# Development of Techniques Enabling Suborbital Small Object Capture Around
Low Gravity Asteroids

Leonard D. Vance111PhD Candidate, Aerospace and Mechanical Engineering,
University of Arizona, 1130 N Mountain Ave, Tucson, AZ 85721 and Jekan
Thangavelautham222Assistant Professor, Aerospace and Mechanical Engineering,
University of Arizona, 1130 N Mountain Ave, Tucson, AZ 85721

###### Abstract

The exploration of small body asteroids provides direct access to the
primitive building blocks of our solar system. Sample and return missions
enable chemical and radioisotope studies which not only provide evidence for
the formation of the solar system, but also a basic understanding of where
resources might be found for future exploitation. The touch-down and sample
techniques established by Hayabusa-2 and OSIRIS-REx accomplish this mission by
physically touching down on the asteroid and collecting samples into a basket
extended via a probe from the bottom of the spacecraft. This technique has
been demonstrated to work, but contains a high cost in both mission operations
as well as the size and complexity of the collection mechanism itself. This
paper explores an alternative sample and return technique by exploiting the
recent discovery of regolith particle ejections from Bennu. Particles ejected
from the surface of Bennu are typically 1 cm in size and spend several hours
in flight, suggesting the possibility that nanospacecraft deployed from the
mother-spacecraft could chase down, collect and return with the sample with
minimal sensor and delta-V capability. Key aspects of this mission are
developed to reduce risk, and an overall mission concept is developed to
establish plausibility.

##  1 Introduction

Asteroid sample and return missions provide scientists with direct access to
materials formed during contraction of our original solar planetary disc.
These missions will also play an important part in resource prospecting,
especially for water which can be used a propellant for interplanetary
spacecraft [65]. Chemical and isotopic analysis of these samples provide the
gold standard for understanding the age, history and evolution of the overall
solar system. Several missions have therefore attempted this, including
Hayabusa 1 and 2, OSIRIS-REx, and NASA’s Stardust, but the challenges involved
in gathering surface samples has proven more difficult than perhaps originally
expected. The combination of very low gravity, relatively high spin rates, and
the unpredictability of the surface composition have proven difficult to
predict, and the results from initial missions have been mixed. Improvements
in retrieval techniques could plausibly reduce mission risks and costs
significantly. Mother-Daughter architectures have also been explored,
exploring the plausibility of daughter vehicles to undertake local missions in
swarms which would would not be plausible with a single spacecraft [64]. This
also avoid a single daughter craft having to have long-range communications
such as using inflatable antennas [66].

Figure 1: Aspects of the proposed mission. Nano-spacecraft are deployed from a
parking orbit to collect ejected particles and return to the mother spacecraft

In December 2018, OSIRIS-Rex arrived at the near-earth asteroid Bennu as a
prelude to touching down and retrieving such a sample. The initial survey of
the asteroid provided two significant surprises. The first involved the lack
of fine particulate matter on the surface, and the second was the discovery
that Bennu is a “live” asteroid, with low velocity objects erupting regularly
from the surface. The first discovery provides a significant challenge to the
mission as the sample system is designed to collect relatively small
particles, while the second suggests that future sample and returns could use
small deployed spacecraft, capturing the expelled samples mid-flight before
returning to the mother spacecraft.

This paper explores the design of such a mission, which is plausibly less
complex than the sample and return techniques currently utilized. Techniques
developed for such a mission would also enable other high value missions such
as low earth orbit debris capture and potential on-orbit servicing by nano-
spacecraft.

Several areas of novel research must be explored to develop a plausible system
design. To begin with, characteristics of the expelled objects must be
established. Positions, velocities, sizes and eruption frequencies are needed.
It is credible that the movement of these objects in part explain the “double-
top” shape of spinning rubble pile asteroids, so it may be conversely possible
that the “double-top” shape implies the existence of object expulsions, and
therefore indicates opportunities to collect samples without touching the
surface.

The design of a temporarily deployed nano-spacecraft to capture expelled
objects provides significant technical challenges. Each vehicle must be able
to navigate, guide, collect, return and dock with the parent spacecraft within
the timeframe of a typical particle trajectory. To begin with, autonomous
navigation can be established using passive line of sight histories to the
targeted object. The inertial line of sight history between two orbital
objects provides sufficient information to establish estimate ephemeris for
both objects, and this can be realized with a modified Extended Kalman Filter
(EKF) formulation, possibly simplified to a constant gain model to reduce
processor utilization. The resulting implementation of closed loop guidance
using the aforementioned navigation output with Lambert’s method is well known
but computationally expensive, and is thus replaced with research towards a
perceptron based network trained for operation within any particular
asteroid’s gravitational field.

Spacecraft control involves both attitude and position control, and would be
exercised via a six degree of freedom thruster system possibly including
reaction wheels for fine control. The concept of a minimal redundant thruster
configuration for nanospacecraft is explored via null vector analysis, leading
to candidate minimal thruster systems with single failure redundancy. Finally,
a generic grappling system is discussed for a non-cooperative target, with
possible application to wider missions such as debris removal from low earth
orbit. The overall proposed mission is summarized in Figure 1.

##  2 Ejected particle characterization on Bennu

The process of rubble pile asteroid formation was discussed by Michel [13] in
2001, and Guibot and Scheeres [11] in 2003, but innovations in deep space
radar imaging led to the confirmation in 2006 [17], [23] that asteroid (66391)
1999 KW4 has a top-shape with a nearly constant slope of about 35 degrees from
mid latitudes down to the equator. It is a fast-rotator, with a period 2.8 hr
and a small moon just off the equator suggesting recent mass shedding [23].
Subsequent radar observations of (341843) 2008 EV5 [18], (29075) 1950 DA [16]
and (10195) Bennu [24] show that top-shapes are common for near-earth
asteroids. Arrival of OSIRIS-REx and Hayabusa-2 confirmed this [31] for both
(10195) Bennu and (162173) Ryugu [9] respectively, providing detailed 3D
models to enable detailed analysis

Centripetally-induced landslides caused by solar-pressure induced YORP spin-up
have been proposed to explain current observations [11], [8]. As more data
have become available, these ideas have been developed analytically [2], [3],
[6], [10], [22] and avalanche failure mechanisms have been explored [29],
[27]. Particle models [19], [26], [30], some including cohesion [14], [15],
have been applied to the problem. There have also been attempts to simulate
low speed collisions as a possible origin for top-shapes. Simulations of
impacts between two near equal mass rubble piles, where friction and porosity
is included [5], can produce final populations where top-shapes are common.
Additionally, the direct measurements of the YORP induced angular acceleration
on Bennu [1], [20], and its sensitivity to variations in configuration [21]
provide an understanding of the possible chaotic nature of Bennu’s
accelerating spin rate.

Figure 2: Particle frequency and velocity distributions derived from OSIRIS-
REx observations provide a basis for the proposed mission

The discovery in 2019 by NASA’s OSIRIS-REx mission that the 500-metre diameter
carbonaceous rubble pile Bennu is an active asteroid, regularly ejecting
objects from the surface [4], suggests the possibility that objects can be
captured in flight without approaching the asteroid surface. Some 599 objects
were observed by OSIRIS-REx over a 45 day observation period following its
initial approach to the asteroid, providing a mean of 13 objects observed per
day. When a simulation experiment is executed accounting for the obscuration
of most of the actual particles leaving the surface, the actual launch
frequency is about 26 particles per day with a mean launch velocity of 9.5
cm/s as shown in Figure 2. The resulting flight time for an average particle
is about 1.5 hours given Bennu’s very low gravity. This ensures that particles
will be available for sampling on a near daily basis.

