Position-Velocity Diagrams for the Maser Emission coming from a
  Keplerian Ring
Accepted by The Astrophysical Journal
Preprint typeset using LATEX style emulateapj v. 03/07/07
POSITION–VELOCITY DIAGRAMS FOR THE MASER EMISSION COMING FROM A KEPLERIAN RING
Lucero Uscanga,
Centro de Radioastronomı́a y Astrof́ısica, Universidad Nacional Autónoma de México and
Apartado Postal 3-72, 58089 Morelia, Michoacán, Mexico
Jorge Cantó,
Instituto de Astronomı́a, Universidad Nacional Autónoma de México and
Apartado Postal 70-264, 04510 México, DF, Mexico
Alejandro C. Raga
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México and
Apartado Postal 70-543, 04510 México, DF, Mexico
Accepted by The Astrophysical Journal
ABSTRACT
We have studied the maser emission from a thin, planar, gaseous ring in Keplerian rotation around
a central mass observed edge-on. The absorption coefficient within the ring is assumed to follow a
power law dependence with the distance from the central mass as, κ = κ0r
−q. We have calculated
position-velocity diagrams for the most intense maser features, for different values of the exponent q.
We have found that, depending on the value of q, these diagrams can be qualitatively different. The
most intense maser emission at a given velocity can either come mainly from regions close to the inner
or outer edges of the amplifying ring or from the line perpendicular to the line of sight and passing
through the central mass (as is commonly assumed). Particularly, when q > 1 the position-velocity
diagram is qualitatively similar to the one observed for the water maser emission in the nucleus of the
galaxy NGC 4258. In the context of this simple model, we conclude that in this object the absorption
coefficient depends on the radius of the amplifying ring as a decreasing function, in order to have
significant emission coming from the inner edge of the ring.
Subject headings: galaxies: individual (NGC 4258) — galaxies: nuclei — masers
1. INTRODUCTION
The a priori probability of seeing a thin disk nearly
edge-on is very small. It is given by p ≃ 0.125 (h/R)2,
where h is the thickness of the disk and R is its radius.
Typically h/R ≃ 0.01 and thus p ≃ 1.25 × 10−5. Sur-
prisingly however, the maser emission observed in sev-
eral cosmic sources has been successfully modeled as
coming from a ring or truncated disk in Keplerian ro-
tation (around a massive object) seen edge-on. For in-
stance: circumstellar disks in star-forming regions as in
S255 (Cesaroni 1990) and MWC 349 (Ponomarev et al.
1994), and also circumnuclear disks around black holes
of galactic nuclei as in NGC 4258 (Watson & Wallin
1994; Miyoshi et al. 1995). In general, the maser emis-
sion from a Keplerian disk observed edge-on produces a
triple-peaked spectrum (Elmegreen & Morris 1979); but
Ponomarev et al. (1994) showed that, there is a transi-
tion from triple- to double-peaked spectra as the width
of the amplifying ring decreases.
NGC 4258 is a Seyfert 2/LINER located at a dis-
tance of 7.2 ± 0.3 Mpc (Herrnstein et al. 1999). The
water maser emission (22 GHz) toward this galaxy was
first detected by Claussen et al. (1984). Shortly after-
wards it was shown that the water masers are confined
in a very small region (∼1.3 pc) at the center of NGC
4258 (Claussen & Lo 1986). Subsequently, Nakai et al.
(1993) discovered water maser emission with velocity off-
Electronic address: l.uscanga@astrosmo.unam.mx
Electronic address: raga@nucleares.unam.mx
sets ±1000 km s−1 from the already known emission at
the galactic systemic velocity of ≃472 km s−1. They sug-
gested that the high-velocity emission could arise from
masers orbiting a massive central black hole, or ejected
in a bipolar outflow. Using the Very Long Baseline Array
(VLBA), Miyoshi et al. (1995) simultaneously observed
the systemic and high-velocity water maser emission in
NGC 4258, finding that the spatial distribution and line-
of-sight velocities of the water masers trace a thin molec-
ular ring in Keplerian rotation around a massive black
hole of 3.6×107 M⊙ seen nearly edge-on. The position-
velocity (PV) diagram for the maser emission shows dis-
tinct Keplerian orbits (with deviations < 1%) defined by
the high-velocity maser emission that arises on the ring
diameter perpendicular to the line of sight, as well as a
line traced by the systemic maser emission that arises
from material on the inner edge of the amplifying ring,
this linear dependence is a consequence of the change in
the line-of-sight projection of the rotation velocity.
