### Abstract

The origin of the Baryonic Tully-Fisher Relationship
(BTFR) M_{Bar} ∝
V^{4} remains a quandary since empirically discovered thirty years ago.
This relationship has been considered an exact function of galactic dynamics underpinning
Modified Newtonian Dynamics (MOND) or a ɅCDM baryonic nuisance constraint. Since
the BTFR is a central tool in astrophysical research, it is prudent to fully
understand its origin and basis for existence. This work leverages recent
availability of extended, “complete” rotation curves for the MW and M31 exhibiting
significant velocity declines just beyond their galactic disks. By modeling
outer velocities as Keplerian, galactic dynamic mass is equal to RV^{2}/G, where R is disk radius, and V, rotational
velocity. This equation is also governed by the general Newtonian relationship R
∝ M_{Dyn}/V^{2} and a dynamical relationship,
R ∝ V^{2}. This constrains galaxy morphologies to a
narrow band of physically permissible R-V for any given dynamic mass M_{Dyn}
and directly leads to the fundamental relationship M_{Dyn} ∝ V^{4}. This is linked to baryonic mass (BTFR) via the ratio of
dynamic/baryonic mass considered constant for the entire disk galaxy
population, recognized as a foundational element to explain the BTFR. Note that
the BTFR only relates the baryonic mass component to galactic rotation, and not
angular momentum and energy contributions. This model is superior to mainstream
propositions that rely on unproven beliefs rather than a solid mechanical
foundation. More importantly, this proposal is advantaged in that it is amenable
to the scientific method and subject to falsification.

### Introduction

A “state model” for rotationally supported galaxies is presented. The model is based on the application of simple mechanical law with respect to morphology and kinematics as obtained from very recent astrophysical data. Results indicate disk galaxies operate in a very confined window that serves as a “necessary precondition” to properly define the state (Curiel 2016). By definition, any dynamical galactic formation and evolution processes stemming from an incomplete or inaccurate state description must be suspect.

In this paper we report a significant breakthrough in
establishing a basic physical model for total galactic dynamic mass (M_{Dyn}).
The model is based on extended, accurate rotation velocity profiles for the
Milky Way (MW) and the Andromeda galaxy (M31). We then extend this model to an
irregular dwarf galaxy Andromeda IV (And IV) to span a wider range of galactic
mass. The model is designated Rotation Curve-Spin Parameter (RC-SP) model and
is based on a classical mechanical interpretation of dynamic mass. This
approach does not introduce new hypotheses and defines galactic state
parameters in terms of observables - rotation velocity, disk radius, and dynamic
mass. To maintain convention between the RC-SP model and the two mainstream
theories, ɅCDM and MOND, we define dynamic/baryon mass ratios as baryonic fraction
(f_{b}) and mass discrepancy (D), respectively* (Milgrom 2016)*.

We define each RC-SP parameter, and how as an integrated set, they are responsible for the phenomenon known as “missing mass.” The empirical specificity and precision of the BTFR faithfully reflects this mechanical solution relating rotation velocity (V) and disk radius (R) if and only if dynamic mass is treated as a point mass equivalent (e.g., Keplerian decline beyond the galactic disk proper). In following sections, we take a step-by-step approach in how each RC-SP galactic parameter is determined.

### Dynamic Mass (M_{Dyn})

A main issue for ɅCDM (and MOND) is the inability to properly define a galaxy’s observed dynamic mass properties in pure physical terms. RC-SP overcomes this deficiency by obtaining total galactic dynamic mass directly from extended rotation curves without need for further interpretation. Figure 1 and 2 demonstrate the process for the MW using two independent literature sources (Sofue 2015).

*Figure
**1**: MW “Grand Rotation Curve” data (black and gray). Average
rotation velocity (horizontal blue dash), Keplerian velocity profile fit (green
dash) and disk radius (black dash).Following the RC-SP method, the intersection
(open circle) is a RC-SP physical constraint defining the galactic state. The
average/flat rotation velocity is determined from compiled data (Pato 2015).Image source - Fig.3 (Sofue 2015)*

Figure 1 defines three RC-SP critical parameters; average
rotation velocity (V), break-point radius where Keplerian decline becomes
apparent (R), and (baryonic and dynamic) mass specified by the Keplerian
profile (M_{Dyn}). All needed RC-SP galactic parameters are obtained
from this single plot using the following three-step process:

1. Establish galactic dynamic mass (green dash) by modeling the outer galactic rotation curve as a conventional Keplerian “point-mass” equivalent,

2. Estimate average disk “flat” velocity (blue-dash) and extend this velocity until it intersects with the Keplerian curve,

3. Obtain associated disk radius (black-dash).

The RC-SP radius (R) is closely related to the outer “edge” of the HI gas disk where the break-point in velocity occurs. This is fortuitous as HI gas tends to accurately track gravitational potential well beyond the stellar disk proper (Lelli 2016). Using data from another survey, Figure 2 returns the same RC-SP parameters obtained in the previous figure (Huang 2016). We can be confident that these parameters truly reflect the physical state of the MW.