##  3 Own-Ship and Target Navigation with a Single Passive Line-Of-Sight
Sensor

Navigation techniques using passive line-of-sight measurements are well known
within the guidance, navigation and control community. The ability to infer
ephemeris information of an orbital object by tracking it from a known
position is well understood, as is the ability to infer one’s own ephemeris by
tracking an object with known position and velocity. Here we show that it is
possible to infer both own-ship and tracked-object ephemeris given knowledge
of the gravitational influences both are subjected to.

Figure 3 shows the general nature of the problem. It is well established that
position and velocity data for an overhead satellite can be extracted from
passive line of sight measurements from the ground. Observability of range is
inferred from the gravitational attraction of the orbiting object. Likewise,
as a simple extension, it is also possible to do the same job from an orbiting
satellite. If you know your own position and velocity, you can infer the
position and velocity of another satellite given a time history line of sight
measurements.

Figure 3: The line of sight history between two objects under the influence of
a common gravitational body is sufficient to estimate the position and
velocity of both

With some additional thought, it is possible that the position and velocity of
both the tar-get and observer satellites can be inferred from a single line of
sight history. With some exceptions, it can be asserted that any line of sight
history between two objects gravitation-ally influenced by a common source is
the result of only one specific target and home satellite trajectory.

###  3.1 Method

The Extended Kalman Filter (EKF) is a well-known, pseudo-optimal predictor-
corrector filter widely used for a wide variety of aerospace purposes. This
section shows the adaptation of the basic filter structure for this particular
problem. The established form of the EKF [32] starts by propagating the
current estimate and covariance matrices forward by one timestep.

$\boldsymbol{\hat{x}}(k+1|k)=\boldsymbol{\hat{x}}(k|k)+\int_{t_{k}}^{t_{k+1}}\boldsymbol{f}[\hat{x}(t|t_{k}),\boldsymbol{u}^{*}(t),t]dt$
(1)

$P(k+1|k)=\Phi(k+1|k)P(k|k)\Phi^{{}^{\prime}}(k+1,k)+Q_{d}(k)$ (2)

This followed by calculation of the Kalman gains:

$\boldsymbol{K}(k+1)=\boldsymbol{P}(k+1|k)[\boldsymbol{H}^{{}^{\prime}}_{x}(k+1)*\boldsymbol{P}(k+1|k)\boldsymbol{H}^{{}^{\prime}}_{x}(k+1)+\boldsymbol{R}(k+1)]^{-1}$
(3)

The state vector estimate can then be updated:

$\hat{x}(k+1|k+1)=\hat{x}(k+1|k)+\boldsymbol{K}(k+1)\\{\boldsymbol{z}(k+1)-\boldsymbol{h}[\boldsymbol{\hat{x}}(k+1|k),\boldsymbol{u}^{*}(k+1),k+1]-\boldsymbol{H}_{u}(k+1)\delta\boldsymbol{u}(k+1)\\}$
(4)

Finally, also update the covariance matrix:

$\boldsymbol{P}(k+1|k+1)=[\boldsymbol{I}-\boldsymbol{K}(k+1)\boldsymbol{H}_{x}(k+1)]\boldsymbol{P}(k+1|k)$
(5)

The definition of these parameters are consistent with normal EKF usage:

$\boldsymbol{\hat{x}}$ = state estimate
$k$ = current time step
$k+1$ = next time step
$\boldsymbol{f}$ = function propagating state estimate $\hat{\boldsymbol{x}}$
in time
$t$ = time
$\boldsymbol{u}^{*}$ = nominal input
$\boldsymbol{P}$ = Covariance matrix
$\boldsymbol{\Phi}$ = Linearized time propagation matrix
$\boldsymbol{Q_{d}}$ = State error propagation matrix
$\boldsymbol{K}$ = Kalman gain matrix
$\boldsymbol{h}$ = function taking state vector to measurements
$\boldsymbol{H_{x}}$ = Jacobean of $\boldsymbol{h}$ w/r to state vector
$\boldsymbol{\hat{x}}$
$\boldsymbol{H_{u}}$ = Jacobean of $\boldsymbol{h}$ w/r to external inputs
$\boldsymbol{I}$ = Identity matrix

The function $\boldsymbol{u^{*}}$ and its derivatives are zero for the
purposes of this exercise since there are no external forces acting on the
system, and as such, (1) through (5) can be simplified to give

$\boldsymbol{\hat{x}}(k+1|k)=\boldsymbol{\hat{x}}(k|k)+\int_{t_{k}}^{t_{k+1}}\boldsymbol{f}[\hat{x}(t|t_{k}),t]dt$
(6)

$P(k+1|k)=\Phi(k+1|k)P(k|k)\Phi^{{}^{\prime}}(k+1,k)+Q_{d}(k)$ (7)

for the propagation equations, and

$\boldsymbol{K}(k+1)=\boldsymbol{P}(k+1|k)[\boldsymbol{H}^{{}^{\prime}}_{x}(k+1)*\boldsymbol{P}(k+1|k)\boldsymbol{H}^{{}^{\prime}}_{x}(k+1)+\boldsymbol{R}(k+1)]^{-1}$
(8)

$\hat{x}(k+1|k+1)=\hat{x}(k+1|k)+\boldsymbol{K}(k+1)\\{\boldsymbol{z}(k+1)-\boldsymbol{h}[\boldsymbol{\hat{x}}(k+1|k)\\}$
(9)

$\boldsymbol{P}(k+1|k+1)=[\boldsymbol{I}-\boldsymbol{K}(k+1)\boldsymbol{H}_{x}(k+1)]\boldsymbol{P}(k+1|k)$
(10)

for the update equations.

The state estimation vector contains 3 element position, velocity and
accelerations for each object. Since we are tracking one object from a home
vehicle, there are 18 elements in the state estimation vector comprising:

Figure 4: The line of sight history between two objects under the influence of
a common gravitational body is sufficient to estimate the position and
velocity of both

The function $f$ is a simple orbital propagator, utilizing the basic
gravitational law

$\boldsymbol{a}=-\frac{G\,m_{bennu}\boldsymbol{r}}{|\boldsymbol{r}|^{3}}$ (11)

where the r’s are taken from the position estimates in the state vector. The
corresponding matrix $\Phi$ is the Jacobean of this with respect to
$\boldsymbol{\hat{x}}$ , giving an 18 × 18 matrix. The function
$\boldsymbol{h}$ takes state variables and constructs the measurements
$\boldsymbol{z}$, and is therefore of the form:

$\boldsymbol{z}=\boldsymbol{h}(\boldsymbol{\hat{x}})$ (12)

Since the measurements for this system are the normalized line of sight
vectors from the tracking vehicle to the two asteroids, the resulting function
$\boldsymbol{h}$ is

$\boldsymbol{h}=\begin{bmatrix}\frac{x-x_{t}}{\sqrt{(x-x_{t})^{2}+(y-y_{t})^{2}+(z-z_{t})^{2}}}\\\
\frac{x-x_{t}}{\sqrt{(x-x_{t})^{2}+(y-y_{t})^{2}+(z-z_{t})^{2}}}\\\
\frac{x-x_{t}}{\sqrt{(x-x_{t})^{2}+(y-y_{t})^{2}+(z-z_{t})^{2}}}\\\
\end{bmatrix}$ (13)

The Jacobian of this function with respect to each component of the state
estimate vector provides the resulting 6 × 18 $\boldsymbol{H_{x}}$ matrix.

