11institutetext: Data Semantics Laboratory, Kansas State University, USA
22institutetext: Atmosphere and Space Technology Research Associates (ASTRA
LLC), USA 33institutetext: Department of Earth, Planetary, and Space Sciences,
University of California, Los Angeles, USA

# Towards a Modular Ontology for Space Weather Research

Cogan Shimizu 11 0000-0003-4283-8701 Ryan McGranaghan 22 0000-0002-9605-0007
Aaron Eberhart 11 0000-0003-3007-5460 Adam C. Kellerman 33 0000-0002-2315-936X

###### Abstract

The interactions between the Sun, interplanetary space, near Earth space
environment, the Earth’s surface, and the power grid are, perhaps
unsurprisingly, very complicated. The study of such requires the collaboration
between many different organizations spanning the public and private sectors.
Thus, an important component of studying space weather is the integration and
analysis of heterogeneous information. As such, we have developed a modular
ontology to drive the core of the data integration and serve the needs of a
highly interdisciplinary community. This paper presents our preliminary
modular ontology, for space weather research, as well as demonstrate a method
for adaptation to a particular use-case, through the use of existential rules
and explicit typing.

##  1 Introduction

The solar-terrestrial system is complex, and consists of many interconnected
subsystems nonlinearly interacting; energy flows from the surface of the Sun,
through interplanetary space, into the near Earth space environment, and
across the surface of the Earth. The expanse of the system as well as resource
limitations dictate that a heterogeneous collection of data and models must be
seamlessly utilized together [4].

Interactions across the solar-terrestrial system are manifest on Earth in
myriad ways that pose threats to our technological infrastructure and ability
to use the near-Earth space environment [7]. Among the most important, yet
ironically, least well specified impacts is that of the electric power grid.
During periods of enhanced space weather activity, a series of physical
processes beginning with a solar event (e.g., the launch of a coronal mass
ejection (CME) or a high speed stream (HSS) from the Sun) gives rise to
intense electric currents reaching millions of Amperes surrounding the Earth,
which then become electric currents on the ground flowing through electrical
transmission lines. This phenomenon, known as Geomagnetically Induced Currents
(GICs), can disrupt the operation of high-voltage power grid transformers via
overheating and generation of harmonics, potentially leading to failures [6].

Power grid utility operators must stitch together observational and simulation
data to know the appropriate responses to be taken. This is challenged by the
number of disciplines that must cooperate and the intricate linking of the
data across those disciplines involved–from space scientists, to earth
scientists, to power system owners and operators, and power consumers.

What is needed is a semantic understanding of the notion linking solar events
to power grid responses to form the foundation of an effective data
integration system to study space weather and protect the power grid [5].

This paper describes the first modular ontological pattern that links the Sun
to the power grid, specifically to guide decisions for power grid utilities.

##  2 Use-case Scenario

The Convergence Hub for the Exploration of Space Science (CHESS) project aims
to unify the traditionally disparate communities and data sets that span the
Sun, interplanetary space, near Earth space environment, the surface of the
Earth, and the power grid. GICs arise from a series of interactions across
these domains, beginning with the solar cloud of plasma interacting with the
Earth’s magnetic field, creating currents in space and in the upper
atmospheric region known as the ionosphere, which produces the electric field
on the ground through magnetic induction. However, knowledge of many aspects
of this chain is limited However, knowledge of many aspects of this chain is
limited, largely due to the interdisciplinary nature of each interconnected
link. Therefore, the motivating question is, “What is the ontological notion
that links energy generated at the Sun, propagated through interplanetary
space, and efficacious of disturbance to the power grid?” Important in this
question is the responses that each link requires (e.g., conducting a
predictive simulation, monitoring real-time data streams, or taking mitigative
action to protect the grid) such that the ontological design must connect with
ODPs for simulation activities, sensor networks, and the actions that must be
taken by power grid utility operators. We detail the ontology designed for
CHESS, demonstrating the value of the modular design approach to facilitate
interoperability and reusabiliity of existing ontological patterns.

##  3 The Preliminary Ontology

This section first introduces the Data Transformation ontology design pattern
and its instantiation, the Simulation Activity Module, which are key pieces of
the ontology. We then show how the module fits into the modular ontology for
space weather. Next, we describe a mechanism for adapting the domain ontology
to a particular use-case and, finally, provide a worked example.

The OWL files for the pattern, module, and ontology can be found online, as
well as our supplementary data.111See https://github.com/cogan-shimizu-
wsu/DataTransformationPattern. The OWL files were generated using the CoModIDE
plugin [8] and annotated with OPLa [3, 2] using the OPLa Annotator plugin [9];
both plugins are for Protégé222See https://protege.stanford.edu/.

###  3.1 The Data Transformation Pattern

In many fields, the transformation of data from one form to another, whether
that be data-sanitation, analysis, or formatting, is a first-class operation.
The Data Transformation pattern, herein described, allows for the description
of data-driven workflows.

In January 2020, as part of a joint venture for the NSF’s Convergence
Accelerator: Open Knowledge Networks initiative, several teams met to discuss
the data modeling needs for each of the teams. It quickly became apparent that
tracking data workflows, as well as availability of data, and even how that
data might be used at all, was a very important aspect to each team’s use-
case, as we see in Section 2. During this meeting we identified a number of
interesting competency questions that serve to further frame the goals; a
selection follows, with the entire document being found in the supplementary
data.

  1. CQ1.

What datasets are available to view?

  2. CQ2.

What does dataset _X_ contain?

  3. CQ3.

In what ways is dataset _X_ used?

  4. CQ4.

What is the result of dataset _X_ transformed by Algorithm _A_?

  5. CQ5.

What dataset _X_ was used for input to Simulation _S_?

The resultant pattern can be shown in Figure 1. The core concept is
DataTransformation. A DataTransformation always occurs in space and time
(SpatiotemporalExtent) and in some ComputationalEnvironment. Each one also
provides a number of roles: InputRole, OutputRole, and ParameterRole. These
roles allow us to disconnect the Data from the transformation, acknowledging
that the same data set can be used in different datasets and that no
transformation is truly destructive. However, we do say that each instance of
a role (as a transformation may have many, say, input data) may only be
performed by exactly one Data. Further, every Data is an EntityWithProvenance,
meaning that it is generally known from where data is sourced. We also
specify, using the Explicit Typing pattern [10], that Data might have a
DataType, which allows us to specify, in a domain dependent manner, broad
categories of data or format of data.333The explicit declaration of types
allows us to easily change domain dependent information, without impacting an
ontology’s subsumption hierarchy. We posit that this strategy can aid with
downstream alignment and co-reference resolution.

We see in the schema diagram that Algorithm, ComputationalEnvironment, and
EntityWithProvenance are represented by blue dashed boxes. This means that
they are external to the pattern. That is, this pattern acknowledges that
these concepts exist, and are integral to the notion of a DataTransformation,
but makes no claims on their ontological structure. For the latter two, we
recommend the use of their eponymous counterparts in [10]. Algorithm is very
domain dependent and we leave it to be specified in specific use-cases.

####  3.1.1 Axiomatization

This axiomatization was produced using the out-of-the-box axioms depicted in
the Edge Inspector tool in CoModIDE [8], as well as the same axioms used
during the systematic axiomatization process particular to Modular Ontology
Engineering [11]. They are provided in description logic syntax below. The
axioms are segmented to aid reading. Axioms 2-10 are scoped ranges (e.g. axiom
2) and existentials (e.g. axiom 3). Axiom 1 is a subclass. Axiom 11 is a
_structural tautology_ which serves to inform users of the intended use of an
axiom. Axioms 12-16 and 29-32 are scoped domain axioms.

