independent confirmation of the measured time
variation of Newton’s constant of gravitation and it would act as further
testing grounds for the falsification of the ASTG.

##  7 Outflow Anatomy

Briefly, we shall look into the anatomy of the outflow. We say “briefly”
because each of the issues we shall look into requires a separate reading to
fully address them. First, before we do that, it is important to find out when
does the outflow switch-on and also when does it switch-off. That at some
point in time in the evolution of a star, outflows switch-on and off is not
debatable. So, before we even look into them, it makes perfect sense to
investigate this. From figure (2), we see that the anatomy of the outflow has
been identified with four regions, i.e. the Outflow Feed Region, the Outflow
Region and the Shock Ring. After investigating the switching-on and off of the
outflow, we will look into the nature of these regions. Our analysis is
qualitative rather than quantitative. We believe a quantitative analysis will
require a fully-fledged numerical code. Work on this numerical code is
underway.

###  7.1 Switching-on of Outflows

Let us call the loboid described by (40) the outflow loboid and likewise the
loboid described by (42) the outflow feed loboid. From the preceding section,
it is abundantly clear that we are going to have repulsive bipolar regions
whose surface is described by a cone and a outflow loboid section. From this,
we know that the maximum spacial extend of the repulsive gravitational field
region will be given by the maximum spatial length of the lobes which occurs
when: $\cos\theta=1$, i.e. $l_{max}=\epsilon_{1}^{star}\mathcal{R}_{star}$.
Now, to ask the question when does the outflow switch-on amounts to asking
when is $l_{max}$ equal to the radius of the star? because the repulsive
gravitational field will only manifest beyond the surface of the star if and
only if the maximum spatial extent of the region of repulsive gravitation is
at least equal to the radius of the star, i.e.:
$l_{max}~{}\geq~{}\mathcal{R}_{star}$, this means,
$l_{max}~{}=~{}\epsilon_{1}^{star}\mathcal{R}_{star}$; clearly, this will
occur when: $(\epsilon_{1}^{star}=1)$. Therefore, outflows will switch-on when
the condition: $(\epsilon_{1}=1)$, is reached, otherwise when:
$(\epsilon_{1}^{star}<1)$, the repulsive gravitational field is confined
inside the star.

This strongly suggests that if we are to use the ASGT to model outflows, then
we must think of $\epsilon_{1}$ (hence $\lambda_{1}$) as an evolutionary
parameter of the star i.e., this value starts of from a given absolute minimum
value $($say $\epsilon_{1}^{star}=0)$, and as the star evolves, this value
gets larger and larger until such a time that the repulsive gravitational
field is switched on when: $(\epsilon_{1}^{star}=1)$, and thereafter it
continues to grow and as it grows so does the spatial extend of the outflow
(since this parameter controls the spatial size of the region of the repulsive
gravitational field).

If the outflow switches on – as it must, the question is: “Why does it switch
on at that moment when it switches on and not at any other moment? What is so
special about that moment when it switches on that triggers it [outflows] to
switch on?” As we have already argued, this special moment is when
$(\epsilon_{1}^{core}=1)$ for a star that co-rotates with its parent core and
$(\epsilon_{1}^{star}=1)$ for a star that rotates independently of its parent
core. From equation (39 and 47), this means we must have:

$\epsilon_{1}^{core}=\left[\zeta\left(\frac{4\pi^{2}\mathcal{R}_{core}^{3}}{G\mathcal{M}_{core}\mathcal{T}_{core}^{2}}\right)^{\zeta_{0}}\left(\frac{2G\mathcal{M}_{core}}{c^{2}\mathcal{R}_{core}}\right)\right]^{\frac{1}{2-\alpha_{\rho}}}\left(\frac{\mathcal{R}_{core}}{\mathcal{R}_{star}}\right)=1,$
(65)

and for a star that rotates independently of its core:

$\epsilon_{1}^{star}=\zeta\left(\frac{4\pi^{2}\mathcal{R}_{star}^{3}}{G\mathcal{M}_{star}\mathcal{T}_{star}^{2}}\right)^{\zeta_{0}}\left(\frac{2G\mathcal{M}_{star}}{c^{2}\mathcal{R}_{star}}\right)=1,$
(66)

where $(\mathcal{T}_{core},\mathcal{T}_{star})$ are the period of the spin of
the core and the star respectively. If $\mathcal{T}_{core}^{on}$ is the period
of the core’s spin when the outflow switches on and $\mathcal{T}_{star}^{on}$
is the period of the spin when the outflow switches on, then, from the above
equations, it follows that:

$\mathcal{T}^{on}_{core}=\left(\frac{\pi}{c}\right)\left[\zeta\left(\frac{2G\mathcal{M}_{core}}{c^{2}}\right)^{1-\zeta_{0}}\left(\mathcal{R}_{star}^{on}\right)^{\alpha_{\rho}-1}(\mathcal{R}_{core}^{on})^{3\zeta_{0}-\alpha_{\rho}+1}\right]^{\frac{1}{2\zeta_{0}}},$
(67)

$\mathcal{T}^{on}_{star}=\left(\frac{\pi}{c}\right)\left[\zeta\left(\frac{2G\mathcal{M}_{star}^{on}}{c^{2}}\right)^{1-\zeta_{0}}\left(\mathcal{R}_{star}^{on}\right)^{3\zeta_{0}-1}\right]^{\frac{1}{2\zeta_{0}}},$
(68)

where
$(\mathcal{M}_{star}^{on},\mathcal{R}^{on}_{star},\mathcal{R}^{on}_{core})$
are the mass and radius of the star and core at the time the outflow switches
on respectively. From this, it follows that if the Sun were to spin on its
axis once in every $7.70\pm 0.40\,\textrm{hrs}$ (i.e. $39.0\pm
2.00\,\mu\textrm{Hz}$), the bipolar repulsive gravitational field must switch
on and for the Earth, it would require it to spin once on its axis in every
$10.00\pm 2.00\,\textrm{min}$ (i.e. $1.80\pm 0.50\,\textrm{mHz}$). If the
above is correct, then the Earth must spin about one hundred and forty four
times its current spin in order to achieve the bipolar repulsive gravitational
field while the Sun must spin about five thousand six hundred its current spin
rate to achieve a bipolar repulsive gravitation. The spin rate of the Earth is
far less than that needed to cause the bipolar repulsive gravitational to
switch on thus polar bears can smile knowing they will not fly off into space
anytime soon.

We know that outflows are not always present, at some-point in the evolution
of the star, they switch-off. What could cause them to do so? Given the
reality that within the outflow loboid, there is the outflow feed loboid; this
too, grows in size as the outflow loboid grows; at some-point the outflow and
the outflow feed loboid will become equal – leaving the outflow with no feed
point. At this point when the outflow and outflow feed loboids become equal,
clearly, the outflow must switch-off. This occurs when $l_{max}=l_{min}$ and
from (43) this means the condition for this to occur is
$|\lambda_{2}|=2\lambda_{1}^{2}/9$ and given that
$\lambda_{2}=-\lambda_{1}/96$, this means $\lambda_{1}^{off}=9/192$. From
(39), it follows that:

$\lambda_{1}^{off}=\zeta\left(\frac{4\pi^{2}\mathcal{R}_{star}^{3}}{G\mathcal{M}_{star}\mathcal{T}_{star}^{2}}\right)^{\zeta_{0}}\left(\frac{2G\mathcal{M}_{star}}{c^{2}\mathcal{R}_{star}}\right)=\frac{9}{192},$
(69)

this implies:

$\mathcal{T}^{off}_{star}=\left(\frac{192}{9}\right)^{\frac{1}{2\zeta_{0}}}\left(\frac{\pi}{c}\right)\left[\zeta\left(\frac{2G\mathcal{M}_{star}^{off}}{c^{2}}\right)^{1-\zeta_{0}}\left(\mathcal{R}_{star}^{off}\right)^{3\zeta_{0}-1}\right]^{\frac{1}{2\zeta_{0}}},$
(70)

where likewise $(\mathcal{M}_{star}^{off},\mathcal{R}^{off}_{star})$ are the
mass and the radius of the star at the time when the outflow switches off. We
expect that $\mathcal{T}^{on}_{star}>\mathcal{T}^{off}_{star}$. If this is to
hold, then:

$\left(\frac{\mathcal{M}_{star}^{off}}{\mathcal{M}_{star}^{on}}\right)^{1-\zeta_{0}}\left(\frac{\mathcal{R}^{off}_{star}}{\mathcal{R}^{on}_{star}}\right)^{3\zeta_{0}-1}<\left(\frac{9}{192}\right)^{\frac{2}{5}}=0.30.$
(71)

Hence, outflow activity will take place when:
$(\mathcal{T}^{off}_{star}\leq\mathcal{T}_{star}\leq\mathcal{T}^{on}_{star})$.
When $\mathcal{T}_{star}=\mathcal{T}^{off}_{star}$, we have:
$\epsilon_{1}=9\mathcal{R}_{star}^{s}/192\mathcal{R}_{star}$. Using the
approximate relation for an accreting star:
$\mathcal{R}_{star}~{}\sim~{}61\mathcal{R}_{\odot}\left(\mathcal{M}_{star}/\mathcal{M}_{\odot}\right)$,
one arrives at: $\epsilon_{1}=3.32\times 10^{6}$. This means:
$(\epsilon_{1}^{on}=1)$, and: $\epsilon_{1}^{off}=3.32\times 10^{6}$, where
$\epsilon_{1}^{on}$ and $\epsilon_{1}^{off}$, are the values of $\epsilon_{1}$
when the outflow switches on and off respectively, hence thus outflow activity
will take place during which period when:

$1\leq\epsilon_{1}<3.32\times 10^{6}.$ (72)

The emerging picture is that $\mathcal{T}$ gets larger and larger as the star
accretes more and more matter until a peak moment is reached (most probably
when the star stops growing in mass) where upon the spin begins to slow down,
in which process of slowing down the inner cavity inside the lob of the
outflow is created. This inner cavity grows bigger and bigger as the star’s
spin slows down, until such a time when the spatial dimensions of this cavity
is equal to the outflow lobe itself. Once this state is attained, the outflow
switches off because the growing cavity has – eaten up from within, all the
outflow region.

Clearly, the above picture suggests that the spin of a star is what controls
outflows, at some specific state, the outflow switch’s-on; it evolves to some
peak spin-value; thereafter, its spin slows down. This means that during the
outflow process after the begins to slow down, the star loses some spin
angular momentum. This idea resonates with the long held suggestion discussed
earlier that outflows are thought to exist as one means to tame the spin
angular momentum of a star (see e.g. Larson $2003b$). We will not go deeper
than this in our analysis. The aim has been to show that the emergent picture
of outflows from the ASTG is capable (in principle) to answer such questions.
This means in a future study, these are the things to look forward to.

###  7.2 Outflow Feed Region

In the Outflow Feed Region – i.e. the region in figure (2) described OEF and
OGH, clearly, any material that enters this region is going to be channeled
into the Outflow Region because the repulsive radial component of the
gravitational field (aided by the radiation field) is going to channel this
matter radially outward while the azimuthal component is going to going to
channel this outward radially moving material toward the spin axis, hence it
is expected that most of the matter will enter the Outflow Region along the
the spin axis of the star. It is important to state that no matter the
radiation from the star, there will be no reversal of in-falling matter
outside the region of repulsive gravitation due to the radiation field of the
nascent star – we shall discuss this in §$(8)$.

###  7.3 Outflow Region

The Outflow Region is comprised of a section of a cone (OAB & OCD), the
outflow loboid minus the Outflow Feed Region. In this Outflow Region, the
gravitational force is both radially azimuthally repulsive i.e.,
$(g_{\theta}>0)$ and $(g_{r}>0)$. This means, once the repulsive gravitational
force is switched-on and it is in a fully fledged phase, all material found in
this region is going to be channeled out of this region radially along with
most of the matter concentrated along the spin axis. The material will be
concentrated along the spin axis because the repulsive azimuthally
gravitational component will channel toward the spin axis. The repulsive
radial component pushes the material out radially, while the repulsive
azimuthal component of the gravitational force draws this material close the
spin axis hence the bulk of the outflow material must be found along the edge
spin axis.