The $\boldsymbol{R}$ matrix represents the system measurement noise, and this
takes the form of angular uncertainties along the line of sight to the object.
The measurement vector $\boldsymbol{z}$ is the normalized line of sight vector
the target as given by an imaging system, so given an angular uncertainty
$\epsilon$ in each axis, the covariance matrix for measurement errors to the
tracked object, in line of sight coordinates is:

$\boldsymbol{R_{los}}=\begin{bmatrix}1&0&0\\\ 0&\epsilon^{2}&0\\\
0&0&\epsilon^{2}\\\ \end{bmatrix}$ (14)

This must then be rotated into an inertial coordinate frame used by the rest
of the filter. Derivation of the coordinate transform matrix is done by
establishing sequential coordinate frame rotation angles about the z and then
y-axes respectively, providing a coordinate transformation matrix of

$\boldsymbol{T_{los}}=\begin{bmatrix}cos\psi cos\theta&sin\psi
cos\theta&-sin\theta\\\ -sin\psi&cos\psi&0\\\ cos\psi sin\theta&sin\psi
sin\theta&cos\theta\\\ \end{bmatrix}$ (15)

Where the sequential angles of rotation about z and y axes respectively are:

$\psi=tan^{-1}(\frac{y-y_{T}}{x-x_{T}})$ (16)

$\theta=tan^{-1}(\frac{z-z_{t}}{\sqrt{(x-x_{t})^{2}+(y-y_{T})^{2}}})$ (17)

This matrix takes a vector from inertial to the line of sight coordinated
frame with respect to object 1. Using a similarity transformation to convert
from line of sight to inertial coordinates, we have

$\boldsymbol{R}_{inertial}=\boldsymbol{T}_{los}\boldsymbol{R}_{los}\boldsymbol{T}^{T}_{los}$
(18)

Finally, the 18x18 state process noise matrix Q is constructed with
arbitrarily small diagonal elements for the purpose of establishing
feasibility.

$\boldsymbol{Q}=1.0x10^{-30}*\boldsymbol{I}_{18x18}$ (19)

Figure 5: Proof of concept trajectories for both the spacecraft and the
ejected particle around (101955) Bennu

###  3.2 Results

The implementation of the above filter is tested in an environment derived
from the physical environment of (101955) Bennu. The specific intent is to
show whether autonavigation can be performed from a nanospacecraft that is
tasked with capture of an ejected particle from that surface. As proof of
concept, the spacecraft is placed in a circular orbit with an 800m semi-major
axis with 90 degrees inclination. The tracked object ejects from the mid
latitudes of Bennu with a velocity of 17.5 cm/s, and it tracked by the
spacecraft for a 10 hour epoch. The trajectories of both objects relative to
Bennu is shown in Figure 5

Using only inertial line of sight measurements from the spacecraft to the
ejected particle, position and velocity can be established for both
simultaneously. Given a reasonable initialization for covariances, the results
are shown in Figure 6, illustrating good convergence in position and velocity
in all 3 axes, although over a time period of several hours.

Figure 6: Convergence of position and velocity estimates for both spacecraft
and ejected particle over the 10 hour epoch described by the previous figure

##  4 Perceptron Based Orbital Guidance in a Low Gravity Asteroid Environment

###  4.1 Introduction

Lambert’s solution is a well studied approach [46], [47] to orbital guidance,
providing a near universal iterative solution for transition between two
points under the influence of a single gravitational body. The user specifies
starting and end points along with transit time and central body gravitational
parameter $\mu_{m}$, and the algorithm produces the corresponding velocities
at those starting and end locations along with minimum and maximum ranges to
the gravitational body. While the algorithm is well known, accurate and almost
universal in convergence, it requires a significant number of trigonometric
function calls in an iterative framework. In comparison, a feed forward neural
network has low computational cost implementation and can theoretically mimic
the output of much more complex algorithms if trained correctly.

Neural Network guidance could therefore be used by nanospacecraft to
rendezvous and gather the particles emitted by an active asteroid in flight.
The discovery in 2019 that Bennu is an active asteroid, and is regularly
ejecting particles from its surface [45] makes this concept possible. the
frequency, size and velocity of these particle ejections were established to
be 1-10 cm in diameter, 6 to 330 cm/s in velocity and were observed to launch
episodically with an cumulative frequency of approximately 599 particles over
45 days. The escape velocity from Bennu is about 20 cm/s, and the time spent
by these particles in flight typically is measured in hours before they either
escape from gravitational influence, or return to the surface of the asteroid.

In this section, a training set of orbits and their resulting transfer
trajectories are established using randomly selected position and velocity
parameters around asteroid Bennu as inputs to the Lambert algorithm. Seven
separate filters are trained, one each for three axes of starting velocity,
ending velocity plus a minimum transition altitude. Figure 7 defines the basic
outline of the problem, and shows an example of a transfer trajectory derived
using Lambert between two objects in orbit around Bennu. A Levenburg-Marquardt
algorithm is implemented which shows good convergence with a modest number of
hidden nodes and layers. Finally, the resulting trained networks are used to
verify performance in an orbital test case.

Figure 7: Definition of the Problem: A neural network approach to lambert
guidance is defined to reduce processing requirements for a nanospacecraft
attempting to rendezvous an ejected particle around Bennu

###  4.2 Implementation of Levenburg-Marquardt Optimization

Research indicates that perhaps the most common training technique for neural
networks is the Levenberg-Marquardt algorithm [7]. This method effectively
acts as a gradient descent at the beginning of the epoch, and gradually
transitions to solve for zero error effectively using a Newton-Raphson style
method as the system starts to converge. Using the Jacobean derived for use in
gradient descent, the weight update equation can be reformulated as:

$w_{i}=[P^{T}P+\mu I]^{-1}J^{T}(y-y_{train})+W_{i-1}$ (20)

where $\mu$ is a scaler control variable with value $\mu>0$, $w_{i}$ is the
updated weight vector, P is the same Jacobean matrix used in the gradient
descent method, $(y-y_{train})$ is the error vector and $w_{i-1}$ the previous
weight vector. This optimizer approximates a small step gradient descent
method when $\mu$ is large, but becomes closer to a Newton-Raphson solver as
mu approaches zero. $\mu$ is adjusted during an optimization run, increasing
fractionally if the error function increases, but decreasing fractionally when
it is converging. In a nominally converging system, $\mu$ will steadily
decrease as the network converges.

Figure 8: Levenberg-Marquardt Descent for Lambert’s solution converges quickly
and accurately for the velocity-X axis, and this behavior is near identical
for the other two axes, as well as the velocity required at intercept.