By monitoring both systematic and high-velocity wa-
ter maser emission of NGC 4258 over periods of sev-
eral years with different radio telescopes, a signif-
icant centripetal acceleration was observed only for
the maser features near the galactic systemic velocity.
The systemic maser features drift at a mean rate of
∼9 km s−1yr−1 (Haschick et al. 1994; Greenhill et al.
1995; Nakai et al. 1995; Bragg et al. 2000) while the
high-velocity maser features drift by .1 km s−1yr−1
(Greenhill et al. 1995). In a recent spectroscopic
study, Bragg et al. (2000) detected accelerations for
http://arxiv.org/abs/0704.0819v1
mailto:l.uscanga@astrosmo.unam.mx
mailto:raga@nucleares.unam.mx
2 USCANGA, CANTÓ, & RAGA
the high-velocity features in the range of −0.77 to
0.38 km s−1 yr−1. These measurements indicate that
the systemic water masers lie within a relatively nar-
row range of radii, on the near side of the ring at the
proximity of its inner edge, while the high-velocity wa-
ter masers are located near the ring diameter (between
−13.o6 and 9.o3 of the mid-line, Bragg et al. 2000). In
addition, the deviation of the high-velocity masers from
a straight line passing through the systemic masers in
the plane of the sky suggests that the rotating disk is
slightly warped (Herrnstein et al. 1996, 1999, 2005).
Previously, Watson & Wallin (1994) demonstrated
that the maser emission from a rapidly rotating, thin
Keplerian ring viewed edge-on can reproduce the general
features of the observed 22 GHz radiation from the nu-
cleus of NGC 4258, including the high-velocity satellites.
However, it is important to point out that their assump-
tion of a uniform absorption coefficient within the ampli-
fying ring results in a PV diagram for the most intense
masers that is qualitatively different from the observed
one. While their model predicts that the maser emission
at velocities around the systemic velocity of the galaxy
comes mainly from the outer edge of the ring, the ob-
servations indicate that this emission is actually coming
from the inner parts of the truncated disk.
In this paper we show that this discrepancy can be
resolved if the absorption coefficient decreases with dis-
tance from the central mass. The model is presented in
§2. The main results are described in §3. Finally, the
conclusions are discussed in §4.
2. MODEL
We study the maser emission that arises from a thin,
planar, gaseous ring in Keplerian rotation around a mas-
sive central object when it is observed edge-on. The mas-
ing gas is located between R0 and R, the inner and outer
radii of the ring, respectively. For simplicity, we assume
that the disk is transparent to the maser radiation at
radii smaller than R0 and greater than R, although the
inner region is probably thermalized due to the higher
gas density, and actually it would absorb a significant
fraction of the maser radiation produced in the far side
of the ring (see §4). The absorption coefficient is assumed
to follow a power law function of the distance from the
central mass within the amplifying ring as, κ = κ0r
The distances are measured in units of R, and the veloc-
ities are measured in units of vout, the rotation velocity
at the outer edge of the ring (see Figure 1).
For the case of an unsaturated maser and neglecting
the spontaneous emission, the intensity of the maser ra-
diation from a line of sight with impact parameter y at
a velocity vr is
I(vr, y) = I0e
τ(vr,y) , (1)
where the optical depth or gain along the line of sight is
given by
τ(vr , y) = 2 κ0
∫ xmax
(x2+y2)−q/2 exp
−(v − vr)2
xmin =
r20 − y2 for 0 ≤ |y| ≤ r0 ,
0 for r0 < |y| ≤ 1 ,
xmax =
1− y2 .
The line-of-sight velocity component of the gas at the
position (x, y) can be expressed as v = y/(x2 + y2)3/4.
Here I0 and ∆vD are the background intensity and the
Doppler width, respectively, which are supposed to be
uniform inside the amplifying ring. The Doppler width
∆vD is related with the FWHM of the velocity distribu-
tion of the emitting particles as ∆vD = FWHM/
4 ln 2.
We have numerically solved equations (1) and (2), and
we have also calculated the y-positions (impact parame-
ters) of maximum maser intensity for each specific value
of the velocity vr. When we have found two local max-
ima, we have kept both. With this information, we have
constructed the PV diagrams using the positions of the
observer’s line of sight with maximum emission at each
velocity. This way to construct the PV diagrams was
previously used by Uscanga et al. (2005).
