# Spherically symmetric charged black holes with wavy scalar hair

Yves Brihaye<EMAIL_ADDRESS>Service de Physique de l’Univers,
Champs et Gravitation, Université de Mons, Mons, Belgium  Betti Hartmann
<EMAIL_ADDRESS>Department of Mathematics, University College London,
Gower Street, London, WC1E 6BT, UK

###### Abstract

We study standard Einstein-Maxwell theory minimally coupled to a complex
valued and self-interacting scalar field. We demonstrate that new, previously
unnoticed spherically symmetric, charged black hole solutions with scalar hair
exist in this model for sufficiently large gravitational coupling and
sufficiently small electromagnetic coupling. The novel scalar hair has the
form of a spatially oscillating “wave packet” and back-reacts on the space-
time such that both the Ricci and the Kretschmann scalar, respectively,
possess qualitatively similar oscillations.

##  I Introduction

It is well known for some decades now that black holes can carry scalar field
hair if the scalar field model is non-linear. The first example of this type
was given within the Skyrme model minimally coupled to graviy luckock_moss .
While the Skyrme model is often considered as an effective model in the
context of nuclear physics, black holes that carry scalar hair can also be
constructed in models inspired by high energy physics. The SU(2) Yang-Mills-
Higgs model with the Higgs field in the adjoint representation (and hence
being real) possesses hairy black holes Breitenlohner:1991aa ;
Breitenlohner:1994di  which can be thought of as black holes residing inside
the core of magnetic monopoles. However, diverse no-hair conjectures (see
herdeiro2018asymptotically  for a recent review) seem to prevent the existence
of black holes with scalar hair charged under an abelian group. In recent
years and motivated by an increased interest in the phenomenon of
superradiance superradiance , it was realized that black holes can carry
complex valued scalar hair under certain conditions. The first (and best-
known) example was put forward in the context of rotating, uncharged black
holes Hod:2012px ; Herdeiro_2014 . The no-hair conjecture can be cirumvented
in this case because of the harmonic dependence on the time and azimuth
coordinates and assuming the scalar field frequency to be fine-tuned to the
horizon angular frequency of the black hole. This so-called synchronization
condition appears exactly at the threshold of superradiance and allows for so-
called scalar Q-clouds to exist on and close to the black hole horizon.

Now, returning to models that contain U(1) gauge fields, a similar
construction is possible – even when the black hole is non-rotating and
spherically symmetric Herdeiro:2020xmb . In this case, the frequency of the
complex valued scalar field needs to be fine-tuned to the electric potential
on the horizon. It was then realized that when sufficiently strong back-
reaction of the scalar cloud is taken into account that next to the expected
extremal black holes with diverging scalar field derivative on the horizon, a
new type of solution exists that represents a black hole with an inflating
exterior Brihaye:2020vce . On a shell of given thickness outside the horizon,
the scalar field becomes constant and non-vanishing such that the scalar field
potential energy corresponds to a positive cosmological constant. Moreover,
the charge of these black holes gets screened.

In this paper, we point out that additional solutions to the model studied in
Herdeiro:2020xmb ; Brihaye:2020vce  exist which have not been noticed so far
and which have a novel type of scalar hair. These solutions exist at large
gravitational coupling and small gauge field coupling and possess “wavy”
scalar hair in the sense that the scalar field constitutes spatial
oscillations outside the black hole horizon. These oscillations lead to
qualitatively similar oscillations in the curvature of the space-time. We will
discuss these solutions and their properties in the following and point out
that the phenomenon is independent of the choice of the self-interaction
potential of the scalar field.

##  II The model

We study a simple (3+1)-dimensional model in which Einstein gravity is
minimally coupled to an electromagnetic field as well as to a complex valued,
self-interacting scalar field. The Lagrangian density reads

${\cal{L}}=\frac{1}{16\pi
G}R-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-D_{\mu}\Psi^{\dagger}D_{\mu}\Psi-U(|\Psi|)\
,$ (1)

where $R$ is the Ricci scalar, $F_{\mu\nu}$ denotes the field strength tensor
of the electromagnetic potential $A_{\mu}$ and
$D_{\mu}\Psi=\left(\partial_{\mu}-igA_{\mu}\right)\Psi$ denotes the covariant
derivative of the scalar field $\Psi$. The important point in our construction
is that the scalar field is self-interacting. In the following, we will study
two different potentials $U(|\Psi|)$. These read, respectively :

${\rm polynomial\ :}\ \ \
U_{1}(|\Psi|)=\mu^{2}|\Psi|^{2}-\lambda|\Psi|^{4}+\nu|\Psi|^{6}\ \ $ (2)

and

${\rm exponential\ :}\ \ \
U_{2}(|\Psi|)=\mu^{2}\eta^{2}\left[1-\exp\left(-\frac{|\Psi|^{2}}{\eta^{2}}\right)\right]\
.$ (3)

The parameter $\mu$ represents the rest mass of the scalar field. The
potential $U_{1}$ is motivated by studies of scalar fields in flat space-time
and proves essential in the construction of scalar hair on charged black holes
Herdeiro:2020xmb . The potential $U_{2}$ appears in models describing gauge-
mediated supersymmety breaking susy1 ; susy2 . Of course, the polynomial
potential $U_{1}$ is an approximation to the exponential potential $U_{2}$ for
small scalar fields. Note, however, that the potential $U_{1}$ depends a
priori on the independent parameters $\lambda$ and $\nu$ that can be freely
chosen, while the potential $U_{2}$ is fully fixed by choosing the mass of the
scalar field as well as the energy scale $\eta$.

In the following, we will discuss the simplest possible solution that can be
constructed in this model: a static, spherically symmetric, electrically
charged black hole. The symmetries in the space-time as well as the U(1) gauge
symmetry allow the following Ansatz for the metric, gauge field and scalar
field, respectively:

${\rm d}s^{2}=-(\sigma(r))^{2}N(r){\rm d}t^{2}+\frac{1}{N(r)}{\rm
d}r^{2}+r^{2}\left({\rm d}\theta^{2}+\sin^{2}\theta{\rm d}\varphi^{2}\right)\
\ ,\ \ A_{\mu}{\rm d}x^{\mu}=V(r){\rm d}t\ \ ,\ \ \Psi=\psi(r)\exp(i\omega t)\
,$ (4)

where $\omega$ is a real-valued constant.

As is usual in systems with a U(1) gauge symmetry, the phase of the scalar
field can be gauged away. Stating it differently, the resulting equations
depend only the combination $\Omega:=\omega-gV(r=\infty)$. Now, it is well
known that a charged black hole does not per se possess scalar hair, but that
a so-called synchronization condition has to be fulfilled in order for the
black hole to carry a Q-cloud of scalar fields. This condition follows from
the regularity of the matter fields on the horizon and reads $\omega-
gV(r_{h})=0$. In the following, we will choose the gauge $\omega=0$ and hence
the boundary condition for the electric potential reads: $V(r_{h})=0$

The equations of motion following for the variation of the action have been
given in other publications (see e.g. Herdeiro:2020xmb ; Brihaye:2020vce ) and
we refer the reader to these papers. These equations depend on a number of
dimensionless coupling constants that result from appropriate rescalings.
These are :

  * •

For the potential $U_{1}$ we use rescalings $r\rightarrow r/\mu$,
$v=\frac{\sqrt{\lambda}}{\mu}V$, $\psi=\frac{\sqrt{\lambda}}{\mu}\Psi$ such
that the dimensionless couplings read

$\alpha=\frac{4\pi G\mu^{2}}{\lambda}\ \ ,\ \
\beta^{2}=\frac{\nu\mu^{2}}{\lambda^{2}}\ \ ,\ \ e=\frac{g}{\sqrt{\lambda}}\
.$ (5)

  * •

For the potential $U_{2}$ we use the same rescaling for the radial coordinate
as above, but now $v=\frac{V}{\eta}$, $\psi=\frac{\Psi}{\eta}$ which results
in the dimensionless coupling constants

$\alpha=\frac{8\pi G\eta^{4}}{\mu^{2}}\ \ ,\ \ e=\frac{\eta}{\mu}g\ .$ (6)

We have chosen to use the same letters for the dimensionless couplings to
demonstrate the qualitative similarily of our results for both potential.

From the equations of motion we can infer the asymptotic behaviour of the
functions, which reads

$N(r\gg 1)=1-\frac{2M}{r}+\frac{\alpha Q_{e}^{2}}{r^{2}}+.....\ \ ,\ \ v(r\gg
1)=\Phi-\frac{Q_{e}}{r}+....\ \ ,\ \ \psi(r\gg 1)\sim\frac{\exp(-\mu_{\rm
eff,\infty}r)}{r}+....\ \ ,\ \ $ (7)

where the parameters $M$ and $Q_{e}$ denote the (dimensionless) mass and
electric charge of the solution, respectively. Moreover, $\mu_{{\rm
eff},\infty}$ is the effective mass of the scalar field which is given by the
bare mass (rescaled to unity with our choice of rescalings) and the asymptotic
value of $v(r)$ denoted by $\Phi$, i.e. reads $\mu_{{\rm
eff},\infty}=\sqrt{1-\Omega^{2}}$, $\Omega\equiv e\Phi$. Hence, increasing
(decreasing) $\Omega$ will decrease (increase) the effective mass of the
scalar field. Note that the larger the effective mass of the scalar field, the
stronger the scalar field will be localised.

The scalar cloud (Q-cloud) surrounding the charged black hole can be thought
of as made up of $Q_{N}$ scalar bosons, where $Q_{N}$ is the globally
conserved Noether charge :

$Q_{N}=\int{\rm d}r\ \frac{2r^{2}ev\psi^{2}}{N\sigma}\ .$ (8)

Through the coupling to the electromagnetic field each of the scalar bosons
carries a charge $e$. The total electric charge $Q_{e}$ of the solution is
then a sum of the electric charge in the cloud given by $eQ_{N}$ and the
horizon electric charge $Q_{H}$ of the black hole given by
$Q_{H}=v^{\prime}(r_{h})r_{h}^{2}/\sigma(r_{h})$ Herdeiro:2020xmb  . Next to
the electric charge, we can also attribute a mass to the cloud. This is given
by Herdeiro:2020xmb  :

$M_{Q}=\frac{1}{4\pi}\int{\rm d}^{3}x\ \sqrt{-g}\left(T^{i}_{\ i}-T^{t}_{\
t}\right)\ \ ,\ \ i=1,2,3\ .$ (9)

Finally, we the Hawking temperature of the black hole is given by

$T_{H}=(4\pi)^{-1}\sigma(r_{h})N^{\prime}|_{r=r_{h}}\ $ (10)

and is related to the masses $M$ and $M_{Q}$ via the Smarr relation
Herdeiro:2020xmb  :

$2T_{H}{\cal S}=M-\alpha M_{Q}\ \ ,\ \ {\cal S}=\pi r_{h}^{2}\ ,$ (11)

where ${\cal S}$ is the black hole entropy.