This algorithm performs well and converges for all six $\Delta V$ estimates.
Figure 8 shows this convergence using this technique for a modest 2 layer
system of 16 and 8 hidden nodes respectively. All other values of $\Delta V$
show similar excellent results, though the estimate for minimum altitude
during the intercept trajectory has more difficulty converging. Performance
starts to fall off significantly with less that 16 nodes in the first hidden
layer, but is not visibly improved with increases beyond the nominal values
for the two layer system specified as the nominal case. The rms error of this
estimator calculates to 0.9 cm/s, and when implemented as the guidance method,
the resulting trajectories can be seen in Figure 9

Figure 9: The overall performance of the resulting neural network guidance is
shown, comparing the Lambert derived trajectory (yellow) to the neural network
derived trajectory (magenta). Discrete guidance updates using the neural
network are calculated and shown at 1250 second intervals (magenta circles)

###  4.3 Thruster Configuration Optimizations

The size and complexity of a spacecraft is significantly effected by its
propulsion configuration. In the case of a nanospacecraft executing the
mission of collecting ejected particles from an active asteroid, there would
be a requirement for full six degree of freedom motion, and possibly a
requirement for redundancy if one of the thrusters fails. The physical volume
required for plumbing, valves and expansion nozzles encourages the designer to
reduce the individual number of thrusters to the absolute minimum. This
section discusses the minimum configuration needed for full 6 axis control of
a nanospacecraft, and then the mathematical techniques to show controllability
as well as thruster failure redundancy.

This section seeks to establish minimal thruster configurations which maximize
effectiveness and while still providing single thruster failure redundancy.
The actuation logic for a multiple coupled thruster system can be expressed
as:

$\begin{bmatrix}\boldsymbol{F_{b}}\\\ \boldsymbol{L_{b}}\\\
\end{bmatrix}=[\boldsymbol{T}][\boldsymbol{F_{t}}]$ (21)

where $\boldsymbol{F_{t}}$ is a vector containing the thrust values from each
thruster at a given time, $\boldsymbol{F_{b}}$ is the resulting three
dimensional net thrust in the body frame, $\boldsymbol{L_{b}}$ is the body
frame torque vector, and $\boldsymbol{T}$ is the matrix relating thrust values
to net forces and torques.

Figure 10: An example of a fully redundant, six degree of freedom thruster
configuration derived via pseudo-inverse techniques combined with null vector
evaluation. Improvement of placement and orientation via genetic algorithms
provides a path towards optimizing nanospacecraft propulsion systems

In order to get full three dimensional control of a spacecraft in both linear
and rotational space, a minimum of seven thrusters is required, with more
needed if redundancy is required. The classical approach to establishing the
thruster control levels needed uses the pseudo inverse method such that

$\boldsymbol{F_{t}}=[\boldsymbol{T^{-1}}]\begin{bmatrix}\boldsymbol{F_{b}}\\\
\boldsymbol{L_{b}}\\\ \end{bmatrix}$ (22)

In order this technique to work, there must also be at least one positive
definite null space vector to the $\boldsymbol{T}$ matrix, so that thrust
levels derived from the pseudo-inverse method above can be adjusted upwards
until all are non-zero. In the case of testing for redundancy, these two
conditions must be met for a each subset of the $\boldsymbol{T}$ matrix
corresponding to the loss of one thruster.

Preliminary simulations have established minimal configurations of 9 thrusters
which provide full 6 degree of freedom control with single failure redundancy,
but these thruster configurations are asymmetric. Figure 10 shows an example
of such a system configured on a spherical volume which has full six degree of
freedom control for both the nominal configuration as well as any given single
failures. Time history performance of this configuration is shown in the
bottom half of this same figure. The proposed dissertation shall execute the
analysis necessary to establish full six degree of freedom control with
redundancy with a minimum number of thrusters, including placement and
orientations optimized for control effectiveness.

###  4.4 Generic Capture Mechanism

This concept involves the deployment of multiple grappling arms, each
comprised of identical segments with independently controlled single axis
rotational joints. each link includes a sensor to establish contact with the
object to be grappled. Figure 11 provides an overview of one joint of this
proposed grappling system.

Figure 11: Link Concept: The candidate link concept uses a miniature worm
drive to actuate successive links. A wire is included in the outline of each
link which is triggered upon establishing contact with an object.

Upon establishing position and rate matching along center of rotation of the
object to be grappled, the spacecraft starts to extend the adaptive grappler
link by link, stopping each time contact is detected with that link, and then
stepping to the next link. In this fashion, the target object is slowly
enveloped in a low-force, controlled fashion until the entire grappling arm is
deployed. This approach is shown in a planar view in Figure 12.

The control for this concept needs to be expanded to include the algorithms
necessary to establish a rippling motion for the as yet undeployed links so
that 1) the amount of energy necessary to deploy the system is reduced, and 2)
to reduce the probability that grappler might hit obstructions out beyond the
minimal extent of the object being captured. 6-degree of freedom simulation of
this concept and the capture sequence will be executed and optimized for
system performance leading to a conceptual design.

Figure 12: Concept of Operations: Upon establishing stable proximity with the
object to be captured, the spacecraft establishes an approach vector along the
spin axis of the object, and deploys the grappler arms one link at a time,
where each link stops when sensing contact with the object, and subsequent
links then rotate forward, slowly enveloping the object to be captured.

##  5 Conclusions

In light of the complexity and risks associated with the sampling experience
of Hayabusa 1 and 2 as well as OSIRIS-REx, establishing alternative approaches
to obtain samples from small asteroids is a beneficial exercise. The discovery
of regularly ejected particles from Bennu suggests the possibility that most
if not all top shaped near Earth asteroids are active in some similar fashion,
enabling the possibility that sampling could be done by capturing these
particles in flight and returning them to Earth. This dissertation proposal
documents the development of key technologies enabling this mission.

Characterisation of the ejected particle environment is established showing
that the ejection of surface particles from an active asteroid contributes to
the commonly observed ’double-top’ shape. Auto-navigation and Neural network
based guidance is then demonstrated. The concept of achieving auto-navigation
using passive line of sight measurements from poorly known objects under the
influence of a common gravitational field is quantified, with results showing
convergence for both the spacecraft and ejected particle position and velocity
estimates. Significant improvements in performance are plausibly expected when
the spacecraft begins maneuver, as observability improves quickly what is
effectively temporal triangulation.

Perceptron based guidance is demonstrated by showing convergence of a trained
neural network to mimic the output of the iterative Lambert’s solution.
Comparison of trajectories showing both solutions demonstrates the promise of
this technique. Spacecraft control is explored by showing the existence of a
10 thruster placement for redundant 6 degree of freedom control, thus
indicating the means to establish optimization. The development of a universal
grappling scheme for non-cooperative targets and consistent with known
projectile sizes is then explored and control algorithms are developed for its
use. Finally, a demonstration of these technologies is executed on a air-
bearing table contoured to mimic the low gravity environment of a small
asteroid. The overall resulting mission design is conceptually designed and
risk reduced for key technological components.