Each of the properties included in this pattern have scoped domain and range,
as we want to limit the ontological commitments made by this pattern. Each of
the “performs” properties also have inverse existential and functionality.
This means that only exactly one data can perform a role provided by a
DataTransformation.

DataTransformation $\displaystyle\sqsubseteq\textsf{Algorithm}$ (1)
DataTransformation
$\displaystyle\sqsubseteq\forall\textsf{occursInCE.ComputationalEnvironment}$
(2) DataTransformation
$\displaystyle\sqsubseteq\exists\textsf{occursInCE.ComputationalEnvironment}$
(3) DataTransformation
$\displaystyle\sqsubseteq\forall\textsf{occursDuringSTE.SpatiotemporalExtent}$
(4) DataTransformation
$\displaystyle\sqsubseteq\exists\textsf{occursDuringSTE.SpatiotemporalExtent}$
(5) DataTransformation
$\displaystyle\sqsubseteq\forall\textsf{providesInputRole.InputRole}$ (6)
DataTransformation
$\displaystyle\sqsubseteq\exists\textsf{providesInputRole.InputRole}$ (7)
DataTransformation
$\displaystyle\sqsubseteq\forall\textsf{providesOutputRole}$ (8)
DataTransformation
$\displaystyle\sqsubseteq\exists\textsf{providesOutputRole}$ (9)
DataTransformation
$\displaystyle\sqsubseteq\forall\textsf{providesParamaterRole.ParameterRole}$
(10) DataTransformation
$\displaystyle\sqsubseteq\mathord{\geq}0\textsf{providesParamaterRole.ParameterRole}$
(11) $\displaystyle\exists\textsf{occursInCE.ComputationalEnvironment}$
$\displaystyle\sqsubseteq\textsf{DataTransformation}$ (12)
$\displaystyle\exists\textsf{occursDuringSTE.SpatiotemporalExtent}$
$\displaystyle\sqsubseteq\textsf{DataTransformation}$ (13)
$\displaystyle\exists\textsf{providesInputRole.InputRole}$
$\displaystyle\sqsubseteq\textsf{DataTransformation}$ (14)
$\displaystyle\exists\textsf{providesOutputRole.OutputRole}$
$\displaystyle\sqsubseteq\textsf{DataTransformation}$ (15) Data
$\displaystyle\sqsubseteq\textsf{EntityWithProvenance}$ (16) Data
$\displaystyle\sqsubseteq\forall\textsf{performsInputRole.InputRole}$ (17)
Data
$\displaystyle\sqsubseteq\exists\textsf{performsInputRole${}^{-}$.InputRole}$
(18) Data
$\displaystyle\sqsubseteq\mathord{\leq}1\textsf{performsInputRole${}^{-}$.InputRole}$
(19) Data
$\displaystyle\sqsubseteq\forall\textsf{performsOutputRole.OutputRole}$ (20)
Data
$\displaystyle\sqsubseteq\exists\textsf{performsOutputRole${}^{-}$.OutputRole}$
(21) Data
$\displaystyle\sqsubseteq\mathord{\leq}1\textsf{performsOutputRole${}^{-}$.OutputRole}$
(22) Data
$\displaystyle\sqsubseteq\forall\textsf{performsParameterRole.ParameterRole}$
(23) Data
$\displaystyle\sqsubseteq\exists\textsf{performsParameterRole${}^{-}$.ParameterRole}$
(24) Data
$\displaystyle\sqsubseteq\mathord{\leq}1\textsf{performsParameterRole${}^{-}$.ParameterRole}$
(25) Data
$\displaystyle\sqsubseteq\forall\textsf{participatesInDataTransformation.DataTransformation}$
(26) Data $\displaystyle\sqsubseteq\forall\textsf{hasPayload.Payload}$ (27)
Data $\displaystyle\sqsubseteq\forall\textsf{hasDataType.DataType}$ (28)
$\displaystyle\exists\textsf{performsParameterRole.ParameterRole}$
$\displaystyle\sqsubseteq\textsf{Data}$ (29)
$\displaystyle\exists\textsf{participatesInDataTransformation.DataTransformation}$
$\displaystyle\sqsubseteq\textsf{Data}$ (30)
$\displaystyle\exists\textsf{hasPayload.Payload}$
$\displaystyle\sqsubseteq\textsf{Data}$ (31)
$\displaystyle\exists\textsf{hasDataType.DataType}$
$\displaystyle\sqsubseteq\textsf{Data}$ (32)

###  3.2 The Simulation Activity Module

The Data Transformation pattern is a necessarily general concept. During the
ontology development process, we adapted the pattern to the use-case outlined
in Sections 1 and 2. To do so, we made use of the strategy outlined in [1],
which is called “Template-based Instantiation.” That is, we generally replaced
names of classes and properties, instead of using subclass relationships.
However, to keep the provenance of the pattern’s structure, we use annotations
from the Ontology Design Pattern Representation Language (OPLa) to indicate
the original pattern. Figure 2 shows the new module.

The major addition is concept of SimulatedData and additional specification of
its own provenance. These changes to the pattern allow us to answer the
following competency questions that are more specific to the use-case.

  1. CQ6.

What were the solar wind conditions at the time of a given GIC observation
_G_?

  2. CQ7.

Is this GIC node at risk of exceeding a threshold GIC level in the next _X_
time?

  3. CQ8.

What are the simulated conditions in given the results of a simulation?

The axioms for the additions are as follows.

SimulationActivity
$\displaystyle\sqsubseteq\forall\textsf{used.SimulatedData}$ (33)
SimulationActivity
$\displaystyle\sqsubseteq\forall\textsf{simulated.SimulatedData}$ (34)
SimulatedData
$\displaystyle\sqsubseteq\forall\textsf{wasDereivedFrom.SimulatedData}$ (35)

Figure 1: The schema diagram for the DataTransformation pattern. Yellow boxes
denote classes. Blue boxes with dashed outline denote external patterns or
modules. Purple boxes denote controlled vocabularies. Solid arrows denote
object or date properties, while open arrows denote the subclass relationship.
Figure 2: The schema diagram for the SimulationActivity module. It utilizes
the same notation as previous diagrams.

###  3.3 The Modular Ontology for Space Weather

At a top level, the purpose of the ontology is to tie together Activities,
such as simulations or interpretations; Agents, such as power grid operators;
Solar Events, such as coronal mass ejections; and Responses, such as simulate,
monitor, or update. In addition, it is necessary to record the provenance of
datasets as they are generated and transformed. We present the entire
assembled ontology in Figure 3.

We have previously discussed the pattern used for one Activity: the Simulation
Activity. The other activities are Monitor and Interpretation. The Monitor
activity is currently left as a stub, which is a tiny pattern for
acknowledging further complexity [10]. The Interpretation Activity is used
when an Agent, either a person or computer algorithm, perhaps, interprets some
data to indicate that a Solar Event, or not, has a occurred.

The Agent Role module is based on the Agent Role pattern, equipped with
Explicit Typing for the categories of roles. This set of roles is a controlled
vocabulary (CV) and would be particular to the domain. In the next section, we
discuss how these CVs can be used to add structured domain knowledge.

Agents are responsible for certain Responses. In the same manner, there are
different categories of Responses. Responses also encompass an activity. For
example, a response to some solar event might be to simulate its impact, and
the Activity it encompasses would be exactly that execution.

Figure 3: The schema diagram for the SimulationActivity module. The dashed
grey boxes denote modules, otherwise utilizes the same notation as previous
diagrams.