##  III Black holes with novel scalar hair

Charged black hole solutions with scalar hair of the type that we are
suggesting here have been discussed previously in Herdeiro:2020xmb  and
Brihaye:2020vce , respectively. The system of equations has to be solved
subject to regularity conditions on the horizon where $N(r_{h})=0$ as well as
the requirement of asymptotic flatness and finite energy, respectively. While
$v(r_{h})=0$ is required for regularity, we have the choice to either fix the
value of the electric field on the horizon $\propto v^{\prime}(r_{h})$ or the
effective mass of the scalar field by setting $\Phi$. In this paper, we will
assume the former approach and show that when fixing the gravitational back-
reaction $\alpha$ we find that up to three branches of solutions exist when
varying $v^{\prime}(r_{h})$. The third branch, which has very peculiar new
features as compared to the solutions on the other two branches, had not been
noticed before. We will also demonstrate in the following that this phenomenon
appears for both the polynomial as well as for the exponential potential,
respectively.

###  III.1 Potential $U_{1}$

In order to understand the pattern of solutions, we have first decoupled the
matter fields from the space-time geometry, i.e. we have studied the case
$\alpha=0$ and fixed $\beta$, $r_{h}$ and $e>0$. When varying the electric
field on the horizon $\propto v_{h}^{\prime}$, we find the well known result
that two branches of solutions of scalar Q-clouds exist in
$\Omega\in[\Omega_{\rm min},1]$, where the upper value of the interval results
from the choice of potential parameters, while the value of $\Omega_{\rm min}$
depends on the choice of $\beta$, $r_{h}$ and $e>0$. This is shown in Fig. 1
for $\beta=9/32$, $r_{h}=0.15$ and $e=0.08$, where our numerical results
indicate that $\Omega_{\rm min}\approx 0.668$. The two solutions with equal
values of $\beta$, $r_{h}$, $e$ and $\Omega$ differ in the mass $M_{Q}$ and
Noether charge $Q_{N}$. The branch of solutions with lower (resp. higher)
value of mass $M_{Q}$ is labelled “A” (resp. “B”) and the two branches join at
$\Omega=\Omega_{\rm min}$. On branch A, the mass of the scalar cloud is
increased by adding scalar bosons. The Noether charge $Q_{N}$ (not given here)
shows a qualitatively very similar behaviour to $M_{Q}$. Now, scalar bosons
can be added up to a maximal effective mass of the scalar bosons,
corresponding to a minimal $\Omega$. Then, adding more scalar bosons requires
a decrease of the effective mass (branch B) and the maximum possible mass of
the cloud is reached at $\Omega=1$ on branch B. It is also interesting to note
that the increase in the number of scalar bosons making up the cloud leads to
a decrease in value of the electric field on the horizon, see Fig.1 (right).
We find that $v^{\prime}(r_{h})\approx 81.1$ for a solution on branch A close
to $\Omega=1$, while it decreases to $v^{\prime}(r_{h})\approx 15.7$ for the
cloud with the larges possible $M_{Q}$ and $Q_{N}$.

Now, letting the scalar cloud back-react on the space-time, we find that the
branches A and B exist on modified intervals of $\Omega$, see Fig.1 (left).
This is well understood and has been studied in Herdeiro:2020xmb  and
Brihaye:2020vce , respectively. The new feature in the back-reacting case that
has not been noticed so far is that a third branch of solutions in $\Omega$
exists (labelled “C” in the following). On this branch, the mass $M_{Q}$ as
well as the Noether charge $Q_{N}$ increase further, but now decreasing
$\Omega$. At the same time, the value of $v^{\prime}(r_{h})$ decreases to very
small values, eventually reaching value zero at the end of branch C (see Fig.1
(right)). To state it differently: at the end of branch C the horizon electric
charge $Q_{H}$ of the black hole becomes very small. This is shown in Fig. 2
(left). This means that all the electric charge is now in the cloud and the
black hole is essentially “discharged”.

Figure 1: Left: The dependence of the mass $M_{Q}$ on the parameter $\Omega$
for different values of $\alpha$ with $e=0.08$, $r_{h}=0.15$, $\beta=9/32$.
Right: The dependence of $M_{Q}$ on $v^{\prime}(r_{h})$ for the same set of
solutions.

Figure 2: Left: The dependence of the horizon electric charge $Q_{H}$ on
$\Omega$ for the solutions shown in Fig.1. Right: The dependence of the
Hawking temperature $T_{H}$ on $\Omega$ for the solutions shown in Fig.1 for
$\alpha>0$.

Interestingly, the Hawking temperature $T_{H}$ of the black holes on this
branch C shows a qualitatively different dependence on $\Omega$ than branch A
and B. This is shown in Fig. 2 (right), where we give $T_{H}$ in function of
$\Omega$. Obviously, on branch A and B, the black hole temperature increase
when decreasing $\Omega$, while on branch C, the temperature of the black hole
decreases with decreasing $\Omega$. Now, remembering that along the branches
from A to C, we increase the mass $M_{Q}$ and Noether charge $Q_{N}$, this
means that while on branch A the addition of scalar bosons to the cloud (and
hence the increase of mass $M_{Q}$) leads to an increase of the temperature,
the (further) addition of scalar bosons to the cloud leads to a decrease in
temperature on branch B and C. On branch B this happens for decreasing
effective mass of the scalar field, while on branch C the temperature
decreases with increasing $M_{Q}$ and increasing effective scalar boson mass.

Considering these physical parameters, the question is what distinguishes the
solutions on branch C from solutions on branch A and B. This is indicated in
Fig. 3, where we show the profiles of the scalar field function $\psi(r)$, the
electric field $\sim v^{\prime}(r)$ as well as the metric function $N(r)$ for
a solution on branch A (left) and a solution on branch C (right). For both
solutions, we have chosen $\alpha=0.09$, $e=0.08$ and $r_{h}=0.15$, but the
solution on branch A has $v^{\prime}(r_{h})\approx 60$, while the solution on
branch C has $v^{\prime}(r_{h})\approx 0.1$.

Obviously, for the solution on branch A (similarly for a solution on branch B)
the scalar function $\psi(r)$ is a monotonic function of the radial variable :
$\psi(r)$ decreases from its maximal value $\psi(r_{h})$ to zero at infinity,
where the fall-off is determined by the effective mass, i.e. by $\Omega$ (see
discussion above). Similarly, the function $v^{\prime}(r)$ determining the
electric field of the solution decreases monotonically and $N(r)$ increases
monotonically from its value zero on the horizon to unity asymptotically. This
is very different for a solution on branch C. Here, the scalar field is nearly
constant (and non-vanishing), while $v^{\prime}(r)$ is very close to zero on
an interval $r\in[r_{h},r_{c}]$. The value $r_{c}>r_{h}$ denotes the
approximate value of the radial coordinate at which the metric function $N(r)$
attains very small (but non-vanishing) values. Interestingly, instead of
forming an extremally charged black hole (which is a solution to the model for
$\psi\equiv 0$), the non-linear interaction between the curvature of space-
time, the electric field and the scalar field now leads to spatial
oscillations in the scalar field around zero on a finite interval of the
radial coordinate. These spatial oscillations are smaller in the other fields,
but are present, as can been seen for an amplified $v^{\prime}(r_{h})$ and the
metric function $\sigma(r)$, respectively, in Fig. 4 (left). In particular,
the solutions have significant amplitude in the scalar curvature as given by
the Ricci scalar $R$, see Fig. 4 (right). We do not give the Kretschmann
scalar $K=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ here, but would like to
state that the oscillations are also present for $K$. This makes us believe
that these are genuine oscillations that are not an artefact of the choice of
coordinates.

After a finite number of oscillations which decrease in amplitude for
increasing $r$, the scalar field becomes identically zero for $r>r_{a}$. The
solution for $r\in(r_{a}:\infty)$ clearly correspond to a Reissner-Nordström
solution : $\sigma\equiv 1$ and the Ricci scalar $R\equiv 0$, while the
electric field $\sim Q/r^{2}$. We hence find a novel black hole solution which
possesses

  * •

an inflating exterior for $r\in[r_{h}:r_{c}]$ – a phenomenon that was already
noticed and discussed in Brihaye:2020vce ,

  * •

“wavy” scalar hair on a intermediate interval $r\in(r_{c}:r_{a}]$ with metric
function $N(r)$ close to zero, and

  * •

vanishing scalar field with the electric and metric fields showing a Reissner-
Nordström behaviour for $r\in(r_{a}:\infty)$.

Figure 3: Left: The profiles of $\psi$ , $v^{\prime}$ and $N$ for a solution
on branch A with $\alpha=0.09$, $e=0.08$, $r_{h}=0.15$ and
$v^{\prime}(r_{h})\approx 60.0$. Right: The profiles of $\psi$ , $v^{\prime}$
and $N$ for a solution on branch C with the same values of $\alpha$, $e$ and
$r_{h}$, but $v^{\prime}(r_{h})=0.1$.

Let us insist here that the oscillations in the fields are not a numerical
artefact. We did perform several checks in this direction, e.g. we verified
the independence of the solution on the interval of integration and/or the
given tolerance. Note that since our numerical integrator colsys  uses an
adaptive grid scheme, changing the interval of integration implies also the
change of the discrete points at which the equations are evaluated. Moreover,
we find full branches of solutions that show a continuous dependence on the
parameters. Also, because the new branch of solutions is connected to the
already known branches, we don’t think that these solutions are radially
excited solutions similar to those observed in many non-linear models that
possess soliton and/or black hole solutions. These latter type of solutions
typically would appear as new, disconnected branches with physical parameters
very different in value.

Figure 4: Left: Functions $v^{\prime}(r)$ and $\sigma(r)$ for a solution on
branch C with $\alpha=0.09$, $e=0.08$, $r_{h}=0.15$. Right: The corresponding
Ricci scalar $R$ for this solution.