Where the cone meets the outflow loboid i.e., along AB and CD, there is going
to be rings. Considering the ring AB, it is clear that this ring (as CD) must
be a shock front since on this ring, along the radial line OA, the in-coming
material will meet the outgoing material with equation but opposite radial
forces. This equal and but opposite forces must create (radially) a stationery
shock. This shock is going to have a ring structure – let us call this the
Shock Ring. As the rings AB & CD, EF & GH will be rings too, but not shock
things. These rings EF & GH are the mouth of the outflow and matter enters in
to the outflow region via this opening.

###  7.4 Shock Rings and Methanol Masers

Given that $(1):$ AB & CD are shock rings, $(2):$ that methanol masers
(amongst other pumping mechanisms) are thought to arise in shock regions and
$(3):$ the observations of Bartkiewicz et al. ([$2005$]) where these authors
discovered a ring distribution of $6.7\,\textrm{GHz}$ methanol masers, it is
logical to assume that this shock ring may well be a hub of methanol masers
arising from the shock present on this ring. Recent and further work by these
authors strongly suggests that a Ring of Masers is a natural occurrence in
star forming regions as (Bartkiewicz et al. [$2009$]).

This ring distribution of masers components, they believe strongly suggests
the existence of a central source – this is the case here, the central source
must exists and it is the forming star. They found an infrared object
coinciding with the center of the ring of masers within $78\,mas$ and this
source is cataloged in the 2MASS survey as 2MASS183451.56-08182114. They
believe this is an evolving evolving protostar driving this masers via
circular shocks – this is in line with the the present. Very strongly, the
Bartkiewicz Ring of Masers suggests – in our opinion that; our outflow model
may very well contain an element of truth, that our model contains the
possible seeds of resolution of this puzzling occurrence of Ring Masers.

About this shock ring; when viewed from the projection as shown in figure (2),
the distance of the shock ring from the star will be:

$l_{sh}=l_{max}(3)^{-\frac{3-\alpha_{\rho}}{4-2\alpha_{\rho}}},$ (73)

and the radius of this shock ring will be:

$\mathcal{R}_{ring}=l_{max}(1.5)^{-\frac{3-\alpha_{\rho}}{4-2\alpha_{\rho}}}.$
(74)

Clearly, for an isolated system, depending on the orientation relative to the
observer, this ring can appear as a linear structure, a circular or an
elliptical ring.

At present more than $500$ $6.7\,\textrm{GHz}$ methanol masers sources are
known to exist (Malyshev & Sobolev [$2003$]; Pestalozzi et al. [$2005$]; Xu et
al. [$2003$]) and are associated with a very early evolutionary phase of high
mass star formation. The methanol maser emitting at the $6.7\,\textrm{GHz}$
frequency first discovered by [$1991$] is the second strongest centimeter
masing transition of any molecule (after the $22\,\textrm{GHz}$ water
transition) and is commonly found toward star formation regions. It is
typically stronger than $12.2\,\textrm{GHz}$ methanol masers (discovered by
Batrla et al. [$1987$]) observed toward the same region. Methanol masers have
become well established tracers or sign spots of high mass star formation
regions. It is thought that methanol masers occur in the very early stages of
massive star formation.

While methanol masers are found in regions of massive star formation, some
have been found with no associated high mass star formation actively (see e.g.
Ellingsen et al. [$1996$], Szymczak et al. [$2002$]. Besides this non-
association, some methanol masers are and have been observed to exist in close
spatial proximity of massive stars. This has lead to the classification of
methanol masers into Class I and Class II. Class I masers emit at the
frequencies $25.0$, $44.0$, $36.0\,\textrm{GHz}$ etc while class II methanol
masers emit at $6.7$, $12.2$, $157.0\,\textrm{GHz}$ etc methanol masers is
classified as Class II. Class I methanol masers are often observed to exist
apart from the continuum sources , while Class II are observed to exist very
close, albeit, both classes often co-exist in the same star forming region
inside an HII regions (e.g. Sobolev et al. [$2004$]). Clearly,
$l_{sh}=l_{sh}(t)$ and $\mathcal{R}_{ring}=\mathcal{R}_{ring}(t)$ and as the
star evolves, $l_{sh}$ and $\mathcal{R}_{ring}$ get larger. This means in the
case of young stars, if this ring is a hub of methanol masers, it is expected
that methanol masers will be found closer to the star for young HMS and
likewise, for more evolved massive stars, methanol masers will be found
further from the nascent star. If this is correct, then it may explain the
aforesaid; why Class II methanol masers are mostly found close to the nascent
star and why Class I methanol masers are found existing further from the
nascent star.