###  3.4 Adding Domain Knowledge with Existential Rules

Although the precise specification that an ontology provides is very helpful
for modeling a complex scenario, it is also occasionally difficult to
visualize exactly which connections are purely logical and which are essential
for a user interacting with the ontology. In order to provide a, perhaps more
intuitive, representation we show a First-Order Predicate Logic (FOPL) Rule
statement that corresponds to entailments of some sample data that would
comprise the ontology.444The abbreviations are as follows: NOAA is the
National Oceanic and Atmospheric Administration, SPWC is the Space Weather
Prediction Center, and GOES is Geostationary Operational Environment
Satellites. As we are writing the rule in FOPL, we may omit many of the
description logic classes that are conceptually important, but play a
diminished role in the semantics for a domain expert, and instead focus on the
connecting predicates that form the relevant entailment. In this example, we
show how a SolarEvent, such as a “Solar Flare”, leads to a definite chain of
Responses. Every solar event defined in the ontology must be represented in
this way. This means that if another solar event is known by the ontology, a
well-defined sequence of responses to that event must also exist.

$\displaystyle hasSolarEventType(\text{solarevent,``Solar Flare"})\to$
$\displaystyle\exists\text{sequentialresponse,response1,response2,response3,response4,response5,}$
activity1,activity2,activity3,activity4,activity5 $\displaystyle
requiresResponse(\text{solarevent,sequentialresponse})\wedge$ $\displaystyle
hasFirstResponse(\text{sequentialresponse,response1})\wedge$
$\displaystyle\qquad encompasses(\text{response1,activity1})\wedge$
$\displaystyle hasResponseType(\text{response1},$
$\displaystyle\qquad\text{``Simulate radiation conditions at Earth using
historical analog data"})\wedge$ $\displaystyle
hasNextResponse(\text{response1,response2})\wedge
encompasses(\text{response2,activity2})\wedge$ $\displaystyle
hasResponseType(\text{response2},$ “Update radiation conditions using NOAA,
SWPC, $\displaystyle\qquad\text{ and GOES satellites observations"})\wedge$
$\displaystyle hasNextResponse(\text{response2,response3})\wedge
encompasses(\text{response3,activity3})\wedge$ $\displaystyle
hasResponseType(\text{response3},$ $\displaystyle\qquad\text{``Based on
radiation conditions run ionospheric models"})\wedge$ $\displaystyle
hasNextResponse(\text{response3,response4})\wedge
encompasses(\text{response4,activity4})\wedge$ $\displaystyle
hasResponseType(\text{response4},$ “Based on ionospheric currents from
ionospheric simulations, $\displaystyle\qquad\text{ simulate geomagnetic
field"})\wedge$ $\displaystyle
hasNextResponse(\text{response4,response5})\wedge
encompasses(\text{response5,activity5})\wedge$ $\displaystyle
hasResponseType(\text{response5},$ “Based on geomagnetic field from
simulations, $\displaystyle\qquad\text{ simulate geomagnetically induced
currents"})$

###  3.5 Example Triple Data

Below we reproduce a fragment of what the populated ontology would look like,
corresponding to the rule in the previous section. Some triples are omitted
for brevity and clarity, however no new data was created besides instances and
strings used to fill out the ontology, so this can be viewed as a correct
representation.

    :Activity rdf:type owl:Class .
    :Response rdf:type owl:Class .
    :ResponseType rdf:type owl:Class .
    :SequentialResponse rdf:type owl:Class .
    :SolarEvent rdf:type owl:Class .
    :SolarEventType rdf:type owl:Class .
    :encompasses rdf:type owl:ObjectProperty .
    :hasFirstResponse rdf:type owl:ObjectProperty .
    :hasNextResponse rdf:type owl:ObjectProperty .
    :hasResponseType rdf:type owl:ObjectProperty .
    :hasSolarEventType rdf:type owl:ObjectProperty .
    :requiresResponse rdf:type owl:ObjectProperty .
    :asString rdf:type owl:DatatypeProperty .

    :solarEvent rdf:type           owl:NamedIndividual ,
                                   :SolarEvent ;
                :hasSolarEventType :solarEventType ;
                :requiresResponse  :sequentialResponse .
    :solarEventType rdf:type owl:NamedIndividual ,
                             :SolarEventType ;
                    :asString "Solar Flare" .

    :sequentialResponse rdf:type owl:NamedIndividual ,
                                :SequentialResponse ;
                        :hasFirstResponse :response1 .

    :response1 rdf:type owl:NamedIndividual ,
                        :Response ;
               :encompasses :activity1 ;
               :hasNextResponse :response2 ;
               :hasResponseType :responseType1 .
    :response2 rdf:type owl:NamedIndividual ,
                        :Response ;
               :encompasses :activity2 ;
               :hasNextResponse :response3 ;
               :hasResponseType :responseType2 .
    :response3 rdf:type owl:NamedIndividual ,
                        :Response ;
               :encompasses :activity3 ;
               :hasNextResponse :response4 ;
               :hasResponseType :responseType3 .
    :response4 rdf:type owl:NamedIndividual ,
                        :Response ;
               :encompasses :activity4 ;
               :hasNextResponse :response5 ;
               :hasResponseType :responseType4 .
    :response5 rdf:type owl:NamedIndividual ,
                        :Response ;
               :encompasses :activity5 ;
               :hasResponseType :responseType5 .

    :responseType1 rdf:type owl:NamedIndividual ,
                            :ResponseType ;
                   :asString "Simulate the radiation conditions
                              at Earth using historical analog data" .
    :responseType2 rdf:type owl:NamedIndividual ,
                            :ResponseType ;
                   :asString "Update radiation conditions using NOAA
                              SWPC GOES satellites observations" .
    :responseType3 rdf:type owl:NamedIndividual ,
                            :ResponseType ;
                   :asString "Based on radiation conditions run
                              ionospheric models" .
    :responseType4 rdf:type owl:NamedIndividual ,
                            :ResponseType ;
                   :asString "Based on ionospheric currents from
                              ionospheric simulations, simulate
                              geomagnetic field" .
    :responseType5 rdf:type owl:NamedIndividual ,
                            :ResponseType ;
                   :asString "Based on geomagnetic field from
                              simulations, simulate geomagnetically
                              induced currents" .

    :activity1 rdf:type owl:NamedIndividual ,
                        :Activity .
    :activity2 rdf:type owl:NamedIndividual ,
                        :Activity .
    :activity3 rdf:type owl:NamedIndividual ,
                        :Activity .
    :activity4 rdf:type owl:NamedIndividual ,
                        :Activity .
    :activity5 rdf:type owl:NamedIndividual ,
                        :Activity .

##  4 Conclusion

This paper has introduced a modular ontology for supporting space weather
research, especially in understanding the interactions between agencies and
their responsibilities, as well as tracking the usage and provenance of data
through observations and simulations. In doing so, we presented a novel
Numerical Transformation pattern, which is further modified to fit our use-
case via template-based instantiation to produce the SimulationActivity
module. This is one of the core patterns that constitute the entire modular
ontology. We then go to show how existential rules, making use of instance
data in controlled vocabularies, can be used to add and frame domain knowlege.
Finally, we discuss a worked example.

The next steps for this work will be the further development of the modules
left as stubs, namely Observation Data, which has obvious connections to the
SSN/SOSA555See https://www.w3.org/TR/vocab-ssn/. and the Monitoring activity.
We will also be exploring deployment and integration of real world data.

##### Acknowledgements.

The authors acknowledge partial support from the National Science Foundation
under Grant Numbers 1936677 and 1937152, as well as thorough discussion with
Sean Gordon, Lisa Kerr, and Philip Murphy.