We show the results using the following values for the
model parameters which seem to be appropriate for mod-
eling maser emission in the galaxy NGC 4258. The back-
ground intensity is I0 = 1.3 × 10−5 Jy beam−1, corre-
sponding to a radio continuum source with a temper-
ature of 106 K (Watson & Wallin 1994). The dimen-
sionless inner radius r0 = 0.51, using the estimated val-
ues for the inner and outer radii of 4.1 and 8.0 mas re-
spectively, given by Miyoshi et al. (1995). The Doppler
width ∆vD = 0.007vout which combined with an outer
rotation velocity of 770 km s−1 (Miyoshi et al. 1995),
gives a Doppler width ≃ 5 km s−1, similar to the value
used by Watson & Wallin (1994). We have used some
representative values of the exponent q, specifically q =
0, 1/2, 15/8, 5 for Models I, II, III and IV, respec-
tively. In Model I, we study the simplest situation of a
uniform absorption coefficient. In Model III, we choose
q = 15/8, that corresponds to the density dependence
with the radius of an accretion disk, i.e., Frank et al.
(1992). Finally, in Models II and IV, we explore two
other different values of the exponent q in order to study
how it changes the results. In all the models, the value
for the absorption coefficient κ0 is mainly determined
by the requirement that the intensity at the peak of the
central component (13 Jy beam−1) is compatible with
the observational data when the background intensity is
1.3×10−5 Jy beam−1. Other values of I0, r0, ∆vD, and
κ0 give qualitatively similar results.
We present the results in the next section; but let us
first discuss briefly some important concepts in order to
understand these results. In general, the observed emis-
sion at a given velocity coming from a specific position in
a nebula has contributions of the whole material along
the line of sight. However, when the velocity gradient
along the line of sight is greater than the dispersion ve-
locity (thermal or turbulent) of the emitting material,
the main contribution to the emission is actually coming
from a narrow region around the point with a line-of-
sight velocity equal to the observation velocity. The esti-
mated width of the region is 2l, where l is the correlation
distance defined as
l ≡ ∆vD
|dv/dx|
, (3)
here dv/dx is the line-of-sight velocity gradient. In this
approximation, known as Sobolev’s approximation or the
approximation of high velocity gradient, the observed in-
tensity is given by the following expression
I(vr) = I0(vr)e
−τ(vr) + S(vr)(1 − e−τ(vr)) , (4)
where I0 is the background intensity, S is the source func-
tion and τ is the optical depth given by
τ(vr) = κ(2l) , (5)
where κ is the absorption coefficient.
For the case of maser emission, the value of κ is in-
trinsically negative and τ is also negative, therefore the
factor e−τ(vr) becomes an amplification factor. Because
of this reason, the relative contribution at a given veloc-
ity of the correlation region is even more important with
respect to the remainder of the emitting material than
in the case of non-maser emission. Consequently, the ap-
proximation given by equations (4) and (5) is suitable
for maser emission.
As shown in the next section, for a gaseous ring of in-
ner radius R0 and outer radius R in Keplerian rotation
and seen edge-on, the emission either comes preferen-
tially from the inner or outer edges of the ring or from
the line perpendicular to the line of sight and passing
through the ring center. In the first two cases, it is easy
to show that the expected PV diagram will be a straight
line. When the emission comes from the outer edge, the
slope of the straight line is equal to one (measuring the
distances in units of the outer radius of the ring and the
velocities in units of the rotation velocity at that point),
whereas if the emission comes from the inner edge, the
slope of the straight line is equal to 1/r
0 . On the other
hand, when the emission arises from the line perpendic-
ular to the line of sight, the PV diagram will be a curve
with the form 1/y1/2, where y is the impact parameter
of the observation (see Figure 2).
3. RESULTS
The PV diagrams for the maser emission peak are
point-symmetric, consequently we only discuss positive
velocities from now on (see Figures 3 and 4).
• Model I (q = 0) – With this value of the expo-
nent q, we are considering the simplest situation,
a uniform or constant absorption coefficient. The
strongest maser emission either comes mostly from
the outer edge of the ring at velocities lower than
1, or from the mid-line of the ring perpendicular
to the line of sight and passing through the cen-
tral mass at greater velocities. The filled squares,
circles, and triangles mark the regions of strongest
maser emission at each velocity.