In order to understand these new solutions further, we have also studied the
case of varying back-reaction and found up to three branches – consistent with
the above analysis. An example is shown in Fig.5 for fixed $r_{h}=0.15$,
$e=0.08$ and $\Omega=0.8$ (corresponding to $\Phi=10$). Here, we give the
temperature $T_{H}$ (left) as well as the electric field on the horizon $\sim
v^{\prime}(r_{h})$ (right) in dependence on $\alpha$. As is obvious from this
figure, branch C is an extension of branch B, but – as mentioned above – has
different properties. E.g. while on branch B, the temperature decreases with
$\alpha$, it increases with $\alpha$ on branch C. Branch A and B are connected
to each other at $\alpha=0$, while branch A and C seem to start, respectively
end more or less at $T_{H}=0$. Hence, we find a (nearly) closed curve in the
$\alpha$-$T_{H}$-plane. Fig.5 (right) also demonstrates that there exists a
gap in $v^{\prime}(r_{h})$ for which no black hole solutions with scalar field
exist at all. Increasing $\alpha$, this gap increases slightly.

Figure 5: Left: We show the temperature of the black holes, $T_{H}$, in
dependence on $\alpha$ for $r_{h}=0.15$, $e=0.08$ and $\Omega=0.8$
(corresponding to $\Phi=10$). Right: The $\alpha$-dependence of the electric
field on the horizon $\sim v^{\prime}(r_{h})$ for the same family of
solutions.

###  III.2 Potential $U_{2}$

Studying the potential $U_{2}$, which is a priori better motivated physically
as it appears in certain gauge-mediated supersymmetry breaking models, we find
that the qualitative phenomenon of a third branch of solutions consisting of
black hole solutions with wavy scalar hair persists. In the following (and not
to repeat ourselves), we will put the emphasis on additional features of the
solutions, in particular, we want to emphasize the role of the gauge coupling
$e$ here. For that we have fixed $\alpha$ and varied $e$. Our results are
given in Fig. 6 for $\alpha=0.0012$ and $r_{h}=0.15$. This figure demonstrates
that the phenomenon described above appears only for values of $e$
sufficiently small. For large $e$, branch B extends all the way back to
$\Omega=1$, where it terminates. In this latter case, the black hole has
temperature significantly larger than zero and $v^{\prime}(r_{h})>0$. Only
when $e$ is small enough are we able to find branch C. For our choice of
parameters, only $e=0.02$ (in comparison to $e=0.03$ and $e=0.04$) allows the
existence of this new branch.

Figure 6: Left: We show the dependence of the temperature $T_{H}$ of the black
holes on $\Omega$ for $\alpha=0.0012$, $r_{h}=0.15$ and different values of
$e$. Right: Same as left for the electric field on the horizon $\sim
v^{\prime}(r_{h})$.

Knowing the qualitative behaviour of the solutions on branch C, we have then
checked how the solutions evolve along this branch C. So, in Fig. 7, we show
the profiles of the functions as well as the Ricci scalar $R$ in the interval
of $r$ where oscillations appear for $\alpha=0.0012$, $e=0.02$ and three
values of $v^{\prime}(r_{h})$. As is obvious from Fig. 6 (right) decreasing
$v^{\prime}(r_{h})$ leads to a decrease in $\Omega$ and hence an increase in
the effective mass of the scalar field. So, not suprisingly, we see that the
scalar field function oscillations (which are more or less equal in amplitude)
appear on a smaller interval of the radial coordinate $r$, i.e. are stronger
localized when moving along the branch C. At the same time, the minimal value
of $N(r)$ decreases and gets closer to zero and accordingly, the amplitude of
the Ricci scalar $R$ increases. Hence, the solution has a stronger spatial
variation in the scalar curvature when moving along the branch.

Figure 7: We show the evolution of solutions on branch C for $\alpha=0.0012$,
$e=0.02$ when varying $v^{\prime}(r_{h})$: profile of $\psi(r)$ (top left),
profile of $v^{\prime}(r)$ (top right), profile of $N(r)$ (bottom left) and
profile of the Ricci scalar $R$ (bottom right). Note that we are only showing
the interval of $r$ where oscillations of the respective profiles appear.

##  IV Conclusions

In this paper, we have studied a novel type of scalar hair which appears for
spherically symmetric black hole solutions coupled minimally to a U(1) gauge
field and a self-interacting complex valued scalar field. This scalar hair
appears for sufficiently large gravitational coupling and sufficiently small
gauge coupling. The subtle non-linear interaction between the curvature of
space-time, the scalar field and the electric field than leads to spatial
oscillations of the scalar field. Considering our results, it appears that the
solution is a result of the existence of a black hole horizon and a “barrier”
at spatial infinity which comes from the effective mass of the scalar field.
The decrease of the horizon electric field pushes the electric charge into the
Q-cloud outside the horizon such that at the end of the branch the horizon
electric charge of the black hole is essentially zero. The electrically
charged and gravitating cloud then forms a shell outside the horizon that gets
“squeezed” between the horizon and the barrier due to the scalar field
effective mass and consequently develops oscillations. These oscillations are
also present in the curvature scalars such as the Ricci scalar and the
Kretschmann scalar, respectively, and since they appear well outside the
horizon they might have observational consequences such as influencing the
motion of objects in the vicinity of the black hole. This is currently under
investigation and will be discussed elsewhere. As a final note, let us remark
that due to the similarity of the effects of a U(1) gauge field and rotation
in the context of the synchronization condition, it would be interesting to
see whether the scalar field oscillations are also present for the solutions
studied in Herdeiro_2014 . If so, there might be a new observational Ansatz to
observe black holes with scalar hair.

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# Panchromatic HST/WFC3 Imaging Studies of Young, Rapidly Evolving Planetary
Nebulae. II. NGC 7027

Paula Moraga Baez  School of Physics and Astronomy and Laboratory for
Multiwavelength Astrophysics, Rochester Institute of Technology, Rochester, NY
14623, USA  Joel H. Kastner  Chester F. Carlson Center for Imaging Science and
Laboratory for Multiwavelength Astrophysics, Rochester Institute of
Technology, 54 Lomb Memorial Drive, Rochester, NY 14623, USA  Bruce Balick
Department of Astronomy, University of Washington, Seattle, WA 98195, USA
Rodolfo Montez, Jr  Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, Cambridge, MA 02138, USA  Jesse Bublitz  Green Bank Observatory, 155
Observatory Road, Green Bank, WV 24944, USA

###### Abstract

The iconic planetary nebula (PN) NGC 7027 is bright, nearby ($D\sim 1$ kpc),
highly ionized, intricately structured, and well observed. This nebula is
hence an ideal case study for understanding PN shaping and evolution
processes. Accordingly, we have conducted a comprehensive imaging survey of
NGC 7027 comprised of twelve HST Wide Field Camera 3 images in narrow-band and
continuum filters spanning the wavelength range 0.243–1.67 $\mu$m. The
resulting panchromatic image suite reveals the spatial distributions of
emission lines covering low-ionization species such as singly ionized Fe, N,
and Si, through H recombination lines, to more highly ionized O and Ne. These
images, combined with available X-ray and radio data, provide the most
extensive view of the structure of NGC 7027 obtained to date. Among other
findings, we have traced the ionization structure and dust extinction within
the nebula in sub-arcsecond detail; uncovered multipolar structures actively
driven by collimated winds that protrude through and beyond the PN’s bright
inner core; compared the ionization patterns in the WFC3 images to X-ray and
radio images of its interior hot gas and to its molecular outflows; pinpointed
the loci of thin, shocked interfaces deep inside the nebula; and more
precisely characterized the central star. We use these results to describe the
recent history of this young and rapidly evolving PN in terms of a series of
shaping events. This evolutionary sequence involves both thermal and ram
pressures, and is far more complex than predicted by extant models of UV
photoionization or winds from a single central progenitor star, thereby
highlighting the likely influence of an unseen binary companion.

Planetary nebulae (1249), Stellar mass loss (1613), Jets (870), Circumstellar
matter (241)

##  1 Introduction

A planetary nebula (PN) is generated when an intermediate mass star ( 0.8-8
M⊙) evolves off the asymptotic giant branch (AGB), having ejected its dusty
AGB envelope at mass loss rates exceeding $\sim 10^{-5}$ $M_{\odot}$ yr-1
(Höfner & Olofsson, 2018). Within a few tens or hundreds of years after AGB
envelope ejection, depending on the star’s initial mass (e.g., Miller
Bertolami, 2016a), the increasing wind speed of the rapidly transitioning,
mass-losing star — from $\sim$10 km s-1 on the AGB to $\sim$1000 km s-1, post-
AGB — sweeps the AGB material into a dense shell, even as the UV from the
newly exposed, hot ($T_{eff}\sim$ 200 kK) AGB core begins to ionize the AGB
ejecta, forming the PN. The “classical” result of the foregoing post-AGB
evolutionary processes is a spherically symmetric, thick-walled PN with semi-
evacuated interior that is bright in atomic emission lines across the UV
through IR wavelength range (Kwok et al., 1978; Schönberner et al., 2018,
2005; Toalá & Arthur, 2014).

However, a significant fraction of PNe display profoundly axisymmetric and/or
point-symmetric structures (see, e.g., Parker et al., 2006, and references
therein). Furthermore, it has been known for decades that many of these highly
non-spherical objects harbor large masses of residual molecular gas and dust
that are directly descended from AGB envelope material (e.g. Kastner et al.,
1996; Bublitz et al., 2019). The present consensus is that the non-spherical
shapes and large residual molecular masses of these PNe are the direct
consequence of the presence and influence of binary companions during AGB
evolution (e.g., De Marco & Izzard, 2017, and references therein). Binary
companions can influence mass-launching and collimation mechanisms, which then
shape PNe by way of ram pressure in collimated flows or jets (e.g., Balick et
al., 2019). The details of these binary-induced PN shaping processes remain
elusive, however; fast mass loss driven by a binary companion’s accretion disk
as well as common envelope evolution are leading hypotheses to explain the
origin of collimated outflows (e.g., García-Segura et al., 2018; Zou et al.,
2020, 2022).

Hubble Space Telescope (HST) emission-line imaging studies of the most
recently formed, and hence most rapidly evolving, PNe provide particularly
effective means to understand PN ionization and shaping processes (e.g., Sahai
& Trauger, 1998). In HST’s Cycle 27, NGC 7027 was one of the first two PNe to
be targeted for a comprehensive, contemporaneous set of emission-line images,
from near-UV through optical to near-IR, with the Wide Field Camera 3 (WFC3)
(Kastner et al., 2020). We selected NGC 7027 for such a study because it is
among the youngest and most rapidly evolving PN within $\sim$1 kpc of the Sun
($D=0.89$ kpc; Masson, 1989). Indeed, NGC 7027 represents an “industry
standard” for emission-line searches and surveys, thanks to the sheer number
of bright lines it displays across the electromagnetic spectrum, from radio to
X-ray (e.g., Zhang et al., 2005, 2008; Montez & Kastner, 2018; Neufeld et al.,
2020).