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# A Lyman-$\alpha$ transit left undetected: the environment and atmospheric
behavior of K2-25b

Keighley E. Rockcliffe Department of Physics and Astronomy, Dartmouth College,
Hanover, NH 03755, USA  Elisabeth R. Newton Department of Physics and
Astronomy, Dartmouth College, Hanover, NH 03755, USA  Allison Youngblood
Laboratory for Atmospheric and Space Physics, University of Colorado Boulder,
Boulder, CO 80303, USA  Vincent Bourrier Observatoire Astronomique de
l’Université de Genève, Chemin Pegasi 51b, CH-1290 Versoix, Switzerland
Andrew W. Mann Department of Physics and Astronomy, University of North
Carolina at Chapel Hill, Chapel Hill, NC 27599, USA  Zachory Berta-Thompson
Department of Astrophysical & Planetary Sciences, University of Colorado
Boulder, Boulder, CO 80303, USA  Marcel A. Agüeros Department of Astronomy,
Columbia University, New York, NY 10027, USA  Alejandro Núñez Department of
Astronomy, Columbia University, New York, NY 10027, USA  David Charbonneau
Center for Astrophysics - Harvard-Smithsonian, Cambridge, MA 02139, USA

###### Abstract

K2-25b is a Neptune-sized exoplanet ($3.45$R⊕) that orbits its M4.5 host with
a period of $3.48$ days. Due to its membership in the Hyades Cluster, the
system has a known age ($727\pm 75$ Myr). K2-25b’s youth and its similarities
with Gl 436b suggested that K2-25b could be undergoing strong atmospheric
escape. We observed two transits of K2-25b at Lyman-$\alpha$ using HST/STIS in
order to search for escaping neutral hydrogen. We were unable to detect an
exospheric signature, but placed an upper limit of
$(R_{\text{p}}/R_{\star})|_{Ly\alpha}<0.56$ at 95% confidence by fitting the
light curve of the Lyman-$\alpha$ red-wing, or $<1.20$ in the blue-wing. We
reconstructed the intrinsic Lyman-$\alpha$ profile of K2-25 to determine its
Ly$\alpha$ flux, and analyzed XMM-Newton observations to determined its X-ray
flux. Based on the total X-ray and extreme ultraviolet irradiation of the
planet ($8763\pm 1049$ erg s-1 cm-2), we estimated the maximum energy-limited
mass loss rate of K2-25b to be $10.6^{+15.2}_{-6.13}\times 10^{10}$ g s-1
($0.56$M⊕ per 1 Gyr), five times larger than the similarly estimated mass loss
rate of Gl 436b ($2.2\times 10^{10}$ g s-1). The photoionization time is about
3 hours, significantly shorter than Gl 436b’s 14 hours. A non-detection of a
Lyman-$\alpha$ transit could suggest K2-25b is not significantly losing its
atmosphere, or factors of the system are resulting in the mass loss being
unobservable (e.g., atmosphere composition or the system’s large high energy
flux). Further observations could provide more stringent constraints.

††software:  astropy (Price-Whelan et al., 2018), batman (Kreidberg, 2015),
calstis (https://github.com/spacetelescope/stistools), emcee (France et al.,
2013), lyapy (Youngblood et al., 2016), matplotlib (Hunter, 2007), numpy
(Harris et al., 2020), scipy (Virtanen et al., 2020)

##  1 Introduction

Features in the exoplanet radius–period diagram are consequences of
exoplanetary formation and evolutionary processes. In particular, the “hot
Neptune desert” and the “radius gap” motivate understanding atmospheric
escape. The hot Neptune desert is the lack of short period planets
(P${}_{\text{p}}\lesssim 3$ days) with radii between that of super-Earths and
Jupiter (Lundkvist et al., 2016; Mazeh et al., 2016; Kurokawa & Nakamoto,
2014; Beaugé & Nesvorný, 2013; Szabó & Kiss, 2011; Davis & Wheatley, 2009;
Lecavelier Des Etangs, 2007). Atmospheric escape, along with orbital
migration, is one of the dominant processes thought to shape the desert (Owen
& Lai, 2018; Mazeh et al., 2016). The second feature, the radius gap, is the
gap in the distribution of planetary radii around $1.5-2$R⊕ (Fulton &
Petigura, 2018; Fulton et al., 2017; Owen & Wu, 2013). The radius gap can also
be attributable to atmospheric escape (Ginzburg et al., 2018; Jin et al.,
2014; Lopez & Fortney, 2013; Owen & Wu, 2013), though Lee & Connors (2021)
argue it can be primordial.

Atmospheric escape occurs on every planet with an atmosphere, but may have
more or less influence on a its evolution depending on the planetary
properties (e.g., bulk density, atmospheric composition, magnetic field) and
environment (e.g., irradiation, stellar wind, impact erosion). The two
atmospheric escape processes that have been mostly explored as shapers of the
exoplanet population are thermal irradiation-driven (photoevaporation) and
core-cooling-driven (core-powered mass loss) processes.

Photoevaporation occurs when a close-in planet with a volatile-dominated
atmosphere receives a large flux of high energy radiation from its host. The
radiation heats the planet’s upper atmosphere leading to the bulk motion of
particles outward in a hydrodynamic outflow (see Owen, 2019, for a review). It
follows that the timescale for this process is closely tied to the evolution
of the star’s high energy radiation. Although a timescale of 100 Myr is
commonly accepted as it traces the period of highest stellar X-ray output
(e.g., Owen & Wu, 2017; Lopez & Fortney, 2013), King & Wheatley (2021) discuss
the possibility of a longer Gyr timescale following the slower decline in
extreme ultraviolet (EUV) output. Alternatively, Ginzburg et al. (2018) found
that the luminosity from a cooling exoplanetary core (core-powered mass loss)
can heat the atmosphere from below and cause hydrodynamic outflow. Exploring
atmospheric escape within a range of host spectral types can distinguish
between photoevaporation and core-powered mass loss (e.g., Ginzburg et al.,
2018). These two processes have different dependencies on stellar and
planetary properties, and different timescales for evolution.

One of the most important tracers of atmospheric escape is the Lyman-$\alpha$
emission line (Ly$\alpha$; $1215.672$ Å). A host star’s Ly$\alpha$ radiation
readily interacts with neutral hydrogen in the planetary atmosphere. While
interstellar hydrogen usually completely absorbs Ly$\alpha$ at the emission
line center, a planet transit can be observed at Ly$\alpha$ if the planetary
hydrogen atoms are accelerated to high velocities such that they attenuate the
wings of the line profile. The exact acceleration mechanisms remain an open
question in the community. Atmospheric modelling work suggests that radiation
pressure and stellar wind interactions change the spectral signature of the
exosphere (Debrecht et al., 2020; McCann et al., 2019; Bourrier et al., 2018a,
2016, 2015; Villarreal D’Angelo et al., 2018; Tremblin & Chiang, 2013; Ben-
Jaffel & Sona Hosseini, 2010; Holmström et al., 2008).

Neutral hydrogen exospheres in two transiting hot Neptunes, Gl 436b (sometimes
referred to as GJ 436b, dos Santos et al., 2019; Lavie et al., 2017;
Ehrenreich et al., 2015; Kulow et al., 2014) and GJ 3470b (Bourrier et al.,
2018a), have been detected. These planets orbit their M dwarf hosts within
$0.03$ AU and are therefore subject to high radiation levels. Observations
with the Hubble Space Telescope (HST) Space Telescope Imaging Spectrograph
(STIS) and the Cosmic Origins Spectrograph (COS) show deep and temporally
asymmetric Ly$\alpha$ attenuation surrounding the white-light transit. These
data suggest an extended comet-like hydrogen tail surrounding the planets.

Table 1: K2-25 system properties.