• Model II (q = 1/2) – The results are qualitatively
similar to those of Model I.
• Model III (q = 15/8) – This value of exponent q
corresponds to the density dependence with radius
of an accretion disk (ρ ∝ r−15/8). The strongest
maser emission either comes mainly from the inner
edge of the ring at low velocities (velocities near
the systemic velocity), or from the outer edge at
velocities close to 1. On the other hand, at ve-
locities greater than 1, the most intense emission
comes predominantly from the mid-line of the ring
perpendicular to the line of sight.
• Model IV (q = 5) – The strongest maser emission
comes mainly from the inner edge of the ring at
velocities lower than 1. At greater velocities, the
most intense emission can either come mainly from
the inner edge or from the mid-line of the ring per-
pendicular to the line of sight.
In summary, from the results of Models I–IV (see Fig-
ure 3), we found that the most intense maser emission
can be around the inner or outer edges of the ring, or
the mid-line of the ring perpendicular to the line of sight
depending on the velocity and also on the value of q. In
fact, the PV diagrams are qualitatively different when
q < 1 or q > 1. In the first case, for q < 1 (including
the simplest situation with a uniform absorption coeffi-
cient, q = 0) and vr < 1, the PV diagram corresponds
to a straight line with slope 1; for vr > 1, the diagram
corresponds to a Keplerian curve. In the second case,
for q > 1 and vr < 1, the PV diagram corresponds to a
straight line with a slope that depends on the inner ra-
dius of the ring. At velocities close to 1 the slope changes
to 1; for vr > 1, the diagram corresponds to a Keplerian
curve and also a straight line with a slope that depends
on the inner radius under circumstances such as in Model
It is also important to realize that when q > 1, the
optical depth or gain presents two local maxima within
a certain velocity range. Either local maxima may be a
global maximum. For vr < 1, the local maximum can be
either at the inner and/or outer edges of the ring, while
for vr > 1, they are located at the inner edge and/or
mid-line of the ring (see Figure 5).
As shown in the top panels of Figure 5 (Model III,
q = 15/8), the relative difference between the two lo-
cal maxima is not very significant. However, when the
value of q is higher (like in Model IV, q = 5 shown in
the bottom panels) the relative difference becomes more
important.
In order to estimate the velocity vc at which the global
maximum of the optical depth changes its locus, we have
calculated analytical approximations for the largest value
of the optical depth or gain that corresponds to the max-
imum intensity at the inner and outer edges of the ring,
and also at the mid-line of the ring perpendicular to the
line of sight. The detailed calculations are presented in
the Appendix. The following equations give the local
maximum depth as a function of the velocity in each
neighborhood
τ(vr) ≃
πκ0∆vD
1− r0v2r
inner edge , (6)
τ(vr) ≃
πκ0∆vD
1− v2r
outer edge , (7)
τ(vr) ≃
r mid-line . (8)
The velocity vc is estimated by combining equations
(6) and (7), or equations (6) and (8) according to the
value of vc (when vc < 1 or vc > 1, respectively). The
results are
2(1−q)
0 − 1
2(1−q)
0 − r0
for vc < 1, (9)
4 USCANGA, CANTÓ, & RAGA
2(1−q)
0 − v
c + r0v
c = 0 for vc > 1, (10)
which are presented in Figure 6 for some representative
values of the exponent q.
The bottom plot of Figure 6 shows vc as function of
the inner radius r0 for different values of q, from equation
(9a). For q < 1, there is no solution to equation (9a).
When q = 1, vc = 0 for any value of r0. That is, the
optical depth has a maximum and its locus is around
the outer edge of the ring, and vc is meaningless as we
have defined it. When q > 1, the optical depth presents
two local maxima, and vc is different from zero and its
value depends on r0. This velocity corresponds to the
value at which the locus of the global maximum changes
from the inner edge to the outer edge of the ring. As a
consequence, there is a slope change in the PV diagrams
at velocities lower than the rotation velocity at the outer
edge of the ring. For instance, when q = 15/8 and r0 =
0.51, the slope change occurs at vc = 0.906. In other
words, the locus of the global maximum of the optical
depth changes from the inner to the outer edge of the
ring at this velocity vc.