*Figure
**2**: MW rotation curve from a second source indicating an
average flat rotation velocity of 230 km/s (horizontal blue dash) transitioning
to a Keplerian decline (green dash) at 40 kpc (black dash. The dynamic mass is estimated
at 5x1010Mʘ (green dash), as in Figure 1. Image source - Fig. 11 (Huang 2016)*

The above figures exhibit a wide spectrum of velocity-radius combinations that can satisfy the observed dynamic mass. Although the MW exhibits a particular combination (230 km/s and 40 kpc), there are infinite velocity-radius combinations possible raising two serious concerns: Why does the MW exhibit this particular combination of rotation velocity and radius and not others? Are all disk galaxies similarity constrained to specific velocity-radius combinations similar to what is observed for the MW? These two questions must be answered in order to fully appreciate the mechanical solution proposed herein.

Next, we apply the same process to M31 and demonstrate the observed Keplerian decline is a universal feature for massive bright spirals. We then extend this approach to irregular dwarf galaxy And IV to demonstrate the universal nature of these galactic parameters. For example, Figure 3 provides the extended rotation curve for M31 from the same source that provided Figure 1 (Sofue 2015). In the figure below, M31’s RC-SP parameters are determined from the physical rotation curve.

*Figure
**3**: The RC-SP constraints are shown for M31’s rotation
curve (black-link). As with the MW, the open circle specifies the intersection
of the average velocity (horizontal blue dash), disk radius (vertical black
dash) and matched Keplerian velocity profile (green dash). The intersection is
denoted by the open circle, specific to M31. Image source - Fig.5 (Sofue 2015), inset Fig.11 (Chemin 2009)*

Sofue derived an average rotation velocity that is
significantly lower than 265 km/s selected for the RC-SP parameter. Rather, we
rely on the rotation curve shown in the inset from which M31’s dynamic mass was
estimated at 10x10^{11}M_{ʘ} (Chemin 2009). Curiously, Sofue’s
virial mass estimate (r<200 kpc) M_{200} = 13.9±2.6x10^{11}M_{ʘ}
is close to the RC-SP value of 14x10^{11}M_{ʘ}. This simple example
illustrates the state of uncertainty and confusion existing today related to
this fundamental galactic parameter. By defining dynamic mass in physically
observed concrete terms, RC-SP allows little room for wide-ranging estimates
plaguing theoretically-based “missing mass” models. This precision is
demonstrated as we apply The RC-SP prescription to an irregular dwarf galaxy
Andromeda IV (And IV). This galaxy had its rotation curve measured to just
beyond peak velocity, and includes an estimate of baryonic and total dynamic
mass (Karachentsev 2015). From these measures, a complete physical description
of the galaxy is possible under RC-SP, as presented in Figure 4 below.

*Figure
**4**: Andromeda IV RC-SP rotation velocity (blue dash), disk
radius (vertical black dash) and Keplerian velocity profile (green dash) are
shown in relationship to the rotation curve data (black open squares). Image
source - Fig. 8 (Karachentsev 2015)*

Figure 4 presents measured rotation velocity data on the
left, extending to ~8 kpc. From the data, the rotation curve has peaked and may
be transitioning to a decline beyond that radius. The same study also provided
a total dynamic mass estimate equal to 0.034x10^{11}M_{ʘ} (D=9.1,
f_{b}=0.11). This situation provides a unique opportunity to demonstrate
the utility of RC-SP over a wide range of galactic mass. In the above figure,
the observed dynamic mass is modeled as a point-mass equivalent (green-dash)
according to the three-step process described above. As with the two massive
spirals, the average (peak) velocity is proportional to V_{Peak}/V_{Bar}
= √D. The intersection of V_{Peak}-M_{Dyn} establishes the disk
radius (R). The open triangle denotes the singular solution meeting all three
physical constraints defining this dwarf galaxy. Interestingly, a recent
comparison of cold dark matter N-body simulations to observation indicate less
massive galaxies (V<100 km/s) do not have the highly extended HI gas disk
radii that is commonly assumed (Maccio 2016). Modeling the dynamic as a
Keplerian decline suggests And IV’s HI gas disk extends to 8 kpc and not far beyond.