Properties (Symbol) | Value | Units
---|---|---
Earth-system distance (d) | $45.014\pm 0.165$ | pc
Age ($\tau$)${}^{\text{a}}$ | $727\pm 75$ | Myr
Right ascension ($\alpha$) | 04:13:05.62 | hh:mm:ss
Declination ($\delta$) |  $+$15:14:51.9 | dd:mm:ss
Spectral type | M$4.5$ |
Bolometric luminosity (L${}_{\text{bol}}$) | $8.16\pm 0.29\times 10^{-3}$ | L⊙
Stellar mass (M⋆) | $0.2634\pm 0.0077$ | M⊙
Stellar radius (R⋆) | $0.2932\pm 0.0093$ | R⊙
Stellar rotation period (P⋆) | $1.88\pm 0.02$ | days
Barycentric radial velocity (v⋆) | $38.64\pm 0.15$ | km s-1
Epoch (t0) | $2457062.57965\pm 0.0002$ | BJD
Transit duration | $0.79^{+0.09}_{-0.17}$ | days
Planetary mass estimate${}^{\text{b}}$ (M${}_{\text{p}}$) | $7^{+10}_{-4}$ | M⊕
Planetary mass measurement${}^{\text{c}}$ | $24.5^{+5.7}_{-5.2}$ | M⊕
Planetary radius (R${}_{\text{p}}$) | $3.4492^{+0.1099}_{-0.1110}$ | R⊕
Orbital period (P${}_{\text{p}}$) | $3.48456322^{+9.7\times 10^{-7}}_{-9.5\times 10^{-7}}$ | days
Semi-major axis (a) | $0.0288\pm 0.0003$ | AU

aafootnotetext: Age estimate of the Hyades cluster as determined by Douglas et
al. (2019).

bbfootnotetext: Mass estimate from Kain et al. (2020) using the mass-radius
relation implemented in MRExo (Ning et al., 2018; Kanodia et al., 2019).

ccfootnotetext: Mass measurement from Habitable Zone Planet Finder radial
velocity observations (Stefansson et al., 2020)

.

Note. — Parameters are from Mann et al. (2016) and Thao et al. (2020).

Figure 1: The distribution of radii (in Earth radii) as a function of orbital
period (in days) for exoplanets within the NASA Exoplanet Archive as of March
2021.

K2-25b is a Neptune-sized short-period exoplanet within the $727\pm 75$ Myr-
old Hyades cluster (Gossage et al., 2018) that was discovered using photometry
from the K2 mission (Mann et al., 2016). K2-25b orbits around its young M4.5V
host with a period of $3.48$ days. We review the physical properties of the
K2-25 system in Table 1, and compare K2-25b to the Neptune-sized planets with
detected neutral hydrogen exospheres in Figure 1.

Determining how system properties like planetary atmospheric composition and
youth impact mass loss is an important part of understanding atmospheric
escape. Young exoplanets are particularly important for this study because
they experience extreme stellar environments and directly probe the theorized
$\sim 100$ Myr timescale for photoevaporation. Constraining the mass loss rate
and timescale for photoevaporation in young planets will test and improve
exoplanet demographic studies. This motivates a detailed study of K2-25b and
its radiation environment. Gaidos et al. (2020) previously analyzed infrared
transmission spectra for K2-25b and did not detect escaping helium; here, we
consider neutral hydrogen escape. We obtained HST/STIS Ly$\alpha$ observations
of K2-25b (HST-GO-14615; PI: Newton), which we present in Section 2. Analysis
of the Ly$\alpha$ light curves is presented in Section 3. We measure the
Ly$\alpha$ and X-ray flux of K2-25, and estimate the energy-limited mass loss
rate for K2-25b in Section 4. We conclude by discussing the implications of
our results in Section 5.

##  2 Lyman-$\alpha$ line observations and analysis

We observed K2-25 with the $52$ x $0.1$″ aperture using HST’s STIS. We used
the G140M grating with a spectral range of 1140-1741 Å and a resolving power
of $\sim$10,000. We observed with the far-ultraviolet multi-anode microchannel
array (FUV-MAMA) in TIME-TAG mode. Two HST visits occurred on 23 March 2017
(Visit 1) and 31 October 2017 (Visit 2), each corresponding to a transit of
K2-25b. Eight science exposures were taken for each visit, each spanning the
observable window of an HST orbit (about 2000 s). TIME-TAG mode observations
individually stamp the arrival time for each photon detected by the
instrument. These time-stamps allowed us to split the full exposures, prior to
extraction and reduction, into three sub-exposures of $684$ s or $625$ s each.

Figure 2: An example of the two-dimensional raw data taken by HST/STIS during
a full orbit within Visit 1. The spectral trace and the two background
locations are highlighted in gray. In Visit 1, the geocoronal emission
encroaches on the blue-wing of the Ly$\alpha$ line.

We used the calstis pipeline (v3.4) to reduce these data. We automatically
located the extraction apertures by determining the centroid of the Ly$\alpha$
red wing. We used the calstis pipeline to extract the background in two
regions $15$ pixels above and below the extraction location. We used these
extraction regions to fit the spatial direction with the default third-order
polynomial, and this fit was used to remove the background from the target
spectrum. We did not find the choice of polynomial order to be important.
Cycle 27 calibration files were used to assign wavelengths to each pixel and
then convert flux in counts to specific flux values (erg s-1 cm-2 Å-1).

Errors from calstis are based on $\sqrt{\text{N}}$, which is inaccurate for
small $N$ (the total counts). Equation 1 approximates the confidence limit for
a Poisson distribution, corresponding to a Gaussian 1$\sigma$ limit (Gehrels,
1986). We used Equation 1 to recalculate errors for our spectra from the total
counts (the GROSS spectrum in the calstis data products):

$\sigma\approx 1-\sqrt{\text{N}+0.75}$ (1)

The error at each pixel was then converted to flux units using the flux
conversion factors we inferred from the calstis data products.

We looked for the HST “breathing effect” (thermal changes in the telescope
optics and focal plane) which can cause variability in flux throughput over
the course of an HST orbit (Bourrier et al., 2013; Huitson et al., 2012; Sing
et al., 2008; Brown et al., 2001; Bowers, 1997). We created six light curves
for the Ly$\alpha$ line (three per visit) by grouping consecutive spectra. We
folded the light curves on HST’s orbital period, and fit a sloped line to the
flux as a function of orbital phase for each one. We also looked at the
behavior of each exposure individually. In all cases, a sloped line fit and a
horizontal line fit were similar as shown by their Bayesian Information
Criteria ($|BIC_{slope}-BIC_{flat}|\sim 2$), so we did not include a
systematics correction in our analysis.

The raw data are contaminated with geocoronal airglow from solar Ly$\alpha$
photons scattered off of the Earth’s atmosphere (Figure 2). Though largely
removed by background subtraction, contaminated regions should be treated with
caution, as evidenced by the increased scatter in these regions as seen in
Figure 3 (gray regions). The contamination is of particular concern in Visit 1
because the geocoronal emission coincides with the blue-wing of K2-25’s
Ly$\alpha$ profile.

Given K2-25’s systemic velocity, the Ly$\alpha$ line center is not observable
at Earth due to the high neutral hydrogen column density, and for this reason
we consider the Ly$\alpha$ wings. The individual spectra in the two panels of
Figure 3 temporally resolve each transit. If significant exospheric neutral
hydrogen was present at high enough speeds, the wing of the Ly$\alpha$ profile
would be attenuated during the exosphere transit. For comparison, one
observation of the warm Neptune Gl 436b yielded a $56.2$% transit in the blue-
wing of Ly$\alpha$ starting 2 hours before and ending between 10 and 25 hours
after the optical transit time of the planet (Lavie et al., 2017). No deep
transit is obvious for K2-25b during either Visit 1 or 2 (see Figure 3): all
spectra taken during and outside of the planet transit are visually
consistent.