The top plot of Figure 6 also shows vc as function of
the inner radius r0 using specific values of q and ∆vD in
equation (9b); in this case, 5 and 0.007vout, respectively.
As an example, when ∆vD = 0.007vout, q = 5 and r0 =
0.51, then vc = 1.077. Stated differently, at that velocity
vc, the largest value of both the optical depth and the
intensity changes its locus from the inner edge to the
mid-line of the ring perpendicular to the line of sight.
The remarkable water maser emission in the nucleus
of the galaxy NGC 4258 traces a PV diagram where the
detected emission around the systemic velocity of the
galaxy comes from the inner edge of the amplifying ring;
this emission delineates a straight line just as the straight
line that connects points C and D in Figure 2 (see Figure
3 of Miyoshi et al. 1995). According to our model re-
sults, this implies that the absorption coefficient within
the molecular ring of NGC 4258 is not uniform, instead
it must be a decreasing function of the distance from the
central mass, i.e., κ = κ0r
−q with q > 1. Moreover, the
observed red/blue-shifted emission at high velocities that
arises from the mid-line of the ring perpendicular to the
line of sight traces a Keplerian curve such as is indicated
by the model results (see Figure 7). Simply stated, when
q > 1 the PV diagram is qualitatively similar to the one
observed for the water maser emission detected in the
nucleus of NGC 4258.
As an example, in Figure 7 we show a comparison be-
tween the results of Model III (q = 15/8) and the water
maser emission in NGC 4258. The detected emission
arises from the inner edge of the amplifying ring and the
mid-line perpendicular to the line of sight. The locus of
the observed maser emission coincides with the locus of
the most intense maser emission as indicated by the sizes
of the circles in Figure 7. The model results indicate that
there is emission coming from the outer edge of the ring
at velocities close to 1, nevertheless the sizes of the circles
indicate that this emission is very weak. Maybe maser
emission is not detected from this locus for this reason.
Additionally, our model results also indicate that the
intense maser emission at the inner edge of the ring ex-
tends neither to velocities very different from the sys-
temic velocity nor to impact parameters very different
from zero, as is indicated by the size of the circles in the
PV diagram shown in Figure 7. Furthermore, accord-
ing to the size of the circles, the other locus of intense
maser emission is the mid-line of the ring perpendicular
to the line of sight, precisely the locus of the red/blue-
shifted maser emission at high velocities that describes
Keplerian curves in the PV diagram.
4. DISCUSSION AND CONCLUSIONS
In our model, we have assumed that the gas in the re-
gion inside the masing ring is transparent to the maser
radiation. This implies that the most intense maser emis-
sion at low velocities (velocities near the systemic veloc-
ity) comes mainly from the outer edge of ring (for q < 1)
or from the inner edge (for q > 1), either the near or
far side of the ring, as is indicated in Figure 4. Mea-
surements of positive acceleration of the maser emission
around the systemic velocity show that this emission cer-
tainly comes from the near side of the ring at the prox-
imity of its inner edge (e.g., Greenhill et al. 1995). If we
suppose that the gas inside the masing ring is thermal-
ized probably due to its higher density then an important
fraction of the maser emission from the backside of the
ring would be absorbed and the detected emission would
come from the front side of the ring at the outer or inner
edge depending on the value of q. For instance, consider-
ing absorption and emission from the gas located inside
the masing ring, the difference in the intensity for a line
of sight that passes through both the inner absorbing
region and the front side of the masing ring from the in-
tensity for a line of sight that passes through both the
backside of the masing ring and the inner absorbing re-
gion is S(1−e−τ2)(eτ1−1) where S is the source function
of the gas inside the masing ring, τ2 is the optical depth
in this region, and τ1 is the optical depth for the front
side of the ring. If τ2 >> τ1, then the detected emission
would be the radiation amplified by the front side of the
masing ring.
Also, we have made a simplifying assumption about
the geometry of the masing ring in NGC 4258, consider-
ing that the amplifying ring is strictly flat. Despite the
observations indicate an apparent warp in the maser dis-
tribution of this galaxy, Kartje et al. (1999) presented a
model in which the disk does not require to be physically
warped in order to the masing gas become exposed to
the central continuum radiation. In this scenario, dusty
clouds provide the shielding of the high-energy contin-
uum, which is required for the gas to remain molecu-
lar. They found that a flat-disk model of the irradiated
ring could be applied to a source like NGC 4258 only
if the water abundance is higher than the value implied
by equilibrium photoionization-driven chemistry. A very
important result from their study (based on radiative
and kinematic considerations) was that, even if the disk
in NGC 4258 is warped, the maser-emitting gas must be
clumpy, instead of homogeneous as in the scenario pre-
viously proposed by Neufeld & Maloney (1995).