In the next section, we apply the concept of spin parameter to equate disk galaxy morphology and kinematics to angular momentum and total energy (from which baryonic content can be derived). We quantify these properties for our three example galaxies utilizing RC-SP derived galactic parameters and extend these findings to the general disk galaxy population. This approach is motivated by a recent investigation into galactic surface density distribution (Marr 2015).

### Angular Momentum (J) and Total Energy (E)

The RC-SP model is based on classical methods and heavily relies on the dimensionless spin parameter equation. This equation relates the contribution of angular momentum and energy to the total dynamic mass of the galaxy. The Peebles derivation is given below (Peebles 1969):

Where *J* = Angular Momentum, *E* = Total Energy, *M _{Dyn}* = Dynamic mass, and

*Total E = K + PE*.

The above equation is typically not solved as a
stand-alone equation due to difficulty in quantifying *J* and *E.* To overcome this
deficiency, an alternative form of the spin parameter was developed by Bullock
to dimension the spin parameter equations with direct observables; rotation
velocity and disk radius (Bullock 2001). This alternative is consistent with
Peebles under the added constraint of an isothermal halo and extended flat
rotation velocity:

Where *J* = Angular
Momentum, *M _{Dyn}*

_{ }= Dynamic Mass,

*V*= Rotation Velocity at radius =

*R*.

It has been shown that Peebles and Bullock equations are meaningfully equivalent and can be equated (Knebe 2008):

Solving for *E:*

J and E are determined by inserting RC-SP defined rotation
velocity (V), disk radius (R) and dynamic mass (M_{Dyn}) with results shown
in Table 1 for the Milky Way, M31, and And IV. The last column provides the
total galactic dynamic mass using the generalized model discussed in later in
this section.

*Table
**1**: RC-SP galactic properties by column; (1) dynamic mass
directly from extended rotation curve, (2) “average” disk velocity and (3),
disk radius at the Keplerian decline “break-point” in rotation curve, Columns (4,
5) represent RC-SP total energy and angular momentum for a constant spin
parameter λ=0.423. Column (6) is the general RC-SP derived “state” equation
linking galactic physical morphology and kinematics to dynamic mass.*

To determine the veracity of the RC-SP definitions, in Figure 5, we plot E and J against dynamic mass and to compare Marr’s results encompassing a wide variety of galaxies and find them indistinguishable. Both independently obtained data sets argue for galactic log normal surface densities rather than the simple “single component exponential distributions” often used for the distribution of the entire distribution of the galactic disk (CGM notwithstanding). The importance of a log normal surface density is that it physically permits galactic self-gravitation and naturally flat rotation profiles.

*Figure
**5**: Log E-J/M _{Dyn} relationship based on galactic
log normal surface density distribution. Solid circles for MW and M31 are RC-SP
E-J values and observed dynamic mass as obtained from rotation curves. This
data supports the observed mean spin parameter λ=0.423. Image source – top
panel, Fig. 3 (Marr 2015)*

Figure 5 verifies RC-SP dynamics conforms to Marr, lending strong support for log normally distributed galactic surface densities. The precision fit between the two data sets is due to highly constrained spin parameter (λ=0.423±0.014). In this analysis, Marr (and RC-SP) defines surface density in terms of the total contribution, baryons, angular momentum and energy (not just stellar and/or gas component often used to model exponential distributions). It is a simple exercise to sum individual surface density contributors to arrive at a total surface density that can be accurately described by a particular log normal distribution. Similar to Marr’s dynamic mass equation, RC-SP also links galactic dynamic mass to disk radius and rotation velocity with 1/G as the constant of proportionality:

Where *R*
is the disk radius, *V*
is the RC-SP rotational velocity and Newton’s constant, *G*.

Column (6) in Table 1 provides the modeled dynamic mass
estimates for the MW and M31 based on the above equation. Within 60 kpc, the
RC-SP MW dynamic mass, M_{Dyn}=4.9x10^{11}M_{ʘ} is
comparable to an earlier well-established dynamic mass estimate equal to 4.0±0.7
x10^{11}M_{ʘ (Xue 2008)}. Both are consistent
with another recent estimate, 4.25x10^{11}M_{ʘ} obtained by
treating galactic “missing mass” as a perfect fluid composed of dark matter (Potapov 2016). There is no doubt that RC-SP parameter definitions accurately
describe the MW. Rules of simple proportionality require nearly all other
rotationally supported galaxies also follow the RC-SP prescription. With a
slight transformation, it is not coincidental that the RC-SP dynamic mass and Newton’s
general equation for circular velocity are equivalent:

Ironically, dark matter was initially introduced to reconcile galactic rotation curves with Newtonian mechanics only to have it removed! ɅCDM’s issue is one of timing. Dark matter solutions became popular in the 1970’s based on incomplete and inaccurate galactic rotation curves. As such, ɅCDM proponents built much of their theory from incomplete data sets, including flat velocities to the outer edge of the dark matter halo (~ 200-400 kpc, depending on ɅCDM model).