High resolution imaging of the $6.7$ and $12.2\,\textrm{GHz}$ methanol masers
has found that many exhibit a simple elongated linear or curved spatial
morphologies (Norris et al. [$1988$]; Norris et al. [$1993$]; Minier et al.
[$2000$]) and as already stated, depending on the orientation of the observer
relative to the star forming system, the ring may appear as a linear
structure. These linear structures have lengths of $50$ to
$1300\,\textrm{AU}$. Because of this, one of the possible interpretations that
has been entertained for sometime is that the masers originate in the
circumstellar accretion disc surrounding the newly formed star (Edris et al.
[$2005$]) and besides this; because of their strong association with outflows
(see e.g. Plambeck & Menten [$1990$]; Kalenskii et al. [$1992$]; Bachiller et
al. [$1995$]; Johnston et al. [$1992$]), other than originating from the
circumstellar disk, also, it has been entertained that methanol maser may
originate from outflows (see e.g. Pratap & Menten [$1992$]; de Buizer et al.
[$2000$]). Clearly, the outflow origin of methanol masers resonates with the
present ideas. If the ideas herein are correct, then, this reading would of
value to researchers seeking an outflow origin of methanol masers.

Further, if viewed from the same view as in figure (2), and if as argued above
that masers are found on the ring, one will expect to observe a linear
alignment of masers above and below the the nascent star. This would explain
the observed linear alignment of methanol masers and also the observed linear
alignment of masers above an below the IRAS source found in molecular cloud
G69.489-0.785 (see Fish [$2007$]). Given Fish’s observations of blue and red-
shifted masers in the ON1-region (Fish [$2007$]), the suggested model of this
ring of masers is interesting as it may offer an explanation of this
unexplained and puzzle of red and blue-shifted masers at opposite sides of the
IRAS source associated with ON1.

###  7.5  Collimation Factor

We can calculate the collimation factor of the outflow since we know the
extent ($l_{max}$) and the breath of the outflow which is the size of the
shock rings i.e., the collimation factor could be:
$q_{col}=\mathcal{R}_{ring}/l_{max}$, which can also be written as:

$q_{col}=(1.5)^{\frac{3-\alpha_{\rho}}{4-2\alpha_{\rho}}},$ (75)

(this has been deduced from equation 74). Now, it is believed that the most
stable density profile is one with a density index $(\alpha_{\rho}=2)$, this
means molecular clouds in a state different from this density profile will
tend to it. Using this assumption, we see that as: $(\alpha_{\rho}\longmapsto
2)$ from $(\alpha_{\rho}=0)$, i.e. $(\alpha_{\rho}:0\longmapsto 2)$, then we
will have: $(q_{col}\longmapsto\infty)$. For this setting, generally:
$(q_{col}>1)$. We also realize that now as: $(q_{col}\longmapsto\infty)$,
when: $(\alpha_{\rho}:3\longmapsto 2)$, then: $(q_{col}>1)$, and if:
$(\alpha_{\rho}:0\longmapsto 2)$. For this setting, generally: $(q_{col}\geq
1.36)$. This means we are going to have two categories of collimation factor
value i.e. $(1<q_{col}<1.36)$ for $(\alpha_{\rho}:0\longmapsto 2)$ and
$(q_{col}\geq 1.36)$ for $(\alpha_{\rho}:3\longmapsto 2)$. Because of
projection effects, it is very difficult to measure the true collimation
factor.