Figure 3: The spectra of K2-25 during Visits 1 (left) and 2 (right). The
different colors indicate the average spectrum prior to transit (yellow), at
times near the white-light transit (green), and after the transit (pink). The
gray shaded region indicates the location of the geocoronal emission region
that was removed from the spectra. Velocities are in the host star’s reference
frame.

##  3 K2-25’s Light Curve

We created two Ly$\alpha$ light curves from our observations, one for the
Ly$\alpha$ blue-wing and the other for the red-wing, in order to
quantitatively investigate the presence of a transit. Our analysis uses the
sub-exposures obtained from the TIME-TAG mode data. To obtain fluxes at each
point in time, we summed over the stellar reference frame velocities $-165.2$
to $-29.5$ km s-1 for the blue-wing, and $44.3$ to $229.3$ km s-1 for the red-
wing (Figure 4).

We normalized all of the sub-exposure fluxes to the out-of-transit data using
a linear fit to the full exposure data points that are beyond $\pm 5$ hours of
mid-transit. Both visits were combined into a single light curve as a function
of planetary orbital phase (time from mid-transit). With sufficient signal and
cadence, light curves such as these can constrain the radius and characterize
the behavior of the potentially escaping neutral hydrogen.

We compared the Ly$\alpha$ red-wing light curve to K2-25’s N V emission to
look for evidence of non-transit related variability (e.g., flares). M dwarf
flares enhance N V flux more than Ly$\alpha$ and therefore should be more
indicative of activity influencing our light curves (Loyd et al., 2018). Two
potential flares in N V were indicated in Gaidos et al. (2020); improved error
analysis does not support the presence of significant variability. Inclusion
of the two points discarded in Gaidos et al. (2020) does not impact the
results of that work. Since no flaring was evident in N V or Ly$\alpha$ we
used all data in our analysis.

We assumed the exosphere can be represented by a transiting opaque disk, and
model the light curves with BATMAN (Kreidberg, 2015). The S/N of our data does
not warrant considering more complicated transit shapes (e.g., asymmetry).
While Gl 436b does produce a clear asymmetric transit shape, GJ 3470b’s
transit is relatively symmetric (see Figures 2 and 5 in Bourrier et al., 2016,
2018a). The full BATMAN transit model consists of eight parameters: the mid-
transit time ($t_{0}$), period ($P_{\text{p}}$), ratio of the planetary radius
to stellar radius ($R_{\text{p}}/R_{\star}$), semi-major axis ($a$),
inclination ($i$), eccentricity ($e$), argument of periastron ($\omega$), and
the limb-darkening coefficients ($u$). We only fit for the transit depth,
fixing all other parameters to the values listed in Table 2. The limb-
darkening coefficients were assumed to be $0$, but are irrelevant given the
sparse sampling of our data.

There is a dip in the flux of both wings at $\sim 1$ hour after the fixed
transit time, lasting for one orbit. At this time, we do not explore
asymmetric transits. Instead, we fit the same transit model, but vary the mid-
transit time. The best fit results in a transit in both the blue and red-wings
are centered at $\sim 1$ hour which does not overlap with the transit
ephemeris (the errors on the transit ephemeris are fractions of a minute, and
Kain et al. (2020) found no evidence of transit timing variations). As we are
not aware of a model that would produce a signal like this, we assume this dip
is not indicative of an exosphere and fix the mid-transit time to that
expected from the white-light transit.

Table 2: Light Curve Parameters.

Parameter | Value | Units
---|---|---
$t_{0}^{\text{a}}$ | $0$ | hours
$P_{\text{p}}$ | $3.48456322$ | days
$a$ | $0.0288$ | AU
$i$ | $88.164$ | degrees
$e$ | $0.27$ |
$w$ | $98.0$ | degrees

aafootnotetext: The uncertainties on each transit epoch where we obtained
observations are fractions of a minute, which allows us to fix this value.

The ratio of the planetary radius to the stellar radius was left to vary with
a uniform prior and a lower limit of 0. We fit the model to the red and blue
light curves using the Markov Chain Monte Carlo algorithm emcee (Foreman-
Mackey et al., 2013). Figure 5 shows the blue- and red-wing Ly$\alpha$ light
curves and samples from the posterior distribution. The upper limits on
$R_{\text{p}}/R_{\star}$ for the blue- and red-wing light curves are,
respectively, $1.20$ and $0.56$ at $95$% confidence.

Figure 4: The Ly$\alpha$ average profiles for Visits 1 and 2. The integrated
regions of the blue- and red-wing are shaded in blue and red, respectively.
Velocities are in the host star’s reference frame. Figure 5: Light curves for
Ly$\alpha$’s blue-wing (top) and red-wing (bottom). Each panel has 16 filled
circles, representing all of the full exposures from Visits 1 (green) and 2
(orange). The 48 empty circles are the sub-exposures. Samples from the
posterior distributions are shown as light blue lines. The white-light transit
time is shown by the gray shaded region.

##  4 K2-25’s High Energy Environment

Photoevaporation in the energy-limited regime is controlled by the entire
X-ray to extreme ultraviolet (XUV) spectrum (5-1170 Å, Owen & Jackson, 2012;
Murray-Clay et al., 2009; Yelle, 2004; Lammer et al., 2003). To investigate
K2-25b’s radiation environment, we determine the star’s Ly$\alpha$, EUV, and
X-ray flux. We then use these results to estimate the energy-limited mass-loss
and photoionization rates for K2-25b.

###  4.1 The intrinsic Lyman-$\alpha$ line

We reconstructed the host’s Ly$\alpha$ emission, modeling the intrinsic
Ly$\alpha$ profile and the attenuation from the interstellar medium (ISM). We
based our reconstruction on the combined spectrum from all HST orbits because
there was no planetary signal in the data. To produce the combined spectrum,
we performed a weighted average of all 16 spectra. We weighted the spectra by
the associated error, using an average “error spectrum” for the two visits,
which was applied to all eight spectra from that visit. We used this method to
avoid biasing fluxes to lower values. While the error bars for the spectra
from within each visit are consistent with each other, the changing location
of the geocoronal emission means the errors changes between the two visits
(see Figure 3). We compute the average error spectrum by averaging the errors
at each pixel for the eight spectra from a single visit.

We used the model developed in Youngblood et al. (2016) that describes the
transmission profile of a local low-mass star with nine parameters. The model
and fitting algorithm are available in the Python package
lyapy111https://github.com/allisony/lyapy.

The emission profile was composed of a narrow and broad Gaussian, each
characterized by an amplitude (A${}_{\text{n}}$ and A${}_{\text{b}}$), full-
width half-maximum (FW${}_{\text{n}}$ and FW${}_{\text{b}}$), and heliocentric
velocity centroid (v${}_{\text{n}}$ and v${}_{\text{b}}$). The interstellar
medium attenuation was modeled with a Voigt profile approximation (Harris,
1948) described by a Doppler broadening parameter ($b$), line of sight H I
column density (N${}_{\text{H I}}$), and velocity centroid (v${}_{\text{H
I}}$). The interstellar medium’s D I content was characterized by the D/H
ratio, which was left to vary. The velocity centroid for deuterium absorption
was assumed to be the same as neutral hydrogen.

This model assumes that all of the interstellar neutral hydrogen exists in one
cloud. While the LISM Dynamical Model (Redfield & Linsky, 2008) outputs three
clouds in the direction of K2-25, the low S/N of our spectra are not
sufficient to constrain multiple ISM components (Youngblood et al., 2016).