The next section focuses on galactic baryonic content
using mainstream definitions - ɅCDM baryonic fraction (f_{b}) and MOND mass
discrepancy (D). To this point, we have incorporated the spin parameter to
define dynamic mass in terms of angular momentum and total energy. If it was
not already apparent, it is evident that galactic baryons are only part of the
RC-SP solution. In this galactic model, baryons can be considered “along for
the ride,” as their morphology and kinematics are completely determined from
the formation process (baryons reveal the absolute magnitude of galactic total mass).
Angular momentum and energy is specific to baryonic mass, and as baryonic
content increases, so does total dynamic mass in direct proportion.

### Baryon Mass (M_{Bar})

This section establishes “fiduciary” baryonic mass
estimates for our three example galaxies. For convenience, we fix the
dynamic/baryon mass ratio to the ɅCDM cosmic average f_{b}=0.17. By
fixing the baryonic fraction to the cosmic average, dynamic mass is not added
or removed from the universe maintaining the balance determined from the cosmic
microwave background and other sources (a precise baryonic content is not
critical to obtaining the holistic RC-SP solution).

Estimating the baryon content (stars, gas, etc.) of disk galaxies is a continually improving field. The general trend is to add additional baryonic mass to the galactic inventory, effectively removing it from the “missing mass” deficit. One recent example is the discovery of the Circum-Galactic Medium (CGM), a massive halo embedding the disk into an estimated stellar disk’s worth of baryonic mass. This discovery effectively “closes the Galaxy’s baryon census” justifying the fiduciary cosmic average value (Nicastro 2016). Table 2 shows the current baryon census by component for each of our three example galaxies:

*Table
**2**: Milky Way and M31 fiducial baryonic inventory fixed at
fb=0.17 in relation to the RC-SP determined dynamic mass. And IV - observed
values (Karachentsev 2015)*

In the lower portion of Table 2, dynamic/baryon mass
ratios are reported for both mainstream paradigms through the identity D ≌ 1/f_{b}. This table
is only included for completeness as RC-SP model is driven from total dynamic
mass, not singled out components as is often the case.

The next section offers strong support for a mean baryonic fraction governing disk galaxies, independent of baryon content. This constraint directly contradicts ɅCDM claims of a non-linear relationship between dark matter halo mass (~dynamic mass) and baryon content. This basic ɅCDM tenet was formulated before accurate assessments of galactic baryonic and dynamic mass were available and has been confirmed to be in error.

### Baryonic Fraction (M_{Dyn}/M_{Bar})

Unlike ɅCDM or MOND, RC-SP relies exclusively on empirical data. In this regard, RC-SP has more in common with MOND as it is baryonically based and requires a mean baryonic fraction, independent of galactic mass. Figure 6 provides incontrovertible proof for constant galactic baryonic fraction independent of rotation velocity (dynamic mass proxy).

*Figure
**6**: Observed baryonic fraction for a wide range of observed
galactic rotation velocities (proxy for dynamic mass). Open circles represent
RC-SP MW and M31 RC-SP galactic parameters. The open triangle is the observed
value for irregular dwarf galaxy And IV (f _{b}=0.11 and V≈45 km/s). For
practical purposes, R1 is equivalent to RC-SP disk radius R. Image source -
Fig.7 (Bradford 2015)*

Data presented in Figure 6 indicate a mean baryonic fraction clustering near the cosmic average for an extremely wide range of rotation velocities at odds with ɅCDM theory. The relatively large scatter is due to the use of baryons to define the fraction, exclusive of angular momentum and energy.

It has been known for almost twenty-years that the empirically
tight BFTR relationship is only physically possible if disk galaxies are
governed by a cosmically “universal” baryon fraction (McGaugh, The Baryon
Fraction Distribution and the Tully-Fisher Relation 1997). With this knowledge,
we can extend the RC-SP approach under the two following constraints; adherence
to Newtonian mechanics, and total galactic dynamic mass treated as a Keplerian “point-mass”
equivalent (e.g., the classic definition of a “particle”). We see these
constraints directly apply within a few kpc beyond the galactic disk (~ HI gas
radius). Inside this radius, disk dynamics dominate and are responsible for the
observed dynamic mass as determined from extended rotation curves. In the next
section, we find these constraints lead to an analytical solution to the origin
of the BTFR, M_{Bar} ∝
V^{4}.