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NKU-2021-SF1

Thermodynamics and Heat Engines of Black Holes with Born-Infeld-type
Electrodynamics

Leonardo<EMAIL_ADDRESS>
Departamento de Ciencias Físicas,
Facultad de Ingeniería y Ciencias
Universidad de La Frontera, Casilla 54-D
Temuco, Chile.
Sharmanthie<EMAIL_ADDRESS>
Department of Physics, Geology & Engineering Technology
Northern Kentucky University
Highland Heights, Kentucky 41099, U.S.A.

Abstract

In this paper we have studied electrically charged black holes in a new model
of nonlinear electrodynamics introduced by Kruglov in Ref. [1]. There are two
parameters for the theory and the black hole could have up to two horizons.
Thermodynamics is studied in the extended phase space where the pressure is
proportional to the cosmological constant. First law and the Smarr formula are
derived. There are phase transitions similar to the Van der Waals liquid-gas
phase transitions. Black hole is also studied as a heat engine and have
discussed how the parameters in the nonlinear electrodynamics theory effect
the efficiency of the heat engine.

Key words: Static black hole; non-linear electrodynamics; heat engine; anti-de
Sitter space; efficiency

##  1 Introduction

The Born-Infeld theory [2] is a model of non-linear electrodynamics which was
proposed to avoid singularities of the Maxwell model. This theory is
characterized by a non-linear parameter $b$ which represents the value of the
electromagnetic field strength at the origin of a point source. In the limit
$b\rightarrow\infty$ the Maxwell electrodynamics is recovered. This theory has
received attention from different contexts [3, 4, 5, 6, 7, 8, 9, 10, 11]. In
recent times alternative theories of nonlinear electrodynamics have been
presented: some of them are also characterized by a nonlinear parameter [12,
13, 14, 15, 16, 17, 18]. Other models have emerged in the context of regular
black holes [19, 20, 21, 22, 23, 24]. However, as shown in Ref. [25] they
cannot include a nonlinearity parameter that allows to obtain the Reissner-
Nordström solution in the limit where it tends to infinity, as in the case of
the nonlinearity parameter $b$ of the Born-Infeld electrodynamics, or as in
the black hole solution with nonlinear electrodynamics given in Ref. [26],
where the Reissner-Nordström case is recovered when the nonlinearity parameter
tends to zero.

Recently a black hole coupled to a Born-Infeld type electrodynamic model has
been proposed in Ref. [27] that, in addition to the parameter $\beta$ which is
related to the maximal electromagnetic field strength, depends on another
extra parameter, denoted by $\sigma$, and where the Born-Infeld black hole
corresponds to one of the values of $\sigma$. This new family of solutions has
two characteristics that make it interesting in order to deepen its study. On
the one hand, depending on the values of the extra parameter, it is possible
to have solutions of an electric field that can be amplified (where therefore
the regularity at the origin is lost) or screened with respect to the electric
field of the Reissner-Nordström solution. This is something that deserves
attention, for a better understanding about vacuum polarization in the context
of black holes itself [28, 29], as well as its relationship with the deviation
of the electric field with respect to the Coulomb field. On the other hand, it
is also worth analyzing the model from the point of view of the energy
conditions that the respective energy-momentum tensor satisfies [30]. In
particular, fulfilment of the dominant energy condition depends on the values
that the parameter $\sigma$ takes. In respect of the latter, the relationship
that the screening or the amplification of the electric field may have with
the energy conditions that the corresponding energy-momentum tensor fulfills
can be studied. More details on this will be presented in a later work.

One of the goals in this paper is to study the electrically charged black hole
in this new electrodynamics in the extended phase space with
$P=-\frac{\Lambda}{8\pi}$. In the extended phase space, first law of black
hole thermodynamics has an extra tern corresponding to $VdP$. Here $M=U+PV$ is
the enthalpy and not the internal energy $U$. Many black holes have
demonstrated phase transitions and critical behavior between large and small
black holes in this context. Such phase transitions are similar to van der
Waals liquid-gas phase transitions. There are large number of work related to
this topic: we would only give few of such work as references as [28, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. There is a review by Kubiznak et.al.
[43] which gives more references and a comprehensive review on the topic.

It should be mentioned that several works have considered AdS black holes as
heat engines [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]. Here the central
idea is that if the extended phase space is considered one can extract
mechanical work from a AdS black hole when one consider it as a heat machine.
To compute the efficiency of the heat engine one must define a thermodynamic
cycle in phase space. In this paper we will present a Born-Infeld-type black
hole given by Kruglov [1] as a heat engine choosing one of the cycles proposed
by Johnson in Ref. [44]. This cycle is composed of two isobaric and two
isochoric paths.

The paper is organized as follows: in section 2, the black hole considered is
presented. In section 3 thermodynamics of the black holes are given and in
section 4 P-V criticality is discussed. In section 5 the black holes are
studied in the context of heat engines and finally in section 6 the conclusion
is given.

##  2 Black holes with Born–Infeld-type electrodynamics

In general the action of the 3+1-dimensional Einstein theory coupled with a
nonlinear electrodynamics model is given by

$S=\int d^{4}x\sqrt{-g}\left[\frac{(R-2\Lambda)}{16\pi
G}+\mathcal{L(F)}\right]$ (1)

Here, $g$ is the determinant of the metric tensor and
$\Lambda=-\frac{3}{l^{2}}$ is the cosmological constant and $\mathcal{L(F)}$
is the the Lagrangian depending on the invariant
$\mathcal{F}=F_{\mu\nu}F^{\mu\nu}$. The variation of the action with respect
to the gravitational field yields the Einstein equations

$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi T_{\mu\nu}$ (2)
$T_{\mu\nu}=g_{\mu\nu}\mathcal{L(F)}-F_{\mu\rho}F_{\nu}^{\,\,\rho}\mathcal{L},_{\mathcal{F}}\,.$
(3)

And the variation with respect to the electromagnetic field gives

$\nabla_{\mu}(F^{\mu\nu}\mathcal{L},_{\mathcal{F}})=0$ (4)

where $\mathcal{L},_{\mathcal{F}}$ is the derivative of $\mathcal{L(F)}$ with
respect to $\mathcal{F}$.

In the present study we will restrict ourselves to the nonlinear
electrodynamic model described by the Lagrangian density

$\mathcal{L}(\mathcal{F})=\beta^{2}\left[1-\left(1+\frac{\mathcal{F}}{\beta^{2}\sigma}\right)^{\sigma}\right]\,.$
(5)

This Lagrangian density can be obtained from the electrodynamic model
introduced by Kruglov in Ref. [1] and whose Lagrangian density is given as

$\mathcal{L(F)}=\frac{1}{\beta}\left[1-\left(1+\frac{\beta\mathcal{F}}{\sigma}-\frac{\beta\gamma\mathcal{G}^{2}}{2\sigma}\right)^{\sigma}\right]$
(6)

where $\mathcal{F}=F_{\mu\nu}F^{\mu\nu}$,
$\mathcal{G}=\frac{1}{4}F_{\mu\nu}\tilde{F}^{\mu\nu}$ and
$\beta,\gamma,\sigma$ are parameters of the theory. In particular by choosing
$\gamma=0$ and doing the replacement $\beta\rightarrow 1/\beta^{2}$ in Eq.
(6), let us obtain Eq. (5).