Table 3: The best-fit parameter values with 1$\sigma$ uncertainties for the intrinsic Ly$\alpha$ profile.  Parameter | Value | Units | Description
---|---|---|---
v${}_{\text{n}}$ | $38.19^{+2.70}_{-2.17}$ | km s-1 | narrow component velocity centroid
log10A${}_{\text{n}}$ | $-13.75^{+0.08}_{-0.07}$ | erg s-1 cm-2 Å-1 | narrow component amplitude
FW${}_{\text{n}}$ | $163.40^{+15.64}_{-16.27}$ | km s-1 | narrow component FWHM
v${}_{\text{b}}$ | $38.56^{+2.87}_{-2.45}$ | km s-1 | broad component velocity centroid
log10A${}_{\text{b}}$ | $-14.72^{+0.21}_{-0.26}$ | erg s-1 cm-2 Å-1 | broad component amplitude
FW${}_{\text{b}}$ | $447.17^{+178.28}_{-78.07}$ | km s-1 | broad component FWHM
log10N${}_{\text{H I}}$ | $18.14^{+0.08}_{-0.08}$ |  | ISM H I column density
b | $8.62^{+3.22}_{-4.01}$ | km s-1 | ISM Doppler broadening parameter
v${}_{\text{H I}}$ | $12.91^{+2.83}_{-2.80}$ | km s-1 | ISM velocity centroid
D/H | $6.95^{+6.40}_{-4.37}\times 10^{-6}$ |  | ISM deuterium-to-hydrogen ratio

We fit the stellar emission and ISM attenuation to the observed Ly$\alpha$
spectrum ($-350<v<390$ km s-1). lyapy uses the MCMC sampler (emcee; Foreman-
Mackey et al., 2013) to explore the parameter space. The chains were
initialized by randomly sampling a normal distribution. The Doppler broadening
parameter had a logarithmic prior. Uniform priors were assumed for all other
varied parameters, including the column density (N${}_{\text{H I}}$). The
posterior distributions were sampled 100,000 times with a burn-in of 20,000
for 30 walkers.

We took the median of each one-dimensional posterior distribution
(marginalized distribution) as the best-fit value and defined the uncertainty
by the 16th and 84th percentiles of the distribution. The best-fit values and
their uncertainties are listed in Table 3. The best-fit column density,
log10N${}_{\text{H I}}=18.14^{+0.08}_{-0.08}$, is consistent with the column
density obtained by the Colorado Model of the Local Interstellar Cloud
(Redfield & Linsky, 2000) for the line of sight to K2-25, log10N${}_{\text{H
I}}=18.17$. There are two other clouds (Hyades, Aur) that are predicted to lie
along K2-25’s sight line (Redfield & Linsky, 2008) which could account for the
differences between our ISM results and the literature. The best-fit
deuterium-to-hydrogen ratio (D/H$=6.95^{+6.40}_{-4.37}\times 10^{-6}$) has
large enough uncertainties such that it is consistent with the accepted
literature value of $1.5\times 10^{-5}$ found by Linsky et al. (2006) (see
also Wood et al. 2004; Hébrard & Moos 2003). The large uncertainties in the
flux within the profile region where deuterium absorption occurs ($\sim-100$
to $-40$ km s-1) result from the overlap with the geocoronal emission region
in Visit 1 (see Figure 3).

K2-25’s reconstructed Ly$\alpha$ profile is shown in Figure 6. K2-25’s
Ly$\alpha$ flux at 1 AU from the star is $1.38^{+0.18}_{-0.13}$ erg s-1 cm-2,
where the uncertainties are $1\sigma$ error bars.

Figure 6: Our reconstructed Ly$\alpha$ profile for K2-25 (dashed blue). The
spectrum created from the sixteen stacked exposures is included in black, with
errors in orange. The best-fit profile - a combination of the Ly$\alpha$
double component Gaussian profile and the ISM’s attenuating Voigt profile - is
depicted as the solid blue line with 1$\sigma$ errors shown in shaded blue.

###  4.2 The EUV spectrum

We estimated K2-25’s EUV spectrum ($100-1170$ Å) using empirical relations
from Linsky et al. (2014). Using updated solar upper atmospheric models from
Fontenla et al. (2014), Linsky et al. (2014) determined that the relationship
between EUV and Ly$\alpha$ flux is roughly constant with Ly$\alpha$ flux. They
obtained EUV to Ly$\alpha$ flux ratios in nine wavebands from $100-1170$ Å
(indicated in Table 4). These relations are accurate within $20$%, as inferred
from the spectra of F5 V - M5 V stars observed with the Extreme UltraViolet
Explorer and Far Ultraviolet Spectroscopic Explorer (Linsky et al., 2014).
Each waveband was summed and scaled to calculate the total EUV flux of K2-25
at 1 AU, $1.40\pm 0.14$ erg s-1 cm-2.

There may be additional, unaccounted for, systematic errors in the EUV
spectrum for K2-25 because stars of its mass and age have not been thoroughly
investigated by the models and observations. For example, Peacock et al.
(2019) showed that the EUV fluxes for Gl 436 from synthetic spectra and from
the relations within Linsky et al. (2014) can differ by a factor of a few.

###  4.3 The X-ray spectrum

K2-25 was observed with XMM-Newton for a broad investigation into stellar
rotation and activity (OBSID 0782061001; PI: Agüeros). The spectrum covers
$5-100$ Å ($0.12-2.48$ keV). We fit the spectrum output from the XMM-
Newton/MOS1 instrument with a one temperature VAPEC model, which characterizes
the emission associated with collisionally-ionized gas (e.g., stellar
coronae). The fit was performed with the HEASARC software
xspec222https://heasarc.gsfc.nasa.gov/xanadu/xspec/, an X-ray spectral fitting
package. The fitted spectrum was integrated to get a total soft X-ray flux
with 1$\sigma$ uncertainty of $5.88^{+1.16}_{-0.56}$ erg s-1 cm-2 at 1 AU from
the star. The ratio of K2-25’s X-ray luminosity to its bolometric luminosity
is $\log_{10}{L_{\text{X}}/L_{\text{bol}}}=-3.28$.

K2-25’s Rossby number, defined as the ratio of the rotation period to its
convective turnover time, is 0.02 according to the mass-convective turnover
time relation from Wright et al. (2018). Its Rossby number places K2-25 in the
saturated part of the rotation-coronal activity relation, where the X-ray
emission appears to decouple from a star’s rotation rate and is roughly
constant. For 20 Hyades members with $M_{*}<$1.2 M⊙ and X-ray detections that
fall in the saturated regime,
$\log_{10}{L_{\text{X}}/L_{\text{bol}}}=-3.17^{+0.11}_{-0.16}$ (Núñez et al.,
in prep.), with uncertainties corresponding to the 16th and 84th percentiles.
K2-25’s X-ray flux is fully consistent with that of other young, rapidly
rotating stars.

Gl 436 and GJ 3470 are field-age stars and have longer rotation periods of
44.09 days and 21.54 days, respectively (Bourrier et al., 2018b; Kosiarek et
al., 2019). Veyette & Muirhead (2018) were able to constrain Gl 436’s age to
$8.9^{+2.3}_{-2.1}$ Gyr. Gl 436 and GJ 3470 have estimated Rossby numbers of
0.78 and 0.47, using the mass-convective turnover time relation from Wright et
al. (2018) and masses from Stassun et al. (2019). These Rossby numbers are
well within the unsaturated regime. K2-25 has an X-ray luminosity that is ten
times larger than GJ 3470 and one hundred times larger than Gl 436, which is
compatible with K2-25’s youth and corresponding rapid rotation.