Also, because of projection effects, the collimation factor that we measure in
real life is not the actual collimation factor but the projected collimation
factor. If we know the actual collimation factor, we will be able to know the
density index since from (75) we can deduce that:

$\alpha_{\rho}=2-\left(\frac{\log q_{col}^{2}}{\log
1.5}-1\right)^{-1}=\frac{\log(q_{col}^{4}/8)}{\log(1.5)}.$ (76)

LMSs are known to have relatively low outflow collimation factors
($q_{col}<2$) while HMSs have significantly high outflow collimation factors
($2<q_{col}<10$), sometimes reach $q_{col}\sim 20$. From (76) the aforesaid
implies, assuming these collimation factors are a good representation of the
real collimation factor, that LMSs cores have density index
$\alpha_{\rho}=1.56$ and HMS cores have density index $\alpha_{\rho}=1.98$.
This is not unreasonable but very much expected. The fact that for HMS forming
cores, we have $\alpha_{\rho}=1.98$ and for LMS forming cores we have
$\alpha_{\rho}=1.56$, means HMS cores are much more dense compared to LMS
forming cores.

##  8 Radiation Problem

While the main thrust and focus of this reading is not on the Radiation
Problem associated with massive stars, but on the polar repulsive
gravitational field and its possible association with the observed bipolar
molecular outflows, we find that the ASTG affords us a window of opportunity
to visit this problem. This so-called radiation problem associated with
massive stars has been well articulated in Paper III. There is no need for us
to go through the details of this same problem here but we shall direct the
reader to Paper III for an exposition of the radiation problem. In the
subsequent paragraphs, we shall – for the sack of achieving a smooth
continuous reading; present the findings of Paper III in nutshell.

In general, a massive star is defined to be one with mass greater than $\sim
8-10\mathcal{M}_{\odot}$ and central to the on-going debate on how these
objects [massive stars] come into being is this so-called radiation problem.
For nearly forty years, it has been argued that the radiation field emanating
from massive stars is high enough to cause a global reversal of direct radial
in-fall of material onto the nascent star. In Paper III, it is argued that
only in the case of a non-spinning isolated star does the gravitational field
of the nascent star overcome the radiation field. An isolated non-spinning
star is a non-spinning star without any circumstellar material around it, and
the gravitational field beyond its surface is described exactly by Newton’s
inverse square law. The supposed fact that massive stars have a gravitational
field that is much stronger than their radiation field is drawn from the
analysis of a non-spinning isolated massive star. In this case, the
gravitational field is (correctly) much stronger than the radiation field.
This conclusion has been erroneously extended to the case of non-spinning
massive stars enshrouded in gas and dust.

It is argued there, in Paper III, that, for the case of a non-spinning
gravitating body where the circumstellar material is taken into consideration,
that at $\sim 8-10\mathcal{M}_{\odot}$, the radiation field will not reverse
the radial in-fall of matter, but rather a stalemate between the radiation and
gravitational field will be achieved, i.e. in-fall is halted but not reversed.
Any further mass growth is stymied and the star’s mass stays constant at $\sim
8-10\mathcal{M}_{\odot}$. This picture is very different from the common
picture that is projected and accepted in the wider literature where at $\sim
8-10\mathcal{M}_{\odot}$, all the circumstellar material, from the surface of
the star right up to the edge of the molecular core, is expected to be swept
away by the all-marauding and pillaging radiation field. There in Paper III,
it is argued that massive stars should be able to start their normal stellar
processes if the molecular core from which they form has some rotation,
because a rotating core exhibits an ASGF which causes there to be an accretion
disk and along this disk the radiation is not powerful enough to pillage the
in-falling material. We show here that in the region:
($\theta:\,[125.3<\theta<54.7]$ & $[234.7<\theta<305.3]$), around a spinning
star the gravitational field in the face of the radiation field, will never be
overcome by the radiation field hence in-fall reversal does not take place in
this region and this region is the region via which the nascent massive star
forms once the repulsive outflow field and the star’s mass has surpassed the
critical $8-10\,\mathcal{M}_{\odot}$. Reiterating, in this region i.e.
($\theta:\,[125.3<\theta<54.7]$ & $[234.7<\theta<305.3]$), infall is never
halted but continues unaborted and unabated.