When $\beta=\gamma$ and $\sigma=\frac{1}{2}$ in Eq. (6), the model becomes the
Born-Infeld electrodynamics. Hence the above new model is named Born-Infeld-
type electrodynamics. A two parameter model was considered in Ref. [14] which
is close to the above mentioned model. In Ref. [27], Kruglov investigated a
magnetic black hole arising from the above model coupled to general
relativity. In Refs. [14, 55, 56, 57], the possible constraints for the
parameters of this model (and also of other nonlinear electrodynamic models)
are analyzed.

Static spherically symmetric black holes for the above theory is given by the
metric $ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$. This ansatz in the Einstein equations (2) and using the
Lagrangian density (5) allows us to obtain the metric function

$\displaystyle
f(r)=1-\frac{2M}{r}+\frac{r^{2}}{l^{2}}-\frac{2\beta^{2}r^{2}}{3}\left[\,{}_{2}F_{1}\left(-\frac{3}{4},-\sigma;\frac{1}{4};-\frac{q^{2}}{2\sigma\beta^{2}r^{4}}\right)-1\right]\,\,\,$
(7)

where $M$ represents the ADM mass, $q$ the electric charge and
${}_{2}F_{1}(a,b;c;z)$ is the Gauss hypergeometric function. In Fig. 1, the
function $f(r)$ is plotted for two values of $\sigma$. For small $\sigma$, the
black hole has one horizon and behaves similar to the Schwarzschild AdS black
hole. For large $\sigma$, the black hole could have two horizons and behave as
the Reissner-Nordström-AdS black hole.

Figure 1: The figure shows $f(r)$ vs $r$. Here $q=0.434,M=0.878$, and
$\beta=0.425$. The plots correspond to two values of $\sigma$ as given in the
figure.

The electromagnetic tensor takes the form
$F_{\mu\nu}=E(r)\left(\delta^{t}_{\mu}\delta^{r}_{\nu}-\delta^{t}_{\nu}\delta^{r}_{\mu}\right)$,
where $E(r)$ is the electric field, that is, $\mathcal{F}=-2E^{2}(r)$, which
together with Eq. (4) allows us to obtain the corresponding electric field for
the Born-Infeld-type electrodynamics

$E(r)=\frac{2qr^{2}\beta^{2}\sigma}{q^{2}+2\sigma\beta^{2}r^{4}}\left(1+\frac{q^{2}}{2\sigma\beta^{2}r^{4}}\right)^{\sigma}\,\,\,.$
(8)

The electric field asymptotically behaves as the one for the
Reissner–Nordström-AdS black hole as,

$E(r)=\frac{q}{r^{2}}-\frac{q^{3}(1-\sigma)}{2\sigma\beta^{2}r^{6}}+O(1/r^{10})\,\,\,.$
(9)

In Fig. 2, the electric field is plotted for two values of $\sigma$. For small
$\sigma$, the electric field is finite at $r=0$ similar to Born-Infeld
electrodynamics. For large $\sigma$, the electric field is infinite at $r=0$
similar to Coulomb electric field.

Figure 2: The figure shows $E(r)$ vs $r$. Here $q=0.411$ and $\beta=0.425$.
The plots corresponds to two values of $\sigma$ as given in the figure.

##  3 Thermodynamics of the black hole with Born–Infeld-type electrodynamics

In this section, we will derive thermodynamic quantities, first law and the
Smarr formula for the Born-Infeld-type black hole. We will be studying
thermodynamics in the extended phase space where the thermodynamic pressure is
given by $P=-\Lambda/(8\pi)=3/(8\pi l^{2})$. The conjugate quantity to $P$ is
$V=\frac{4\pi r_{+}^{3}}{3}$. The Hawking temperature is given by (after
replacing $l$ in terms of $P$),

$T=\frac{1}{4\pi}\left|\frac{df(r)}{dr}\right|_{r_{+}}=\frac{1}{4\pi
r_{+}}\left[1+8\pi
Pr_{+}^{2}+2r_{+}^{2}\beta^{2}\left(1-\left(1+\frac{q^{2}}{2r_{+}^{4}\beta^{2}\sigma}\right)^{\sigma}\right)\right]\,\,\,$
(10)

where $r_{+}$ is the black hole horizon. The entropy of the black hole, $S=\pi
r_{+}^{2}$ is conjugate to the temperature.

The black hole horizon $r_{+}$ satisfies $f(r_{+})=0$. Then from Eq. (7), we
can arrive at a formula for $r_{+}$ as a function of the parameters $l$, $q$
$\beta$, $\sigma$ and the ADM mass $M$. And from here we obtain the following
expression for $M$

$M=\frac{r_{+}}{6}\left[-2r_{+}^{2}\beta^{2}\,_{2}F_{1}\left(-\frac{3}{4},-\sigma;\frac{1}{4};-\frac{q^{2}}{2\sigma\beta^{2}r_{+}^{4}}\right)+8\pi
Pr_{+}^{2}+2r_{+}^{2}\beta^{2}+3\right]\,\,\,$ (11)

The electric potential $\Phi$ on the outer horizon, which is conjugate to the
charge $q$ is computed as,

$\displaystyle\Phi(r_{+})=\int_{r_{+}}^{\infty}E(r)dr=\frac{\beta^{2}r_{+}^{3}}{2q}\left[\left(1+\frac{q^{2}}{2\sigma\beta^{2}r_{+}^{4}}\right)^{\sigma}-\,_{2}F_{1}\left(-\frac{3}{4},-\sigma;\frac{1}{4};-\frac{q^{2}}{2\sigma\beta^{2}r_{+}^{4}}\right)\right]\,\,\,$
(12)

We have defined a new quantity $B$ as the conjugate quantity to $\beta$. To
write the first law, we will treat $M$ as the enthalpy given by $M=U+PV$.
Here, $U$ is the internal energy of the black hole.

Rewriting the mass $M$ as

$M(S,P,q,\beta)=\frac{\sqrt{S}}{6\sqrt{\pi}}\left[-\frac{2S\beta^{2}}{\pi}\,_{2}F_{1}\left(-\frac{3}{4},-\sigma;\frac{1}{4};-\frac{\pi^{2}q^{2}}{2\sigma\beta^{2}S^{2}}\right)+8PS+\frac{2S\beta^{2}}{\pi}+3\right]\,\,\,,$
(13)

the differentiation allows us to obtain

$dM=\left(\frac{\partial M}{\partial
S}\right)_{q,P,\beta}dS+\left(\frac{\partial M}{\partial
P}\right)_{S,q,\beta}dP+\left(\frac{\partial M}{\partial
q}\right)_{S,P,\beta}dq+\left(\frac{\partial
M}{\partial\beta}\right)_{S,q,P}d\beta\,\,\,,$ (14)

with

$\left(\frac{\partial M}{\partial
S}\right)_{q,P,\beta}=T\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left(\frac{\partial
M}{\partial
q}\right)_{S,P,\beta}=\Phi\,\,\,\,\,\,\,\,\,\,\,\left(\frac{\partial
M}{\partial P}\right)_{S,q,\beta}=V\,,$ (15)

which are compatible with the expressions given in Eqs. (10) and (12) and the
volume of the black hole. We have also defined a new quantity $B$ as the
conjugate quantity to $\beta$ following Refs. [28, 58]. Thus, we can compute
the variable $B$ as,