Previous HST imaging using its Wide Field Planetary Camera 2 (WFPC2) and Near-
Infrared Camera and Multi-object Spectrometer (NICMOS) have established the
remarkably complex structure of the nebula (e.g., Latter et al., 2000;
Schönberner et al., 2018; Guerrero et al., 2020, and references therein). Its
distinguishing features include a bright, elliptical inner region, a dusty
equatorial belt, and multiple, point-symmetric collimated outflows, with the
latter confirmed kinematically via ground-based near-infrared and radio
interferometric imaging spectroscopy (e.g., Cox et al., 2002; Nakashima et
al., 2010). Concentric ring-like or spiral structures are observed to surround
the core region of NGC 7027 in HST/WFCP2 H$\alpha$ and V-band images (Guerrero
et al., 2020, and references therein). Radio-regime proper motion analyses of
the inner elliptical shell indicate a kinematic age of $\sim$600 yrs (Masson,
1989; Zijlstra et al., 2008), while nebular expansion analyses that are based
on multi-epoch HST imaging of the extended ring system suggest a dynamical age
in the range 1080–1620 yrs (Schönberner et al., 2018; Guerrero et al., 2020).
Consistent with such a recent departure from the AGB, the nebula harbors an
exceedingly hot ($\sim$200 kK) proto-white dwarf (pWD) at its center (Latter
et al., 2000).

First results from our Cycle 27 HST/WFC3 imaging programs targeting NGC 7027
and a second, well-studied PN — the similarly young (expansion age $\sim$2000
yr), nearby ($D\sim 1.0$ kpc) bipolar PN NGC 6302 — were presented in Kastner
et al. (2020). In Kastner et al. (2022), we presented a detailed overview of
the HST/WFC3 imaging survey of NGC 6302. Here, we present the full suite of
Cycle 27 HST/WFC3 images of NGC 7027, and we highlight compelling results
obtained from the image suite. In § 2, we describe the observations and image
processing that was performed to obtain the final suite of a dozen (mostly
narrow-band) WFC3 images, which are presented and described in § 3. In § 4, we
present key line ratio images that reveal the extinction and ionization
structure of the nebula. In § 5, the focus is shifted onto the central star
(CS) where its extinction, luminosity, and present-day mass are evaluated. In
§ 6, we develop and discuss extinction, ionization, and other lines of
evidence for the presence and structure of nebula-shaping shocks. The
implications of the results for the ongoing structural evolution of NGC 7027
are described in § 7. A summary of the results of our HST/WFC3 imaging study
of NGC 7027 is presented in § 8.

##  2 Observations

Images of NGC 7027 presented in this paper were obtained with both the UVIS
and NIR units of HST’s WFC3 during Cycle 27 in 2019 September. The UVIS (CCD
sensor) channel provides a field of view of 2.7′ x 2.7′ and pixel scale of
0.04′′, and the NIR channel uses a HgCdTe array with field of view of 2.27′ x
2.27′ and pixel scale of 0.13′′. Over the range of wavelengths imaged here,
the point-spread function (PSF) varies from $\sim 0.025^{\prime\prime}$ (near-
UV) to $\sim 0.16^{\prime\prime}$ (NIR). Table 1 summarizes the WFC3 filters
used to obtain images of NGC 7027 as well as the targeted emission line, the
date the image was obtained, and the total exposure time of imaging through
each filter. The UVIS images were obtained in 2-point GAP-LINE dither mode
(DITHER-LINE for quad filter FQ243N), and the IR images were obtained in
2-point DITHERBLOB dither mode. UVIS and IR images were obtained at a position
angle of –16.9∘ and –17.2∘ measured east of north, respectively.

Processing of images is described in the comprehensive HST/WFC3 imaging
targeting NGC 6302 (Kastner et al., 2022) and is briefly described here. The
standard pipeline calibration and processing, using CALWF3 v3.5.0, included
bias correction, dark current subtraction, flat field and shutter shading
corrections, cosmic-ray rejection, and (for UVIS images) CTE correction.
Additional post-pipeline processing was performed using the DrizzlePac package
software111https://drizzlepac.readthedocs.io/en/latest/index.html,
particularly the tweakreg and astrodrizzle modules. Fine image registration
was accomplished by defining the World Coordinate System of the F160W filter
image via the tweakreg module, using Gaia Data Release 2 catalog stars as
positional references. The other WFC3 images were then aligned (using
tweakreg) with the F160W image serving as the reference. This positional
calibration also included geometric distortion corrections and additional
cosmic-ray corrections. The dithered exposures of each long exposure image
were then merged using the astrodrizzle module. This process was implemented
to correct for any misalignment that remained after the initial pass through
HST’s image reduction pipeline. We estimate that, following this image
registration procedure, the WFC3 images of NGC 7027 are aligned to a relative
accuracy of $\leq$0.05′′.

Evidence for pixel saturation is present in the long-exposure F502N and F656N
images at the regions of highest surface brightness within the nebula,
specifically $\sim$10′′ to the northwest of the central star. Pixels that
exceeded the saturation threshold of 65,000 counts (see WFC3 Instrument
Handbook; Dressel 2021) were replaced with count rates from short-exposure
image pixels. This “pixel grafting” procedure was confined to displacements of
0′′ to $-$10′′ (RA) and 0′′ to 10′′ (dec) with respect to the position of NGC
7027’s central star.

Table 1: HST/WFC3 Imaging Survey of NGC 7027: Observation Summary Filter |  $\lambda_{0}$ ($\Delta\lambda$)a | Line Targeted | Date | Exp.
---|---|---|---|---
| (nm) |  |  | (s)
FQ243N | 246.8 (3.6) |  [Ne iv] $\lambda 2425$ | 2019-09-30 | 1140
F343N | 343.5 (25.0) |  [Ne v] $\lambda 3426^{b}$ | 2019-09-30 | 1140
F487N | 487.1 (6.0) |  H$\beta$ $\lambda 4861^{c}$ | 2019-09-30 | 1130
F502N | 501.0 (6.5) |  [O iii] $\lambda\lambda 4959,5007$ | 2019-09-30 |  1000, 30d
F656N | 656.1 (1.8) |  H$\alpha$ $\lambda 6563^{c}$ | 2019-09-30 |  1130, 30d
F673N | 676.6 (11.8) |  [S ii] $\lambda\lambda 6716,6730$ | 2019-09-30 | 1260
F110W | 1153.4 (443.0) | “YJ band” | 2019-09-30 | 556
F128N | 1283.2 (15.9) |  Pa$\beta$ 1.28 $\mu$m | 2019-09-30 | 506
F130N | 1300.6 (15.6) |  Pa$\beta$ continuum | 2019-09-30 | 506
F160W | 1536.9 (268.3) | “H band” | 2019-09-30 | 456
F164N | 1640.4 (20.9) |  [Fe ii] 1.64 $\mu$m | 2019-09-30 | 1306
F167N | 1664.2 (21.0) |  [Fe ii] continuum | 2019-09-30 | 1306

##  3 The HST/WFC3 Image Suite

Fig. 1 presents the complete suite of HST/WFC3 images obtained for NGC 7027,
spanning a range from near-UV ($\sim$245 nm) to near-IR ($\sim$1665 nm). These
images highlight the overall dramatic variation in the morphology of NGC 7027
over this wavelength range. Much of these differences can be attributed to the
varying effects of extinction, but also reflect the intensity of the specific
emission lines and continuum isolated by each of the filter passbands (Table
1). The structures observed in these images will be further discussed in the
following sections; here, we briefly summarize the main features of the
nebular morphology that is highlighted by the HST/WFC3 image suite.

In all images, the overall SE–NW extension of the nebula, which represents its
main fast outflow axis (e.g., Cox et al., 2002), is apparent. At the shortest
(near-UV) wavelengths, the nebula displays the most profound asymmetry along
this (SE–NW) direction. Specifically, in both the F243N and F343N images, the
regions immediately to the NW of the central star appear brighter than the
regions SE of the central star; this asymmetry is a consequence of the smaller
dust extinction toward the (forward-facing) NW regions (Montez & Kastner,
2018). The prominent extinction “hole” located $\sim$3′′ NW of the central
star discussed in the Chandra X-ray imaging study of Montez & Kastner (2018)
is especially apparent in both of these near-UV images. The nebula’s
equatorial dust “belt,” as well as a system of dusty filamentary structures
located $\sim$5′′ directly north of the central star, appear as prominent
features of the F343N, F487N, F502N images and, to a lesser extent, the F656N
and F673N images.

The near-IR images, covering the range from 1.15 $\mu$m (F110W) to 1.664
$\mu$m (F167N), are less affected by extinction due to intranebular dust, and
(as a result) are dominated by the inner elliptical shell and immediately
surrounding (“cloverleaf”) structures within the nebula that were previously
imaged by HST/NICMOS (Latter et al., 2000). The two images with the highest
surface brightness, F502N ($[$O iii$]$) and F656N (H$\alpha$), best reveal the
extended system of ring-like dust structures surrounding the nebula that was
previously imaged by HST using WFPC2 (Schönberner et al., 2018; Guerrero et
al., 2020).

In all of the WFC3 images (to greater and lesser degrees), jet-like
protrusions at position angles (PAs, measured clockwise from N) of $\sim$120∘
(ESE) and $\sim$300∘ (WNW) are seen superimposed on the aforementioned nebular
structures, with the WNW protrusion particularly prominent. Lesser protrusions
appear at PAs of $\sim$10∘ (nearly due N) and $\sim$200∘ (SSW). In addition, a
complex of dusty ejecta is seen oriented nearly due S (most notably in the
F343N, F487N, F502N, and F656N images); this south-directed ejecta complex has
a faint counterpart at PA $\sim$340∘ (NNW) that is most apparent in the F502N
and F656N images.

Figure 1: The complete suite of HST/WFC3 images obtained for NGC 7027. The
images are oriented with N up and E to the left. The field of view in each
image is 30” x 30”.

##  4 Line Ratio Images

###  4.1 Nebular extinction maps from H recombination lines

Figure 2: Top panels: H$\alpha$/H$\beta$ (left) and Pa$\beta$/H$\beta$ (right)
line ratio images, as obtained from the F656N/F487N and (F128N$-$F130N)/F487N
image ratios (respectively). The lower display limits are fixed at the
theoretically predicted ratios of 2.85 and 0.162, respectively, to highlight
regions where extinction is present. Bottom panels: c(H$\beta$) calculated
from the H$\alpha$/H$\beta$ (left) and Pa$\beta$/H$\beta$ (right) line ratio
images.