There are three cases of an embedded spinning nascent star $(1):$ Where the
nascent star is spinning and the circumstellar material is not spinning or
where the spin of the circumstellar material is so small compared to the star
so much that the circumstellar material can be considered to be not spinning.
$(2):$ Where the nascent star is spinning independently of the circumstellar
material which is itself spinning. $(3):$ Where the nascent star is co-
spinning or co-rotating with the circumstellar material. It should suffice to
consider one case because the procedure to show that in the region:
($\theta:\,[305.3<\theta<54.7]$ & $[234.7<\theta<125.3]$), infall is never
halted but continues unaborted and unabated, is the same. Of the three cases
stated, the most likely scenario in Nature is the second case i.e., where the
nascent star is spinning independently of the circumstellar material which is
itself spinning. We shall consider this case.

The ASGP for the case of a star that is spinning independently of its core has
be argued to be given by (58) and in the face of radiation field, the
resultant radial component of the gravitational field intensity is given by:

$g_{r}(r,\theta)=-\frac{G\mathcal{M}_{0}^{eff}}{r^{2}}\left[1-\frac{\kappa\mathcal{L}_{star}}{4\pi
G\mathcal{M}_{0}^{eff}c}+\frac{2\lambda_{1}^{star}\gamma_{1}G\mathcal{M}_{1}^{eff}\cos\theta}{rc^{2}}+3\lambda_{2}^{star}\gamma_{2}\left(\frac{G\mathcal{M}_{2}^{eff}}{rc^{2}}\right)^{2}\left(\frac{3\cos^{2}\theta-1}{2}\right)\right].$
(77)

For the radiation component to be attractive, we must have:
$[g_{r}(r,\theta)<0]$, and for this to be so, the term in the square brackets
must be greater than zero, this implies:

$\left[1-\frac{\kappa\mathcal{L}_{star}}{4\pi
G\mathcal{M}_{0}^{eff}c}\right]r^{2}+\left[\frac{2\lambda_{1}^{star}\gamma_{1}G\mathcal{M}_{1}^{eff}\cos\theta}{c^{2}}\right]r+\left[3\lambda_{2}^{star}\gamma_{2}\left(\frac{G\mathcal{M}_{2}^{eff}}{c^{2}}\right)^{2}\left(\frac{3\cos^{2}\theta-1}{2}\right)\right]>0.$
(78)

This inequality is quadratic in $r$ and can be written as: $(Ar^{2}+Br+C>0)$,
where: $A,B,$ and $C$, can easily be obtained by making a comparison.
Since555In Paper I, we argued that, $r$ can take both negative and positive
values, and further argued that the set up of the coordinate system of the
ASGF is such that [$r>0$ & $\cos\theta>0$] and [$r<0$ $\cos\theta<0$], hence
$r\cos\theta>0$, which implies $(Br>0)$.: $(Br>0)$, for: $(Ar^{2}+Br+C>0)$, to
hold absolutely, we must have: $(Ar^{2}>0\Rightarrow A>0)$ and $(C>0)$. The
condition: $(A>0)$, implies:

$\mathcal{M}(r)>\frac{\kappa_{eff}\mathcal{L}_{star}}{4\pi Gc}.$ (79)

To arrive at the above one must remember that:
$\mathcal{M}_{0}^{eff}=\mathcal{M}_{star}+\mathcal{M}_{csl}(r)=\mathcal{M}(r)$.
As shown in Paper II (see equation $5$ & §$5$ of Paper II), the condition (79)
for: $\mathcal{M}_{star}>8-10\,\mathcal{M}_{\odot}$, leads to the formation of
a cavity inside the star forming core. In this cavity, the radiation field in
powerful enough to halt infall reversal but outside of it, it is not.