An important result of our model shows that the as-
sumption, commonly used, that considers a uniform or
constant absorption coefficient within the masing ring
in Keplerian rotation around the nucleus of NGC 4258
is not appropriate. For example, Wallin et al. (1998)
supposed that κ was constant considering that the locus
of the maser emission from NGC 4258 was determined
mainly by the velocity gradients in a Keplerian velocity
field indicating some uniformity of κ, at least on length
scales comparable to the coherence or correlation length
resulting from the Keplerian velocity gradients. On the
contrary, from our analysis, we conclude that a constant
absorption coefficient would result in a PV diagram qual-
itatively different from the observed one, since the most
intense maser emission would come predominantly from
a narrow region close to the outer edge of the ring instead
of a narrow region close to the inner edge of the ring, as
indicated by the observations. Necessarily, the absorp-
tion coefficient must be a decreasing function of distance
from the central mass (i.e., κ = κ0r
−q with q > 1) to
have significant emission coming from the inner edge of
the amplifying ring and hence explain the form of the PV
diagram delineated by the water masers in NGC 4258.
When comparing our edge-on disk model with the
observations of NGC 4258, it is clear that we need a
κ ∝ r−q radial dependence for the absorption coefficient
with q > 1 (so as to favour the emission from the in-
ner edge of the disk, see above) in order to reproduce
the observations. In reality, the fact that the disk of
NGC 4258 is warped introduces geometrical effects which
might favour the inner disk edge emission (over the one of
the outer edge). One will need to compute more complex,
3D transfer models to see whether or not these geomet-
rical effects are sufficient to explain the PV diagrams of
the NGC 4258 masers without introducing the radially
dependent absorption coefficient which is required by the
edge-on disk models described in the present paper.
J. C. and A. C. R. acknowledge support from CONA-
CyT grants 41320 and 43103, and DGAPA-UNAM. L.
U. acknowledges support from DGAPA-UNAM. We sin-
cerely thank J. M. Torrelles and Y. Gómez for useful
comments, which contributed to improve an earlier ver-
sion of this manuscript. L. U. gives special thanks to
M. R. Pestalozzi and M. Elitzur for valuable comments
on this work. We also thank an anonymous referee for
helpful comments on the manuscript.
APPENDIX
ANALYTICAL APPROXIMATIONS FOR THE OPTICAL DEPTH
In this appendix, we describe how to obtain the analytical approximations for the optical depth given by equations
(6)–(8).
First, we define w = (v − vr)/∆vD then we can change the variables in equation (2) and rewrite it as
τ(vr , y) = 2κ0
∫ wmax
(x2 + y2)−q/2
exp(−w2)dw , (A1)
where
wmin =
0 − vr
wmax =
y − vr
Note that wmin > wmax since r0 ≤ 1. Using the expressions for the line-of-sight velocity v = y/(x2 + y2)3/4 and the
previously defined variable w = (v− vr)/∆vD, we can write x =
(y/(vr + w∆vD))
4/3 − y2
. At zero order around
w = 0 we obtain
(x2 + y2)−q/2
≃ − 2∆vD(y/vr)
2(2−q)/3
(y/vr)4/3 − y2
. (A2)
Additionally
∫ wmax
exp(−w2)dw =
erf(wmax)− erf(wmin)
, (A3)
where erf(w) is the error function, defined as erf(w) ≡ (2/
exp(−t2)dt. Finally, substituting equations (A2)
and (A3) into (A1), we obtain the approximation for the optical depth
τ(vr , y) ≃
πκ0∆vD(y/vr)
2(2−q)/3
(y/vr)4/3 − y2
erf(wmin)− erf(wmax)
. (A4)
Around the inner edge of the ring, vr ≃ y/r3/20 and the maximum value of [erf(wmin)− erf(wmax)] = 2. Substituting
these approximations into equation (A4), we obtain equation (6). Similarly, around the outer edge of the ring, vr ≃ y,
and the maximum value of [erf(wmin)− erf(wmax)] also equals 2. Then, equation (A4) reduces to equation (7) for the
optical depth at the outer edge of the ring.