$B=\left(\frac{\partial M}{\partial\beta}\right)_{S,q,P}=\frac{\beta
r_{+}^{3}}{6}\left[4-\,_{2}F_{1}\left(-\frac{3}{4},-\sigma;\frac{1}{4};-\frac{q^{2}}{2\sigma\beta^{2}r_{+}^{4}}\right)-3\left(1+\frac{q^{2}}{2\sigma\beta^{2}r_{+}^{4}}\right)^{\sigma}\right]\,\,\,.$
(16)

Consequently, these quantities satisfy the first law of thermodynamics

$dM=TdS+VdP+\Phi dq+Bd\beta\,\,\,.$ (17)

From Eq. (13), we could obtain the Smarr formula following the scaling
argument given in Ref. [59]. Carrying out the dimensional analysis, we note
that the quantities obey the following scaling relations $S\propto l^{2}$,
$P\propto l^{-2}$, $q\propto l^{1}$ and $\beta\propto l^{-1}$, where
$l=[\mbox{lenght}]$, therefore the mass is a homogeneous function of degree 1.
Thus, the Euler theorem implies

$M=(2)\left(\frac{\partial M}{\partial S}\right)S+(-2)\left(\frac{\partial
M}{\partial P}\right)P+(1)\left(\frac{\partial M}{\partial
q}\right)q+(-1)\left(\frac{\partial M}{\partial\beta}\right)\beta\,\,\,,$ (18)

So one can write the Smarr formula [60] as

$M=2TS-2VP+\Phi q-B\beta\,\,\,.$ (19)

Recently, some results on thermodynamic relations have been obtained for black
holes with nonlinear electrodynamics. For example, in Ref. [61] the author
considers the solution given in Ref. [19], and uses a different thermodynamic
approach than we consider, where the thermodynamic volume of the black hole is
not the usual $4\pi r_{+}^{3}/3$, thus achieving to obtain a first law without
adding new quantities and according to this the Smarr formula turns out to be
$M=2TS-2VP+\Phi q$. In Ref. [62] a new black hole solution with nonlinear
electrodynamics is presented, which depends on a nonlinearity parameter such
that if it is zero then the Maxwell case recovers. Here, as in Refs. [26, 28]
and our thermodynamic approach, a new term appears in the Smarr formula that
is related to the quantity conjugated to the nonlinearity parameter. For its
part, in Ref. [63], the first law and the Smarr formula are modified when a
black hole with non-linear electrodynamics is considered, due to a new
thermodynamic quantity, the surface tension. Also in the context of black hole
solutions with nonlinear electrodynamics, in Refs. [64, 65], the first law is
modified with a quantity that multiplies $dM$. In Ref. [66] it is considered
that the entropy for regular black holes consists of two terms, one of them
the usual and the other corresponds to a deviation, and when the black hole
solution is obtained by coupling a nonlinear electrodynamic model, then it is
also considered a corrected term for the potential electric. As future work
one could take those thermodynamic approaches that are different from the one
we are considering in the present work and make a detailed analysis for the
case of black holes with Born-Infeld-type electrodynamics.

##  4  P-V criticality of the black hole

In this section, we will study the behaviour of pressure. The equation of
state for a Born–Infeld-type black can be described from Eq. (10) as

$P=\frac{2r^{2}\beta^{2}\left[\left(1+\frac{q^{2}}{2\sigma\beta^{2}r_{+}^{4}}\right)^{\sigma}-1\right]+4\pi
r_{+}T-1}{8\pi r_{+}^{2}}\,\,\,$ (20)

For certain values of parameters in the theory, there are critical values for
P vs $r_{+}$ graph as shown in Fig. 3.

Figure 3: The figure shows $P$ vs $r_{+}$ for varying temperature. Here
$\sigma=0.539,q=0.41$, and $\beta=0.536$

In order to calculate the critical values and to study law of corresponding
states, we first identify specific volume $v=2r_{+}$. Then replacing
$r_{+}=\frac{v}{2}$ in Eq. (20), the equation of state can be rewritten as
follows:

$P=\frac{-1+2\pi
Tv+\frac{v^{2}\beta^{2}}{2}\left((1+A)^{\sigma}-1\right)}{2\pi v^{2}}$ (21)

where $A=\frac{8q^{2}}{v^{4}\beta^{2}\sigma}$. Since $v\propto r_{+}$, the
graph in Fig. 3 will be similar in shape to a graph for $P$ vs $v$. Critical
value $t_{c}$ occurs when the $P$ vs $v$ graph has an inflection point. At the
critical point,

$\frac{\partial P}{\partial v}=\frac{\partial^{2}P}{\partial v^{2}}=0$ (22)

Solving the two equations obtained due to the conditions in Eq. (22), critical
temperature $T_{c}$ and critical pressure $p_{c}$ are given as follows:

$T_{c}=\frac{v_{c}^{4}\beta^{2}\sigma+q^{2}(8-8v_{c}^{2}\beta^{2}(1+A_{c})^{\sigma}\sigma)}{\pi
v_{c}(8q^{2}+v_{c}^{4}\beta^{2}\sigma)}$ (23) $P_{c}=\frac{-2+4\pi
T_{c}v_{c}+v_{c}^{2}\beta^{2}(-1+(1+A_{c})^{\sigma})}{4\pi v_{c}^{2}}$ (24)

Here $A_{c}=\frac{8q^{2}}{v_{c}^{4}\beta^{2}\sigma}$. It is not possible to
solve the equations arising due to Eq. (22) completely for $v_{c}$ due to the
nature of the equations. However, one could use ${\it Mathematica}$ to solve
for $P_{c},T_{c}$ and $v_{c}$ for numerical values for $\beta$ and $\sigma$.
The number $\frac{P_{c}v_{c}}{T_{c}}$ is an interesting number to calculate.
In the Van der Waals fluids, $\frac{P_{c}v_{c}}{T_{c}}=\frac{3}{8}$ is a
universal number for all fluids. It was shown that the Reissner-Nordström-AdS
black hole undergo criticality in Ref. [31] and the value for
$\frac{P_{c}v_{c}}{T_{c}}$ is $\frac{3}{8}$. In the current work, we have
computed $\frac{P_{c}v_{c}}{T_{c}}$ numerically for various values of $\sigma$
and have plotted in Fig. 4. It is observed that the ratio
$\frac{P_{c}v_{c}}{T_{c}}$ increases with $\sigma$. For $\sigma<1$,
$\frac{P_{c}v_{c}}{T_{c}}<\frac{3}{8}$ and for $\sigma>1$,
$\frac{P_{c}v_{c}}{T_{c}}>\frac{3}{8}$.