### The Baryonic Tully-Fisher Relationship

The BTFR is a robust empirical correlation between estimated baryonic mass and observed (flat) rotational velocity depicted in Figure 7 (McGaugh 2011).

*Figure
**7**: RC-SP dynamic mass – rotational velocity relationship
(green dash). Data denotes observed baryonic mass for gas-rich galaxies. MW,
M31 and And IV dynamic (open symbol) and baryonic (closed symbol) masses
included for comparison to data set. McGaugh fits (black solid and red dash)
with extended RC-SP fit (black dash). Image source - Fig. 3 (McGaugh 2011)*

Figure 7 data represents gas-rich galaxy properties in
the lower end of the galactic mass spectrum. Note that RC-SP derived data is
entirely compatible with McGaugh’s BTFR fit. This figure serves as a template
to illustrate the functional RC-SP relationship (open circles) derived from the
dynamic mass and rotation velocity estimates. The magnitude of the offset (D=5.9)
from McGaugh’s fit corresponds to a baryonic/dynamic mass ratio equal to the cosmic
average f_{b}=0.17. Rather than attempting to measure baryon content
directly, RC-SP provides baryonic estimates that again are indistinguishable
from the raw data with the MW and M31 data extending McGaugh’s fit to the range
populated massive star-dominated spirals. This is evidence that RC-SP
accurately describes the physical “galactic state” without resorting to “external”
rotational support, normally ascribed to dark matter or modified Newtonian
dynamics. These physical interrelationships are difficult to grasp from
equations and best shown graphically in Figure 8 below.

*Figure
**8**: RC-SP combined (R, V & M _{Dyn}) physical
galaxy map. Individual gas-rich galaxy data from Table 1 (McGaugh 2011) converted to M_{Dyn} via constant baryonic fraction f_{b}=0.17 [see
Appendix).This data confirms all disk galaxies are fully self-gravitating and
exhibit dynamic masses that can be modeled as Newtonian point mass equivalents
in the far field. The figure demonstrates galaxies follow a remarkably strict
R-V relationship determining total galactic dynamic mass.*

In the above figure, most of McGaugh’s gas-rich galaxies
are plotted. McGaugh’s baryonic data was converted to galactic dynamic mass by multiplying
total baryonic content by D=5.9 (conversely, M_{bar} can be divided by
f_{b}=0.17 to arrive at the same dynamic mass estimate). Galactic radii
were determined from the RC-SP model R=GM_{Dyn}/V^{2}. Per
Figure 8, for a given disk radius (R), galactic dynamic mass (M_{Dyn})
varies inversely to the square of rotation (V). A complementary relationship
ties disk radius to the square of the ration velocity. These galactic
constraints conspire to create the “effective” relationship M_{Dyn} ∝ V^{4}:

The BTFR is then trivially obtained,

Figure 8 reveals a simple relationship between disk radii; rotation velocity and dynamic mass meeting all three imposed physical constraints. Individual galaxy data is summarized in the Appendix including a dynamic mass error analysis between analytic RC-SP model and physical placement in Figure 8 directly from R and V values. We find good agreement with less than twenty-percent offset between the two suggesting a systematic effect due to parameter definitions or perhaps a galactic feature such as pseudo-point mass velocity profiles as opposed to the Newtonian model. As this exercise demonstrates, the RC-SP model is internally and externally self-consistent and obeys all imposed classical mechanics constraints, a statement neither ɅCDM nor MOND proponents can convincingly make.

### Cosmological Implications

An issue remains in the cosmology required to impart disk galaxies with universally constrained morphological and kinematic properties. Two possibilities exist; loosely constrained galaxy formation with natural selection over the course of cosmic history permitting only those galaxies having specific combinations to survive, or an early era, cosmic event resulting in shared galactic properties. The first possibility has been discredited with studies demonstrating both baryonic fraction and spin parameter have remained stable throughout most of observed history (Cervantes-Sodi 2012). The second possibility is strongly favored and argues for a “common ancestry” as opposed to “survival of the fittest.”

### Conclusion

A galactic state model is presented based on a novel technique that expresses total dynamic mass in precise physical kinematic terms, angular momentum and energy or morphologically via radius and rotation velocity. This model employs Newtonian mechanics to solve the problem of galactic “missing mass,” and in doing so, reveals the physical origin of the BTFR. We demonstrate the empirical BTFR is a manifestation of baryonic physics in the disk galaxy setting. This physically and mechanically grounded based solution presents an alternative to ɅCDM and MOND models.

### Appendix

Galaxy Data (McGaugh 2011) and Associated RC-SP Parameters

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