In order to find an approximation for the local maximum of optical depth at the mid-line of the ring perpendicular
to the line of sight, we expand the expression for the velocity along the line of sight around x = 0 to obtain
v ≃ 1
, (A5)
6 USCANGA, CANTÓ, & RAGA
therefore
= −∆vD , (A6)
and thus
5/4 . (A7)
Then, using Sobolev’s approximation
τ = κ0(x
2 + y2)−q/2(2x) , (A8)
and substituting equation (A7) into (A8), we obtain the following expression
τ ≃ κ0
5/2 + y2
)−q/2 4
5/4 , (A9)
since y ≪ 1 and ∆vD is small, then 43∆vDy
5/2 ≪ y2, hence
τ ≈ 4
5/4−q , (A10)
considering that y = v−2r , we finally obtain the approximation for the local maximum of the optical depth at the
mid-line of the ring given by equation (8).
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R  /R00
Observer
Fig. 1.— Schematic diagram of a gaseous disk in Keplerian rotation. The masing gas exists between radii R0 and R. At radii smaller
than R0 and greater than R the disk is transparent to the maser radiation. The observer is on the plane of the disk. All the distances are
measured in units of R, the outer radius of the amplifying ring; therefore the variables x, y, and r0 are dimensionless.
8 USCANGA, CANTÓ, & RAGA
v  = y
1/2v  = 1/y
v  = y/r0
Fig. 2.— PV diagram for the maser emission of a gaseous ring with inner radius R0 and outer radius R in Keplerian rotation observed
edge-on. The straight line that connects points A and B has a slope equal to 1, while the straight line that connects points C and D has a
slope equal to 1/r
Fig. 3.— From top to bottom results of Models I, II, III, and IV. The left panels show PV diagrams for the maser emission peak. The
filled squares, circles, and triangles represent the strongest maser emission which is coming from regions a, b, and c, respectively, indicated
in Figure 4. The straight lines or curves represent the velocity dependences of the regions where this emission arises. The central panels
show PV diagrams for the maser emission peak. Because of the point-symmetric shape of these diagrams, only positive velocities are shown.
The radii of the open circles are proportional to the maximum maser intensity at each position and velocity. The right panels show the
logarithm of the ratio between the maximum intensity and the background intensity as a function of the velocity.
10 USCANGA, CANTÓ, & RAGA
:a v   = y
b : v   = 1 / y
x x x
b b b
Observer Observer
Observer
Model  IIIModels  I  and  II Model  IV
:c  v   = y / r0
Fig. 4.— Schematic representation of the results of Models I, II, III, and IV. The filled squares, circles, and triangles represent the most
intense maser emission that is coming from regions a, b, and c, respectively. These regions are very narrow because the correlation distance
is very small; that is, the width ∆vD is much smaller than the line-of-sight velocity gradient. Besides, the exponential amplification of the
intensity emphasizes small changes in the optical depth.
Fig. 5.— Left : PV diagrams for the maser emission in grey scale with intensity contours overlaid. The darker regions show the locus
of the strongest emission in these diagrams, which potentially could be detected, depending on the sensitivity cutoff of the observations.
Right : Close-up to the PV diagrams showing the maser emission at low velocities.
12 USCANGA, CANTÓ, & RAGA
Fig. 6.— vc as function of the inner radius of the ring, r0. Bottom: For vc < 1, it is computed from equation (9a) for some representative
values of q. For q < 1, there is no solution to equation (9a). Top: For vc > 1, it is computed from equation (9b). This is the solution to
equation (9b), using ∆vD = 0.007vout and q = 5.
Fig. 7.— Comparison between the calculated PV diagram for the maser emission peak in Model III (q = 15/8) and the PV diagram
delineated by the water masers observed in NGC 4258 (Miyoshi et al. 1995). The radii of the open circles are proportional to the maser
intensity at each position and velocity. The dots represent the observed maser spots in NGC 4258. We have subtracted the ring systemic
velocity of 476 km s−1 from the observed local standard of rest velocity of the maser spots in order to compare the observed PV diagram
with the modeled one. The positions and velocities are in units of the outer radius of the ring (8 mas) and the rotation velocity at the
outer edge of the ring (770 km s−1), respectively.