The three H recombination line images obtained as part of the WFC3 image suite
— i.e., H$\beta$, H$\alpha$, and Pa$\beta$, as obtained through WFC3 filters
F487N and F656N and from the F128N$-$F130N difference image, respectively —
provide a means to map dust extinction across the nebula at the $\sim$0.1′′
resolution of HST/WFC3. To do so, we followed the methodology described in
Kastner et al. (2022); i.e., we constructed H$\alpha$/H$\beta$ and
Pa$\beta$/H$\beta$ line ratio images, and compared these per-pixel line ratios
with the theoretically predicted ratios, so as to obtain the spatial
distribution of the extinction parameter $c(H\beta)$ for a given choice of
reddening law. To construct $c(\mathrm{H}\beta)$ images from the
H$\alpha$/H$\beta$ and Pa$\beta$/H$\beta$ line ratio images, we adopt
intrinsic relative intensities for H$\alpha$/H$\beta$ and Pa$\beta$/H$\beta$
of 2.85 and 0.162, respectively, which correspond to a “canonical” values of
nebular electron temperature ($10^{4}$ K) and electron density ($10^{4}$
cm-3), with very little sensitivity to either parameter (e.g., Osterbrock,
1989). We then adopt a uniform value of $R_{V}=3.1$ to generate the
$c(\mathrm{H}\beta)$ images, with the caveat that $R$ depends on grain size,
and grain size would not realistically be uniform across the entire PN.

The resulting $c(\mathrm{H}\beta)$ images are shown in the bottom panels of
Fig. 2. Ideally, if the $R_{\lambda}$ value used is correct, the c(H$\beta$)
images obtained from the two line ratios should be identical. Indeed, the
c(H$\beta$) images, shown in the bottom panels of Fig.4, are nearly identical
in morphology. The key differences in values occur in the nebula’s waist and
the eastern region outlining the north lobe, with the $c(\mathrm{H}\beta)$ map
derived using Pa$\beta$/H$\beta$ having systematically larger values than the
$c(\mathrm{H}\beta)$ map obtained from H$\alpha$/H$\beta$. Specifically,
values across the $c(\mathrm{H}\beta)$ map obtained from Pa$\beta$/H$\beta$
range from $\sim 0.9$ (lobes) to $\sim 2.1$ (nebula waist), whereas the values
in the $c(\mathrm{H}\beta)$ map obtained from H$\alpha$/H$\beta$ lie in the
range $\sim 0.9-1.7$. Other, more subtle differences between the extinction
maps could be due to the relatively large variation in the PSFs between
Pa$\beta$ and H$\beta$, which would affect the resolution of structures across
the Pa$\beta$/H$\beta$ ratio image moreso than in the H$\alpha$/H$\beta$ ratio
image, and any dust scattering of H$\alpha$ emission that may be present.

Overall, the results for $c(\mathrm{H}\beta)$ are consistent with previous
studies that mapped extinction across NGC 7027 (Kastner et al., 2002; Montez &
Kastner, 2018, and references therein). In particular, previous extinction
maps also feature extinction maxima along the waist of the nebula, as well as
a prominent gap or “hole” in extinction located $\sim$5–10′′ to the WNW of the
central star, a region that is also bright in soft X-rays (Montez & Kastner,
2018). The apparent lack of obscuring nebular material at this location in NGC
7027 is likely due to the disruptive effects of a fast, collimated outflow, as
is discussed in detail in § 6.1.

However, in contrast to earlier extinction studies, the $\sim$0.1′′ resolution
of the line ratio images and (hence) $c(\mathrm{H}\beta)$ maps obtained from
the HST/WFC3 images (Figure 2) reveal the dust structures within NGC 7027 in
unprecedented detail. North of the CSPN lies dusty lobe structure consisting
of a series of ripple-like filaments that terminate at the lobe’s northernmost
extension, $\sim$10′′ from the CSPN. In the southern regions of the inner
nebula, opposite the CSPN from the northern filament structures, the
$c(\mathrm{H}\beta)$ map reveals dust rims and arcs that appear to outline
bow-shock-like structures. These features, along with the large knot along the
waist of the nebula, appear more prominent in the $c(\mathrm{H}\beta)$ image
derived from Pa$\beta$/H$\beta$ (Figure 2, lower right panel). All of the dust
structures revealed in the $c(\mathrm{H}\beta)$ maps in Figure 2 appear to
have direct counterparts in near-IR H2 emission (see, e.g., Fig. 3 in Latter
et al., 2000), indicating that these structures constitute dense regions of
dust and molecular gas that have survived the processes of photoionization and
shock-driven ionization that are ongoing in the inner nebula (see next).

In Appendix A, we describe how these c(H$\beta$) images are used to perform
extinction corrections for several of the HST/WFC3 images.

###  4.2 High and low excitation lines: ionization gradients

In Figs. 3 and 4 we present a series of emission line and emission line ratio
images obtained from the WFC3 image suite. Fig. 3 displays the H$\alpha$
emission line and a series of line ratio images obtained for the high-
excitation forbidden lines $[$O iii$]$, $[$Ne v$]$, $[$Ne iv$]$, and the
H$\alpha$ and H$\beta$ recombination lines. Given the wavelength separation,
the $[$O iii$]$/H$\alpha$ ratio image was corrected for extinction (as
described in Appendix A) to mitigate the effects of intranebular dust (see
previous section). As detailed in the Appendix, the $[$O iii$]$/H$\beta$ ratio
line image does not require extinction correction because of the small
wavelength separation between the two emission lines; this ratio image
displays structure essentially identical to that seen in the $[$O
iii$]$/H$\alpha$ ratio image. Note that the only differences between these
line ratio images are dusty structures along the N lobe and the equatorial
belt, both of which are likely artifacts left from the extinction corrections
performed on the $[$O iii$]$ and H$\alpha$ images.

The $[$O iii$]$/H$\alpha$ ratio image appears to trace an elliptical shell of
efficient O+ ionization (relative to H ionization) with a semimajor axis of
$\sim$4′′ ($\sim$3600 au). This shell marks the zone where photons with
energies between 35.1 and 54.9 eV (i.e., those required for ionization of O+
and O++, respectively), are plentiful, so as to specifically generate $[$O
iii$]$ emission. Closer to the central star (CS), inside of the enhanced $[$O
iii$]$/H$\alpha$ region, the O is likely to be more highly ionized (O+2, O+3,
etc) by photons with energies $>$54.9 eV, while outside the shell of enhanced
$[$O iii$]$/H$\alpha$, most photons with energies $>$35.1 eV have been
absorbed by the interior regions of ionized nebular gas.

This inner zone of very high excitation is revealed in the various line ratio
images involving $[$Ne iv$]$ and $[$Ne v$]$ in Fig. 3. The energies required
for ionization of O+ and Ne3+ are 35.1 eV and 97.1 eV, respectively, such that
one expects the $[$Nev$]$ emitting region to lie interior to that of $[$O
iii$]$. That is, photons with energies $>$97.1 eV would be absorbed closer to
the central star, via ionization of Ne+3, producing $[$Ne v$]$ emission; while
photons with energies $<$97.1 eV but $\geq$35.1 eV can escape to ionize O+,
generating $[$O iii$]$ outside of the $[$Ne v$]$ emission region. This
expectation is indeed confirmed by the $[$Ne v$]$/H$\beta$ ratio image; this
image shows a patchy inner ring of bright $[$Ne v$]$ that resides interior to
that of the enhanced $[$O iii$]$, which appears as a bright ring in $[$O
iii$]$/$[$Ne v$]$. However, the $[$O iii$]$/$[$Ne v$]$ ratio image also
reveals prominent, extended $[$O iii$]$ emission that overlaps shocked
regions, suggesting that photoionization effects could be boosted by shock
mechanisms occurring within the nebula (see below).

Figure 3: HST/WFC3 line ratio images of NGC 7027 that are potentially
diagnostic of photoionization. Top right: $[$Oiii$]$/$[$Nev$]$ (F502N/F343N);
bottom left: $[$Nev$]$/$[$Neiv$]$ (F343N/FQ243N); bottom right:
$[$Nev$]$/H$\beta$ (F343N/F487N). The H$\alpha$ (F656N) image is displayed in
the top left panel, for reference. Figure 4: HST/WFC3 line ratio images of NGC
7027 that are potentially diagnostic of excitation by shocks. Top
right:)$[$Oiii$]$/H$\beta$ (F502N/F487N); bottom left: $[$Sii$]$/H$\alpha$
(F673N/F656N); bottom right: [Fe ii]/continuum (F164N/F167N). The H$\alpha$
(F656N) image is displayed in the top left panel, for reference.

Fig. 4 displays emission line and emission line ratio images featuring
H$\alpha$, $[$O iii$]$, and the low excitation forbidden lines $[$S ii$]$ and
$[$Fe ii$]$. Extinction correction is not necessary for $[$S ii$]$/H$\alpha$
because of the close proximity between $[$S ii$]$ (0.676$\mu$m) and H$\alpha$
(0.656$\mu$m) emission lines. These line ratio images are displayed alongside
an image of the ratio of $[$Fe ii$]$ 1.64 $\mu$m forbidden line emission with
respect to neighboring 1.67 $\mu$m continuum emission (we use this F164N/F167N
ratio image in preference to a F164N$-$F167N difference image, due to its
superior signal-to-noise ratio). The 1.67 $\mu$m continuum emission is likely
due to free-free emission and hence (like H$\alpha$) traces ionized H. As
discussed in Kastner et al. (2022), both the 0.676 $\mu$m $[$S ii$]$ and
(especially) 1.64 $\mu$m $[$Fe ii$]$ emission lines are potential tracers of
shock-driven excitation. Fig. 4 reveals that the peaks in $[$S ii$]$ and $[$Fe
ii$]$ emission (with respect to ionized H) in NGC 7027 are highly localized at
the tips of the breakout regions to the southeast and northwest of the central
star, with fainter, enhanced emission from both lines seen surrounding the
bright ring of enhanced $[$O iii$]$/H$\alpha$. Furthermore, the brightest
$[$Fe ii$]$ lies just interior to the peak in $[$S ii$]$/H$\alpha$. As
described in § 6.1, this nested structure likely traces shock excitation at
the outskirts of NGC 7027. Furthermore, the $[$O iii$]$/H$\alpha$ ratio image
also displays a finger-like protrusion to the NW that closely follows the
zones of brightest $[$Fe ii$]$ and $[$S ii$]$ emission. This correspondence
suggests that the same collimated jet or shock mechanism that generates $[$Fe
ii$]$ and $[$S ii$]$ emission in the NW breakout structure could be
contributing to ionization of O+ and, hence, to $[$O iii$]$ emission, which
would be consistent with studies indicating that [O iii] emission can trace
shock zones (in addition to photoionization) in PNe (Guerrero et al., 2013).