Figure 4: The figure shows $\frac{P_{c}v_{c}}{T_{c}}$ vs $\sigma$. Here
$q=0.41$, and $\beta=0.536$ Figure 5: The figure shows P vs V for isothermals
for varying temperatures. Here $q=1,\sigma=0.35$, and $\beta=0.412$

##  5 Born–Infeld-type black hole as heat engine

Black holes could be used as heat engines as demonstrated in several works
mentioned in the introduction [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54].
Since an equation of state is clearly defined with a pressure term and a
volume, the $Pdv$ term could be utilized to produce work. In order to create a
heat engine, one need a thermodynamic cycle: we will use a thermodynamical
cycle consisting of two isobaric and two isochoric paths as given in Fig. 5.
Heat will be supplied along the path $1\rightarrow 2$ and heat is out from the
path $4\rightarrow 3$. According to Ref. [47] the efficiency for the heat
engine for such an engine is given by
$\eta=1-\frac{M_{3}-M_{4}}{M_{2}-M_{1}}$. If we use Eq. (11), the efficiency
is obtained as

$\eta=\frac{8\left(P_{1}-P_{3}\right)\sqrt{S_{3}}\sqrt{S_{4}}\left(S_{3}+\sqrt{S_{4}}\sqrt{S_{3}}+S_{4}\right)}{\sqrt{S_{3}}\sqrt{S_{4}}\left(8P_{1}\left(S_{3}+\sqrt{S_{4}}\sqrt{S_{3}}+S_{4}\right)+3\right)-3\pi
q^{2}+F(S_{3},S_{4})}\,\,\,.$ (25)

where

$\displaystyle
F(S_{3},S_{4})=\frac{1}{\pi\left(\sqrt{S_{3}}-\sqrt{S_{4}}\right)}\left\\{\sqrt{S_{3}}\left[2S_{4}^{2}\beta^{2}\left(\,{}_{2}F_{1}\left(-\frac{3}{4},-\sigma;\frac{1}{4};-\frac{\pi^{2}q^{2}}{2S_{4}^{2}\beta^{2}\sigma}\right)-1\right)\right.\right.$
$\displaystyle\left.+3\pi^{2}q^{2}\Bigg{]}-\sqrt{S_{4}}\Bigg{.}\left[2S_{3}^{2}\beta^{2}\left(\,{}_{2}F_{1}\left(-\frac{3}{4},-\sigma;\frac{1}{4};-\frac{\pi^{2}q^{2}}{2S_{3}^{2}\beta^{2}\sigma}\right)-1\right)+3\pi^{2}q^{2}\right]\right\\}\,\,\,.$
(26)

Note that when $\beta\rightarrow\infty$, $F(S_{3},S_{4})=0$. Hence,
$\eta\rightarrow\eta_{0}$ where,

$\eta_{0}=\frac{8\left(P_{1}-P_{3}\right)\sqrt{S_{3}}\sqrt{S_{4}}\left(S_{3}+\sqrt{S_{4}}\sqrt{S_{3}}+S_{4}\right)}{\sqrt{S_{3}}\sqrt{S_{4}}\left(8P_{1}\left(S_{3}+\sqrt{S_{4}}\sqrt{S_{3}}+S_{4}\right)+3\right)-3\pi
q^{2}}\,\,\,.$ (27)

Here $\eta_{0}$ is the efficiency of the Reissner-Nordström-AdS black hole
solution. Also notice that,
$F(S_{3},S_{4})>0$ and $\eta<\eta_{0}$ , if $\sigma<1$ ($\sigma=\frac{1}{2}$
Born-Infeld case)
$F(S_{3},S_{4})<0$ and $\eta>\eta_{0}$ , if $\sigma>1$

We have plotted $\eta$ vs $\sigma$ in Fig. 6. It is clear that when $\sigma$
increases, $\eta$ increases. We have also plotted $\eta$ vs $\beta$ in Fig. 7.
When $\beta$ increases, $\eta$ also increases. Hence for fixed $\beta$, large
$\sigma$ yields greater $\eta$.

Figure 6: The figure shows $\eta$ vs $\sigma$. Here
$q=1,\beta=0.412,P_{1}=3.117,P_{3}=2.004,S_{3}=1.199$, and $S_{4}=0.537$
Figure 7: The figure shows $\eta$ vs $\beta$. Here
$q=1,\sigma=0.35,P_{1}=3.117,P_{3}=2.004,S_{3}=1.199$, and $S_{4}=0.537$

##  6 Conclusions

In this paper, we have studied black hole solutions in a new non-linear
electrodynamics model introduced by Kruglov [1]. There are basically two types
of black holes for this new model: for small $\sigma$ the black hole behaves
as the Schwarzschild-AdS black hole, and, for large $\sigma$, the black hole
behaves as the Reissner-Nordström-AdS black hole. When $\sigma=1/2$, the black
hole simplifies to the Born-Infeld black hole.

Thermodynamics of the black hole is studied in the extended phase space where
$P=-\frac{\Lambda}{8\pi}$. First law and the Smarr formula are derived: here,
there is a new thermodynamical quantity introduced as a conjugate quantity to
$\beta$. We have shown that the black hole demonstrate phase transitions
similar to van der Waals liquid-gas phase transitions. These black holes are
between small and large black holes. It is observed that the ratio
$\frac{P_{c}v_{C}}{T_{c}}$ is smaller than the universal values for liquids,
$\frac{3}{8}$, when $\sigma<1$. For large $\sigma>1$ the ratio is large than
$\frac{3}{8}$.

In Ref. [28], it was indicated that the quantity $B$ in the case of Born-
Infeld black holes has units of electric polarization per unit volume. Such
quantity is called Born-Infeld polarization. In this same reference it is
shown that $B$ is always positive and also noted that its physical meaning is
not yet fully understood. For the model we have considered, it can be shown
that $B(r_{+})\geq 0$ if $\sigma\leq 1$ and $B(r_{+})<0$ if $\sigma>1$. It
would be interesting to ask about the interpretation of $B$ when its value is
negative. We leave this aspect of the theory for future work.

We have studied these black holes as heat engines. The thermodynamical cycle
considered for the heat engine is a rectangle in $P,V$ space with two
isochoric and two isobaric processes. The efficiency of the heat engine, when
considered the black hole as a heat engine, is studied. At this point, as a
global conclusion, we can point out the relationship between the efficiency
and the electric field, in particular with the values it takes with respect to
the Coulomb field if we consider the same electric charge in both cases. Thus,
via a numerical analysis, we can observe that by varying the parameter
$\sigma$, which determines the regularity of the electrodynamic model, and
keeping $\beta$ fixed (for the same electric charge), the efficiency increases
together with the amplification of the electric field with respect to the
Coulomb electric field. We also observe that in the region of parameter values
corresponding to non-regular electric field solutions, the efficiency takes
its highest values, for this family of Born-Infeld type solutions; unlike what
happens in the region of regularity of the electric field, where the
efficiency decreases as the electric field is increasingly screened with
respect to the Coulomb electric field (the Born-Infeld case belongs to this
region). In this region of regularity, we can also notice that as the value of
the intercept of the curve of the electric field graph decreases, at the same
time the efficiency decreases.

Additionally, let us indicate that within each of the two regions: of
regularity or non-regularity, which are determined by the parameter $\sigma$
and in no case of $\beta$. The electric field can increase or decrease by
varying the parameter $\beta$, with the limiting that it does not reach values
that allow it to leave the region determined by $\sigma$. In summary, in each
region, we can vary one or both $\sigma$ and $\beta$ parameters at the same
time, and if this makes the electric field of the model increase, for example,
then the efficiency value also increases.

It would be interesting to study these black holes in the context of stability
due to perturbations and see how it is related to phase transitions and the
$\sigma$ value.

## Acknowledgments

L.B. was supported by DIUFRO through the project DI19-0052.

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