Now, for the condition: $(C>0)$, to hold (remember $\lambda^{star}_{2}<0$),
this means: $(3\cos^{2}\theta-1<0)$, hence: ($\theta:\,[125.3<\theta<54.7]$ &
$[234.7<\theta<305.3]$). The result just obtained invariably means inside the
cavity created by the radiation field, the region:
($\theta:\,[125.3<\theta<54.7]$ & $[234.7<\theta<305.3]$) will have an
attractive gravitational field, hence matter will still be able to fall onto
the nascent star via this region and this in-falling of matter is completely
independent of the opacity of the material of the core! Hence we expect
spinning massive stars to face no radiation problem at all. Clearly, if:
$(\lambda_{2}>0)$, then in the region: ($\theta:\,[125.3<\theta<54.7]$ &
$[234.7<\theta<305.3]$), the gravitational field was going to cause in-fall
reversal in the cavity hence disallowing for the star to continues is
accretion. This obviously would have been at odds with experience hence thus
we have the strongest reason for setting: $(\lambda_{2}>0)$, otherwise the
ASTG would be seriously at odds with physical and natural reality as we know
it. Beside, the condition $(\lambda_{2}>0)$ is supported by the solar data
(see Paper I). The fact that in the region; ($\theta:\,[125.3<\theta<54.7]$ &
$[234.7<\theta<305.3]$), is a region of attractive gravitation, it is clearly
that the ASGF will form a disk around the nascent star. Although no detailed
study of accretion disks has been made (Brogen et al. [$2007$]; Araya et al.
[$2008$]) and this being due technological challenges in obtaining must higher
resolution observations on the scale of these accretion disks, it has long
been thought that the accretion disk is a means by which accretion of matter
on the nascent stars continues soon after radiation has (significantly)
sounded her presence on the star formation podium (see e.g. Chini et al.
[$2004$]; Beltr$\acute{\textrm{a}}$n et al. [$2004$]). If our investigation
prove correct, as we believe they will, then, researchers have been right to
think that that accretion disk serves a platform for further accretion of mass
by the nascent star.

##  9 Discussion and Conclusion

This reading should be taken more as a genesis that lays down the mathematical
foundations that seek to lead to the resolution of the problem of outflows,
vis, what their origin is. Also, we should say that, if this reading is
anything go by i.e., if it proves itself to have a real direct correspondence
with the experience of physical reality, then not only have we laid down the
mathematical foundations that may lead to the understanding of outflows; but
we have laid a three fold foundation that could lead to the resolution of
three problems, and these problems are:

>   1. (1).
>
> The Origins and Nature of Outflows
>
>   2. (2).
>
> The Radiation Problem thought to exist for HMS.
>
>   3. (3).
>
> The Origin of Linear & Ring Structures of Methanol Masers.
>
>

All this we have arrived at after the consideration of the azimuthal symmetry
arising from the spin of a gravitating body. This symmetry has been applied to
the gravitational field and where upon we have come up with the ASTG. In Paper
I, we did show that the ASTG can explain the perihelion shift of planets in
the solar system and therein, the ASTG as it lays there, suffers the setback
that the “constants” $\lambda_{\ell}$ are unknown. We have gone so far in the
present as to suggest a way to solve this problem but this suggestion is
subject to revision pending any new data.

It should be said that, to the best of what we can remember ever-since we
learnt that the force of gravity is what causes an apple to fall to the ground
and that the very same force causes the moon and the planets to stay in their
orbs; we have never really convinced of gravitation as being a repulsive
force, let alone that it possibly can have anything to do with the power
behind outflows. Just as anyone would find these ideas in violation of their
intuition, we find our-self in the same bracket. But one thing is clear, the
picture emerging from the mathematics thereof, is hard to dismiss. It calls
one to make a closer look at the what the Poisson equation is “saying to us”.

In closing, allow me to say that as things stand in the present – while we
firmly believe we have discovered something worthwhile; it is difficult to
make any bold conclusions. Perhaps we should only mention that work has began
on a numerical model of outflows based on what we have discovered herein. Only
then – we believe; it will be possible to make any bold conclusions.

##  Acknowledgments

I am grateful to my brother George and his wife Samantha for their kind
hospitality they offered while working on this reading and to Mr. Isak D.
Davids & Ms. M. Christina Eddington for proof reading the grammar and spelling
and Mr. M. Donald Ngobeni for the magnanimous support.

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