The $[$Ne v$]$/$[$Ne iv$]$ ratio image in Fig. 3 should also provide a map of
the reach of highly ionizing radiation. However, it is apparent that the
signature of intranebular extinction remains strong in this ratio image,
despite both component images being subject to the extinction correction
procedure described in the Appendix. Specifically, the dark features around
the nebula’s waist and toward its northern reaches in the $[$Ne v$]$/$[$Ne
iv$]$ image correspond to regions where extinction maps reveal large amounts
of dust and dust scattering (see Fig. 2). Intriguingly, the overall morphology
of this image resembles that of the broad-band Chandra X-ray image of NGC 7027
(Montez & Kastner, 2018, see below). This suggests that the regions of
enhanced $[$Ne v$]$ emission (relative to $[$Ne iv$]$) might trace a zone of
heat conduction from the shocked, X-ray-emitting ($\sim 10^{6}$ K) plasma to
the ($\sim 10^{4}$ K) photoionized gas. Alternatively, the $[$Ne v$]$/$[$Ne
iv$]$ image could be affected by scattering, especially in the outer regions
of the point-symmetric structures, given that scattering is more efficient at
smaller wavelengths, and the scattered UV photons can most easily escape
through the low-extinction “holes” formed by the blowouts associated with
regions of shocks (see below).

##  5 Extinction toward the Central Star

The emission line ratio-based approach to measuring intranebular extinction
described in § 4.1 cannot be applied to determine the extinction along the
line of sight directly toward the CSPN of NGC 7027, as the star is a bright
continuum source. Such an extinction estimate is essential in order to
estimate the CSPN’s present effective temperature Teff and luminosity $L$, and
thereby ascertain the present mass and evolutionary state of the CSPN via
comparison with theoretical models (e.g., Miller Bertolami, 2016a). Hence, we
estimated the extinction toward the CSPN via comparison of the star’s spectral
energy distribution with stellar atmosphere models calculated for hot central
stars of PNe. To this end, measurements of the CSPN flux, corrected for local
nebular background, were made for 9 HST/WFC3 images spanning a wavelength
range from $0.3435\ \mu$m to $1.6677\ \mu$m. Fig. 5 displays these observed
fluxes.

Figure 5: The spectral energy distribution of the NGC 7027 CS compared to a
200 kK pure H WD model atmosphere for $\log(g)=7.0$ (dashed blue line) and 198
kK blackbody (solid gray line) before and after correcting fluxes for
extinction. Red stars are uncorrected fluxes obtained from HST/WFC3 image
photometry; red triangles are uncorrected fluxes from Latter et al. (2000).
Extinction-corrected fluxes adopting reddening laws from Cardelli et al.
(1989), Whittet (1992), and Fitzpatrick et al. (2019) are represented as
purple, teal, and green symbols; corrected fluxes from Latter et al. (2000)
are represented as yellow triangles.

The resulting spectral energy distribution (SED) was then compared with a
Tübingen NLTE model atmosphere222http://astro.uni-tuebingen.de/
rauch/TMAF/TMAF.html (Werner et al., 2012) for a model with $T_{eff}=$ 200 kK
(based on the CS effective temperature inferred by Latter et al., 2000). A
pure H model with $\log(g)=7.0$ was selected, for simplicity. Including near-
IR and optical points from Latter et al. (2000), the extinction is estimated
to be $A_{V}$ = $2.63$ using the Cardelli et al. (1989) reddening law with
$R_{\lambda}$=3.1, $A_{V}$ = $2.54$ using the Whittet (1992) reddening law,
and $A_{V}$ = $2.90$ using the Fitzpatrick et al. (2019) reddening law. To
correct the CS fluxes, we adopted the mean of these extinction estimates,
$A_{V}$ = $2.69$. In Fig. 5, uncorrected and extinction corrected flux points
are overlaid with a 198 kK blackbody, as suggested by Latter et al. (2000),
for comparison. Since the extinction correction results for the CS are
consistent with the 198 kK blackbody approximation, the blackbody model was
used to approximate the total flux of the CS for purposes of luminosity
estimation. The total bolometric luminosity obtained for the CS of NGC 7027 is
$\sim 6.2\times 10^{3}$ L⊙, which agrees well with previous results, all
falling within the range $\sim 5.5\times 10^{3}-2\times 10^{4}$ L⊙ (Beintema
et al., 1996; Masson, 1989).

Table 2: Properties of NGC 7027 CS Temperature | Luminosity |  Agea |  Massa
---|---|---|---
(kK) |  (L⊙) | (yrs) |  (M⊙)
155b |  $9.9\times 10^{3}$ b | $\sim 100$ |  $\sim$0.7
198c | $6.2\times 10^{3}$ | $<1000$ |  $\sim$0.7

Table 2 summarizes the estimated age and present-day mass of NGC 7027’s CS
obtained from the preceding results for luminosity and effective temperature
as well as the CS parameters derived by Schönberner et al. (2005), based on
stellar evolution models presented in Miller Bertolami (2016b). These age and
mass estimates are consistent with results in Latter et al. (2000), who
estimated a post-AGB age of 700 yrs and present-day CS mass of $0.7$ M⊙, as
well as with estimates of CS mass and post-AGB age obtained from other post-MS
evolutionary models. For example, stellar evolution models in Weidmann et al.
(2020) yield an age of $\sim 1000$ years and final mass of $0.6-0.65$ M⊙ for a
CS with a luminosity of $6.2\times 10^{3}$ erg s-1 and an effective
temperature of $198$ kK. Similarly, from the models presented in Valenzuela et
al. (2019) we would estimate a final mass of $0.616-0.706$ M⊙, while using the
core mass vs. luminosity relation for M $>$ $0.5$ M⊙ stars in Vassiliadis &
Wood (1994), we estimate a final CS mass of $\sim\ 0.61$ M⊙. All of these
estimates would suggest that the CS is descended from a progenitor of mass
$\sim$3 $M_{\odot}$ (e.g., Miller Bertolami, 2016a).

##  6 Evidence for Shocks in NGC 7027

The presence of strong, wind-collision-generated shocks in NGC 7027 was
revealed by the Chandra X-ray Observatory detection of extended, luminous
X-ray emission from the nebula extending over the energy range $\sim$0.3-3.0
keV (Kastner et al., 2001; Montez & Kastner, 2018). Montez & Kastner (2018)
estimate that the velocities of the outflows that generate the X-ray-emitting
regions of NGC 7027 may be as high as $v_{\mathrm{wind}}\sim 500$ km s-1, on
the basis of the inferred characteristic temperature of the X-ray emitting
plasma ($\sim 3\times 10^{6}$ K). In Fig. 6 we present an overlay of the hard
(1.0$-$3.0 keV) Chandra X-ray image from Montez & Kastner (2018) on the
HST/WFC3 $[$Ne v$]$ and H$\alpha$ images. It is apparent from this Figure that
the hard X-ray emission closely traces the edge of the inner, highly ionized
([Ne v]-bright) region of the nebula, with the exception of the equatorial
region to the ENE and WSW of the central star. These regions correspond to
directions of the largest dust extinction, as seen in Fig. 2, indicating that
the X-ray emission in these regions is likely being heavily absorbed by
cooler, surrounding, CNO-rich gas (see discussion in Montez & Kastner, 2018).
Furthermore, the X-ray surface brightness protrusions to the NW and SE of the
central star closely correspond to regions of high-velocity gas detected in
near-IR H i (Br$\gamma$) emission (Cox et al., 2002).

In this section, we present additional evidence for shocks in NGC 7027,
through line ratio diagnostics obtained from the [O iii]/H$\alpha$ and [S
ii]/H$\alpha$ ratio images along with a comparison of nebula cross sections in
these line ratios and those of [Fe ii] and X-ray surface brightness.

Figure 6: Color composite of Chandra and HST/WFC3 images of NGC 7027 showing
overlay of hard X-rays (1.0$-$3.0 keV; red image and contours), extinction
corrected $[$Ne v$]$ (green image), and extinction corrected H$\alpha$ (blue
image).

###  6.1 Line ratio diagnostics

Excitation diagrams obtained from diagnostic line ratios, long used to
classify AGN (e.g., Kewley et al., 2006), can, in principle, also be used to
distinguish between photoionized regions and fast low-ionization structures
(LISs), or shocked material, in PNe (Danehkar et al., 2018; Montoro-Molina et
al., 2022). Here, we follow the methodology described in Danehkar et al.
(2018) to quantitatively identify the shocked regions in NGC 7027.

A histogram was produced for $\sim 6\times 10^{5}$ pixels in the [O
iii]/H$\alpha$ and [S ii]/H$\alpha$ line ratio images for an area that covers
$30^{\prime\prime}\times 30^{\prime\prime}$. The results are presented in the
left panel of Fig. 7. The dividing line overlaid on the histogram should
distinguish between pixels that belong to photoionized regions and fast LISs,
wherein the photoionized regions are characterized by enhanced [O
iii]/H$\alpha$ and suppressed [S ii]/H$\alpha$ and fast LISs are characterized
by suppressed [O iii]/H$\alpha$ and enhanced [S ii]/H$\alpha$. Danehkar et al.
(2018) defines this nebular photon-shock dividing line to be parallel to a
classification line that differentiates between Seyfert and low-ionization
nuclear emission-line region (LINER) galaxies. The definition was modified to
take into account diagnostic diagrams for axisymmetric simulations of fast
LISs and observations of photoionized gas in (Raga et al., 2008).

The excitation diagram was then used to classify each pixel and then to color
code an H-alpha intensity image of the nebula. The result is shown in the
middle panel of Fig. 7, where blue represents photoionized regions and red
represents fast LISs. The results indicate that the elliptical interior region
of NGC 7027 is dominated by photoionization, while the most extensive shocked
regions, according to this analysis, correspond to the same directions as the
X-ray protrusions (Montez & Kastner, 2018) and the fast flows seen in
Br$\gamma$ emission Cox et al. (2002).

Most PN studies have, by necessity, employed spatially integrated line ratios
to determine where nebulae lie on a given excitation diagnostic diagram such
as that in the left panel of Fig. 7 (e.g., Frew & Parker, 2010; Raga et al.,
2008). To illustrate how this analysis would play out for NGC 7027, we have
indicated the mean line ratios obtained from the NGC 7027 images, $[$O
iii$]$/H$\alpha=2.4\pm 0.7$, $[$O iii$]$/H$\beta=14.3\pm 1.4$ and $[$S
ii$]/$H$\alpha=0.09\pm 0.05$, in the diagnostic diagrams–left and right panels
in Fig. 7. Note that these mean ratios are within the “photoionized” zone of
the diagram, as a consequence of the relatively small surface area of the
nebula that displays LIS-like ratios. Thus, if only spatially integrated line
ratios were available, one might conclude there were no shocks present in NGC
7027. We caution, however, that this young and rapidly evolving PN may
represent an unusual case study in the application of spatially resolved line
ratio diagnostics to distinguish between shocked and photoionized regions (see
discussion in Montoro-Molina et al., 2022).

Figure 7: Diagnostic diagram analysis to distinguish between photoionized and
shocked regions within NGC 7027. The left panel presents an excitation diagram
that differentiates between $[$O iii$]$ and $[$S ii$]$ emission due to fast
LISs vs. photoionization. The color bar indicates the number of pixels in a
given area of the diagram. The mean flux for both line ratios is indicated by
the ‘x’. The nebular photon-shock dividing line, adapted from Danehkar et al.
(2018), is $1.89\log([\mathrm{S\ {\sc
II}}]/\mathrm{H}\alpha)+2.0=\log([\mathrm{O\ {\sc III}}]/\mathrm{H}\alpha)$.
In the middle panel, the H$\alpha$ image of NGC 7027 is mapped with blue and
red pixels representing photoionization and fast LISs, respectively, as
obtained from the diagram in the left panel. The right panel displays a
‘Baldwin, Phillips, and Terlevich’ (BPT) diagram, commonly used to classify
AGN (Kewley et al., 2006), using $[$S ii$]$/H$\alpha$ and $[$O iii$]$/H$\beta$
emission line ratio images. A different nebular photon-shock dividing line,
here adapted from Danehkar et al. (2018) as $1.89\log([\mathrm{S\ {\sc
II}}]/\mathrm{H}\alpha)+2.46=\log([\mathrm{O\ {\sc III}}]/\mathrm{H}\beta)$,
is plotted to differentiate between emission due to fast LISs vs.
photoionization as well. A similar color bar indicates the number of pixels in
a region of the diagram, and an ‘x’ was placed to mark the mean flux of both
emission line ratio images.

###  6.2 Nested shock structure

In Fig. 8, we present a line ratio image overlay for NGC 7027 that compares
the positions of bright $[$S ii$]$ and $[$Fe ii$]$ emission with the X-ray
emission morphology as well as the zones of high extinction (as traced by the
H$\alpha$/H$\beta$ line ratio). The Figure reveals a stratified structure,
wherein the $[$S ii$]$ and $[$Fe ii$]$ emitting regions display bow shock
structures extending well beyond the X-ray emission, and the $[$Fe ii$]$
emission nestles nicely inside the $[$S ii$]$ emission. This set of nested
bright features in X-rays, $[$Fe ii$]$, and $[$S ii$]$ along the (SW–NE)
direction — which also constitutes the primary direction of fast LISs
identified via the diagnostic diagram analysis (Fig. 7) — serves as clear
evidence that the shocks due to fast outflows impinging on the nebular
material are strongest in this particular direction.

Figure 8: Three-color composite of NGC 7027. Red is F656N/F487N
(H$\alpha$/H$\beta$), green is F673N/F656N ($[$Sii$]$/H$\alpha$), and blue is
F164N/F167N ($[$Feii$]$). White and yellow contours trace hard and soft X-ray
emission, respectively. Figure 9: Overlay of line ratio and emission cross
sections through NGC 7027. Bottom panels show the images ( [O iii]/H$\beta$,
[S ii]/H$\alpha$, [Fe ii], and [energy here] X-rays) from which the cross
sections were extracted for the nebula, with the cross sections indicated by
yellow lines. The top panel plots the normalized amplitude for each image’s
cross section vs. the offset from the central star in arcsec (bottom axis) and
cm (top axis).

To further elucidate this structure, we present in Fig. 9 a cross-section of
the nebula in X-rays, $[$S ii$]$, $[$Fe ii$]$ as well as $[$O iii$]$ emission
along this same direction. This cross-section plot confirms the nested
structure of these shock tracers, and reveals the detailed structure of the
particularly strong shocks that are propagating along this direction within
the nebula. As discussed below (§7), the clarity of these shocks compared to
the nebula’s other outflows indicates that the jets emanating along the SW–NE
direction could dominate the future shaping process of NGC 7027, perhaps
eventually forming a bilobed, pinch-waisted PN that appears as a classical
“butterfly” nebula, like NGC 6302 (Kastner et al., 2022).

The nested spatial distributions of the $[$S ii$]$ and $[$Fe ii$]$ emission
lines also should aid in constraining the densities within these regions of
NGC 7027. In a study of the symbiotic system BI Crucis, whose excitation is
largely collisional, Contini et al. (2009, and references therein) point out
that $[$Fe ii$]$ emission should dominate over Fe ii permitted lines if the
gas density of a nebula falls between 10${}^{2}-$104 cm-3, and both $[$Fe
ii$]$ and Fe ii lines should be excited if the gas density lies between
10${}^{6}-$108 cm-3. Similarly, the presence of $[$S ii$]$ emission constrains
the gas density to be $<$104 cm-3 (Acker & Jaschek, 1995). Since NGC 7027’s
optical spectrum displays fairly weak permitted Fe ii and S ii lines compared
to their forbidden emission counterparts (Zhang et al., 2005), we infer a
nebular density of $<$104 cm-3 in the NW and SE lobe structures. However,
these constraints also suggest that the gas from which $[$Fe ii$]$ emission
arises should have a higher density than regions where $[$S ii$]$ emission is
observed, since regions of $[$S ii$]$ emission are collisionally de-excited
(hence quenched) at a density $<$104 cm-3, but regions of $[$Fe ii$]$ emission
can reach densities of $\sim$105 cm-3 without having visible Fe ii lines. The
expectation that $[$Fe ii$]$ should trace denser gas than $[$S ii$]$ is indeed
consistent with the nested emission structure apparent in Fig. 9.

##  7 The shaping of NGC 7027

Previous sections of this paper have developed the deepest-ever look into the
structure and current state of NGC 7027 and its central star. In this section,
we use these data to attempt to peek back into the source and evolution of
mass injection and nebular shaping.

Figure 10: Left panel, three color composite of NGC 7027 generated from
Pa$\beta$/H$\beta$ (red), extinction corrected H$\alpha$ (green), and 1.3 mm
radio continuum (blue; from Bublitz et al., 2022). Right panel, schematic
diagram outlining the structural components discussed in § 7.1, overlaid on a
color composite constructed from the WFC3 image suite by STScI staff. Three
pairs of jets, labeled as Outflows 1, 2, and 3 (see Cox et al., 2002), are
traced with arrows that are color-coded to indicate the relative
redshift/blueshift for each outflow component. The nebula’s outer ring system,
central elliptical shell (long-dashed ellipse), and equatorial belt (short-
dashed arc) are also indicated.

###  7.1 A series of shaping events

Fig. 10 provides an overview of the key structures observed in NGC 7027. The
left panel presents an overlay of HST/WFC3 extinction and H$\alpha$ images, as
well as a 1.3 mm radio continuum image obtained with the NOEMA interferometer
(from Bublitz et al., 2022); a schematic diagram of NGC 7027 outlining the
primary structural features is presented in the right panel, overlaid on a
color composite constructed from the WFC3 image suite by STScI
staff333https://www.nasa.gov/feature/goddard/2020/hubble-provides-holistic-
view-of-stars-gone-haywire. In the schematic diagram, each of the three
primary previously identified outflows (Cox et al., 2002; Nakashima et al.,
2010; Bublitz et al., 2022) is traced by dotted arrows, with color-coded red
and blue extremes to signify their relative redshifts and blueshifts,
respectively. The elliptical shell seen in the WFC3 NIR images and radio
continuum image is traced by a dashed ellipse. The outer ring system, which is
most clearly visible in the H$\alpha$ and [O iii] images (Fig. 1), as well as
the equatorial belt of NGC 7027 are also indicated in the schematic.

We propose that this complex set of structures is most likely the result of a
series of PN shaping events resulting from binary system interactions
involving NGC 7027’s AGB star progenitor and an unseen compact companion.
Binary interactions while the primary star was still on the AGB likely
resulted in the spirals seen surrounding NGC 7027 (in scattered light) in the
$[$O iii$]$ and H$\alpha$ images. Such ring or spiral patterns in AGB ejecta
are potential signatures of the presence of a wide (separation $\sim$1–10 au)
binary system (e.g., Mastrodemos & Morris, 1999; Chen et al., 2020, and
references therein) and, indeed, have been detected in the ejected envelopes
of AGB stars that are suspected binary systems (e.g., R Scl; Maercker et al.,
2012, and references therein). The estimated post-AGB age of NGC 7027’s
central star, $<1$ kyr (§ 5), is significantly less than the estimated age of
the rings, $\sim 2$ kyr (Guerrero et al., 2020), supporting the idea that
these outer rings formed before the CS left the AGB. The nebula’s prominent
equatorial belt might then trace the binary’s orbital plane, where the bulk of
the AGB star’s ejected envelope mass would have accumulated (e.g., Mastrodemos
& Morris, 1999).

Of the three main outflows noted in Fig. 10, Outflow 1 (the focus of the
discussion in § 6.2) displays the most pronounced evidence for strong shocks,
at the present epoch. The fact that Outflows 2 and 3 appear less prominent or
absent in the shock tracers [S ii] and [Fe ii] relative to Outflow 1 may
indicate that Outflows 2 and 3 are older than (were ejected prior to) Outflow
1. Indeed, Outflows 2 and 3 could have occurred during NGC 7027’s post-AGB but
pre-planetary nebula (PPN) phase (Sahai & Trauger, 1998; Sahai, 2000; Akashi &
Soker, 2021), perhaps during repeated close binary encounters that culminated
in the unveiling of the central star (see below). These two pairs of jets have
triangular shapes that may be bow shocks generated by narrow, collimated jets
that were launched when the outer rings surrounding the nebula were closer to
the CS. Furthermore, Outflow 3 displays a striking misalignment relative to
Outflows 1 and 2, wherein Outflow 3 appears to be oriented close to NGC 7027’s
equatorial plane (Bublitz et al., 2022, and references therein). Some PPNe
display a similar combination of recent, collimated outflows — some with
similarly surprising misalignments — impinging on older, more slowly expanding
ring/spiral structures (AFGL 2688 perhaps being the best-studied example of
this phenomenon; e.g., Cox et al., 2000; Balick et al., 2012).

The central elliptical shell of NGC 7027 (Fig. 10) likely consists of the
ejected material from the former AGB star that has been swept up by a fast,
quasi-spherical post-AGB wind. The age of the CS since leaving the AGB has
been estimated consistently to be $\sim$100-1000 yrs (§ 5). However, since
this post-AGB CS age pertains to a single star rather than a central star
subject to binary system interactions that would, presumably, truncate AGB
evolution, this estimate is most likely an upper limit. Given that the
youngest outflow, Outflow 1, may have been ejected as recently as $\sim$100 yr
ago (Cox et al., 2002), it is possible that this outflow and the unveiling of
the CS occurred essentially simultaneously. A thin shell of [Fe ii] emission
is seen surrounding the elliptical shell (Fig. 10), connecting the shell to
the shocked regions of Outflow 1 (as seen in [S ii] and [Fe ii]), perhaps
providing evidence for this simultaneity.

###  7.2 Did NGC 7027 undergo an ILOT?

The foregoing sequence of events in the shaping history of NGC 7027 appears
consistent with a model of formation for bipolar PNe formation via
Intermediate-Luminosity Optical Transient (ILOT) ejection events (Soker &
Kashi, 2012). This (ILOT) model offers an explanation for the formation of bi-
lobed PNe that display evidence for multiple, paired mass ejections. Soker &
Kashi (2012) set out the following criteria to identify PNe that may have been
formed by ILOT events: the PN (1) contains structures or components that
exhibit a linear relationship between velocity and distance; (2) is bipolar in
structure, indicating the presence of a binary system with orbital separation
of $\sim 1$ AU; (3) has lobe expansion velocities exceeding $\sim 100$ km s-1;
and (4) has a total fast outflow kinetic energy in the range $\sim
10^{46}$–$10^{49}$ erg. Soker & Kashi (2012) further point to NGC 6302 as an
exemplar of a bipolar PN that was generated by an ILOT event some $\sim$2000
yr ago (see also Kastner et al., 2022).

Indeed, NGC 7027 would appear to satisfy all four ILOT criteria. With regard
to criteria 1–3: its cloverleaf-shaped H2 emitting region exhibits a velocity
field proportional to radius (Cox et al., 2002); the nebula displays (nascent)
bipolar structure, in the form of its dusty equatorial belt and collimated
outflows (especially the youngest, Outflow 1; Fig. 10); and the presence of
[Fe ii] and X-ray emission, as well as high-velocity Br$\gamma$ emission (Cox
et al., 2002), along Outflow 1 indicates outflow speeds in excess of $\sim
100$ km s-1, at least along this (SE–NW) direction (see § 6.2). NGC 7027 also
appears to meet the fourth (energy) criterion: adopting the mass of $0.1$ M⊙
estimated by Santander-García et al. (2012) for their “high-velocity molecular
blobs”, and an assumed velocity of 100 km s-1, we arrive at a total energy of
$\sim 2\times 10^{46}$ erg for its collimated outflows. With a post-AGB age of
just $\sim$100–1000 yr (§5) and at least one collimated outflow (Outflow 1)
that is of similar dynamical age (Cox et al., 2002), NGC 7027 may therefore be
the youngest known example of an ILOT-generated bipolar PN.

##  8 Summary

The young, rapidly evolving, nearby NGC 7027 is among the best objects for
studying PN shaping and excitation mechanisms. We have obtained and analyzed a
comprehensive suite of narrow-band HST/WFC3 images for this nebula, spanning a
range from near-UV ($\sim$245 nm) to near-IR ($\sim$1665 nm). Our analysis and
main results can be summarized as follows.

  * •

H$\alpha$/H$\beta$ and Pa$\beta$/H$\beta$ emission line ratio images are
constructed and a per-pixel comparison is made with the theoretically
predicted ratios, so as to obtain the spatial distribution of the extinction
parameter c(H$\beta$). Overall, the results for c(H$\beta$) are consistent
with previous studies that mapped extinction across NGC 7027, featuring
extinction maxima along the waist of the nebula, as well as a prominent gap in
extinction located $\sim 5^{\prime\prime}-10^{\prime\prime}$ to the WNW of the
central star; however, the extinction maps reveal the dusty substructures
within NGC 7027 in unprecedented detail. The c(H$\beta$) images are used to
perform extinction corrections for several of the HST/WFC3 images (see
Appendix A).

  * •

Aperture photometry performed on the WFC3 images and near-IR and optical
measurements from Latter et al. (2000) are used to construct the spectral
energy distribution of the CSPN from the near-UV to near-IR. Analysis of these
data yields an estimate of extinction of A${}_{V}=2.69$ toward the central
star. The resulting CSPN luminosity is $6.2\times 10^{3}$ erg s-1 for an
assumed effective temperature of $198$ kK (Latter et al., 2000). These
parameters indicate that the central star is descended from a progenitor of
$\sim$3 $M_{\odot}$, has a present-day mass of $\sim$0.6–0.7 $M_{\odot}$, and
left the AGB within the last $\sim$1000 years.

  * •

Emission line ratio images constructed for [O iii]/[Ne v], [Ne v]/[Ne iv], [Ne
v]/H$\beta$, [O iii]/H$\beta$, [S ii]/H$\alpha$, and [Fe ii]/continuum reveal
the detailed ionization structure of the nebula (§ 4). This structure appears
best explained as a juxtaposition of photoionized and shock-ionized zones
within the nebula. An excitation diagram is used to assess the spatial regimes
of photoionized vs. shocked nebular material (§ 6.1). This analysis reveals
that the outflow structures oriented SW–NE — Outflow 1 in Fig. 10 — display
the strongest shocks. This conclusion is bolstered by the consistent
orientations of the extended X-ray, [S ii], and [Fe ii] emitting regions along
this same direction. The nested spatial distributions of these tracers (and
[Oiii]) reveal the detailed shock structure (and constrain the gas densities)
along Outflow 1.

  * •

A timeline for the shaping of NGC 7027 is proposed, in which a close binary
system has produced the nebula’s ring system and equatorial belt, and
subsequent binary interactions, possibly in a series of short-lived eruptions
(the ILOT scenario), then generate the multipolar jets observed. This model
awaits confirmation through a study of the proper motions of the complex NGC
7027 jet system, via these WFC3 and archival WFPC2 HST images, so as to
establish more precise dynamical ages.

## Acknowledgements

Based on observations made with the NASA/ESA Hubble Space Telescope, obtained
at the Space Telescope Science Institute (STScI), operated by the Association
of Universities for Research in Astronomy, Inc., under NASA contract
NAS5-26555. These observations are associated with program #15953. Support for
this program was provided by NASA through STScI grant HST-GO-15953.001-A to
RIT.

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##  Appendix A Image extinction correction

The $c(\mathrm{H}\beta)$ images obtained from the H recombination line images
(§4.1) in principle provide a means to correct the extinction from the
HST/WFC3 suite images. This correction was performed, initially, on $[$Ne v$]$
and $[$O iii$]$ images using $c(\mathrm{H}\beta)$ images as a starting point
for extinction calculations at 0.343 $\mu$m ($[$Ne v$]$) and 0.501 $\mu$m
($[$O iii$]$). The relation between $c(H\beta)$ and $E(B-V)$ is described by
Howarth (1983) as $c(\lambda)=0.4(R+0.53)E(B-V)$. For $R_{\lambda}=3.1$, this
relation becomes $c(\lambda)=1.45E(B-V)$. Using the reddening law $f(\lambda)$
from Cardelli et al. (1989), the extinction at a given wavelength is then
obtained as $A_{\lambda}=f(\lambda)R_{\lambda}E(B-V)$.

The images in Fig. 11 show results of the extinction corrections performed for
$[$Ne v$]$ and $[$O iii$]$, respectively. The $c(\mathrm{H}\beta)$ images
obtained from Pa$\beta$/H$\beta$ and H$\alpha$/H$\beta$ yield slight
differences in extinction calculation for both $[$Ne v$]$ and $[$O iii$]$. For
$[$Ne v$]$, the $c(\mathrm{H}\beta)$ image derived from H$\alpha$/H$\beta$
produced traces of slight overcorrection within the nebula’s left dusty region
and a possible undercorrection near the nebula’s waist. This becomes clearer
when comparing results generated using the $c(\mathrm{H}\beta)$ image derived
from the Pa$\beta$/H$\beta$ image. Undercorrection in the nebula’s waist
caused by the H$\alpha$/H$\beta$ image can also be identified in $[$O iii$]$
results as remnants of a dust lane.

Figure 11: Comparison of extinction-corrected $[$Ne v$]$ (top panels) and $[$O
iii$]$ (bottom panels) images as obtained using $c(\mathrm{H}\beta)$ images
derived from H$\alpha$/H$\beta$ (left panels) and Pa$\beta$/H$\beta$ (right
panels), for $R_{\lambda}=3.1$. Images are presented in their native rotation
and units (counts s-1, on a log intensity scale), and the axes correspond to
pixel size.

Extinction corrections for $[$Ne v$]$, $[$Ne iv$]$, H$\alpha$, H$\beta$, and
$[$O iii$]$ images were performed using $c(\mathrm{H}\beta)$ images derived
from the Pa$\beta$/H$\beta$ line ratio map. Fig. 12 compares the extinction-
corrected images with their raw HST/WFC3 counterparts. In all but the $[$Ne
iv$]$ image, the extinction corrections recover the full extent of the bright
elliptical shell seen in the (dust-penetrating) WFC3 near-IR images. The
correction is least successful for the $[$Ne iv$]$ image, likely because of
the poor signal-to-noise ratio.

Fig. 13 compares the $[$Oiii$]$/H$\beta$ line ratio image with the extinction
corrected $[$Oiii$]$/H$\alpha$ line ratio image. A few differences between
these images — primarily dust lane surrounding the nebula and ripples in the N
lobe — are apparent. Overall, however, the morphologies are very similar,
verifying that the extinction corrections are reasonable. The extinction
correction process was also performed on $[$Ne iv$]$, H$\alpha$, and H$\beta$
for line ratio images in Fig. 3.

Figure 12: Comparison of original H$\beta$, H$\alpha$, $[$Ne iv$]$, $[$Ne
v$]$, and $[$O iii$]$ images (left panels) with the extinction-corrected
images (right panels). Figure 13: Comparison of extinction corrected
$[$Oiii$]$/H$\alpha$ and $[$Oiii$]$/H$\beta$ line ratio images.