###  4.4 The XUV spectrum

The X-ray to EUV fluxes and errors are reported in Table 4. The sum of the
X-ray and EUV (XUV) irradiation that K2-25b experiences is $8763\pm 1049$ erg
s-1 cm-2, where the error was calculated using the average X-ray flux
uncertainty and $20\%$ uncertainty in each EUV band. This is roughly $70\%$ of
the XUV flux estimated by Gaidos et al. (2020), who based their estimate on a
scaled composite spectrum of GJ 674 and rotation-activity relations, and is
consistent with their uncertainty (see their Table 1). K2-25b’s XUV
irradiation is roughly seven times that of Gl 436b and twice that of GJ 3470b
(Bourrier et al., 2018a, 2016).

Table 4: K2-25’s X-ray and EUV emission. Waveband (Å) | Flux at 1 AU | Uncertainty
---|---|---
| (erg s-1 cm-2) | (erg s-1 cm-2)
$5-100$ | $5.883$ |  $+1.159$/$-0.563$
$100-200$ | $0.444$ | $\pm 0.089$
$200-300$ | $0.389$ | $\pm 0.079$
$300-400$ | $0.344$ | $\pm 0.069$
$400-500$ | $0.008$ | $\pm 0.002$
$500-600$ | $0.013$ | $\pm 0.003$
$600-700$ | $0.018$ | $\pm 0.004$
$700-800$ | $0.021$ | $\pm 0.004$
$800-912$ | $0.027$ | $\pm 0.005$
$912-1170$ | $0.139$ | $\pm 0.028$
Ly$\alpha$ | $1.375$ |  $+0.182$/$-0.126$

###  4.5 Mass loss rate

The mass measurement for K2-25b from Stefansson et al. (2020) indicates its
atmosphere is currently dominated by hydrogen, and is therefore still
susceptible to hydrodynamic escape. We used our XUV flux to estimate the total
mass loss rate of K2-25b, adopting the energy-limited methodology as reviewed
in Owen (2019). The energy-limited approach was originally derived by Watson
et al. (1981) using outflow calculations for a young Earth and Venus. The
equation is:

$\dot{M}=\eta\frac{\pi
R^{3}_{\text{p}}F_{\text{XUV}}}{GM_{\text{p}}K_{\text{eff}}}$ (2)
$K_{\text{eff}}=\frac{(a/R_{\text{p}}-1)^{2}(2a/R_{\text{p}}+1)}{2(a/R_{\text{p}})^{3}}$
(3)

where $R_{\text{p}}$ is the planetary radius, $F_{\text{XUV}}$ is the stellar
XUV flux at the planet’s location, and $M_{\text{p}}$ is the planetary mass.
This formalism assumes that K2-25b is absorbing all of the XUV flux at its
optical-wavelength radius, where the energy input balances the energy
necessary for escaping the planet’s potential.

Two correction factors remain in Equation 2. Erkaev et al. (2007) added the
$K_{\text{eff}}$ factor (Equation 3) to correct for the potential difference
between the planet’s radius and its Roche lobe, which needs to be overcome by
the energetic atmosphere in order to escape. The $\eta$ factor characterizes
the efficiency at which the XUV radiation heats the planet’s atmosphere. The
exoplanet population could exhibit a wide range of heating efficiencies
(0.15-0.6) (Shaikhislamov et al., 2014). $\eta$ has been shown to depend on
the incident flux level – higher flux levels cause radiative cooling to become
a limiting factor (Murray-Clay et al., 2009) – and atmospheric properties. We
report our mass loss rate in terms of $\eta$, (i.e, without assuming a heating
efficiency), which enables comparison to other systems.

We calculated the total mass loss rate to be $10.6^{+15.2}_{-6.13}\eta\times
10^{10}$ g s-1 assuming the planetary parameters in Table 1 and the estimated
planetary mass derived from mass-radius relations. This is about 50% of the
estimate from Gaidos et al. (2020), who use a different method for calculating
total XUV irradiation, but agrees within our uncertainty.

Stefansson et al. (2020) used radial velocity observations from the Habitable-
zone Planet Finder to measure K2-25b’s mass, a challenging prospect given that
the $1.8$ day stellar rotation period is close to half the planetary orbital
period. These authors determine a mass of $24.5^{+5.7}_{-5.2}$M⊕, higher than
our estimate of $7$M⊕ and the estimate of $9.7$M⊕ from Gaidos et al. (2020).
333They also combined K2 and ground-based photometry observations to obtain a
planetary radius of $3.44\pm 0.12$R⊕, in agreement with the value from Thao et
al. (2020) and quoted in Table 1. Using their planet mass and radius gives a
mass loss rate of $3.00^{+0.77}_{-0.71}\eta\times 10^{10}$ g s-1. This is
about 30% of the estimate using the parameters from Table 1 which is within
our uncertainty and does not substantively alter our conclusions (see Section
5).

###  4.6 Photoionization rate

The XUV irradiation of K2-25b photoionizes its exosphere. The average lifetime
of an escaping neutral hydrogen atom contributes to whether or not an
absorption signature should be observable in Ly$\alpha$. Equation 4 gives the
photoionization rate per neutral hydrogen atom given an incident flux (see
e.g. Bourrier et al., 2016):

$\Gamma_{\text{ion}}=\int^{911.8\text{ \AA}}_{0\text{
\AA}}\frac{F_{\text{XUV}}(\lambda)\sigma_{\text{ion}}(\lambda)}{hc}\lambda
d\lambda$ (4)

where:

$\displaystyle\sigma_{\text{ion}}=$ $\displaystyle 6.538\times
10^{-32}\left(\frac{29.62}{\sqrt{\lambda}}+1\right)^{-2.963}$
$\displaystyle\times(\lambda-28846.9)^{2}\lambda^{2.0185}$ (5)

The photoionization rate, $\Gamma_{\text{ion}}$, is given in s-1.
$F_{\text{XUV}}$ is the stellar XUV flux at the location of the planet in erg
s-1 cm-2 Å-1, and $\sigma_{\text{ion}}$ is the photoionization cross section
in cm2 given in equation 5 (Bzowski et al., 2013; Verner et al., 1996). The
integral is over the H-ionizing portion of the electromagnetic spectrum
($\lambda\leq 911.8$ Å).

We calculated the photoionization rate by summing over the XUV flux values
listed in Table 4 multiplied by $\sigma_{\text{ion}}(\lambda)\lambda$
evaluated at the average wavelength of each wavelength bin. This results in a
photoionization rate of $9.73\times 10^{-5}$ s-1 at K2-25b’s semi-major axis,
which corresponds to a neutral hydrogen atom’s lifetime of $2.86$ hours. GJ
3470b was similarly found to have a short neutral hydrogen lifetime of $\sim
0.9$ hours (Bourrier et al., 2018a). In contrast, Gl 436b has a long neutral
hydrogen lifetime of about $14$ hours which allows the leading and trailing
regions of its exosphere to evade rapid ionization (Bourrier et al., 2016).
Short neutral hydrogen lifetimes (large photoionization rates) result in
smaller neutral exospheres and correspondingly shallower and shorter
Ly$\alpha$ transits (as can be seen by comparing the Gl 436b and GJ 3470b
transit signals; Bourrier et al., 2016, 2018a).

##  5 Discussion

Observations of young planets inform our knowledge of exoplanetary evolution,
and the impact of host star age and activity on planetary properties.

###  5.1 Summary

K2-25b is similar in size and proximity to its host star ($3.45$R⊕, $0.0288$
AU) to Gl 436b ($4.26$R⊕, $0.0287$ AU) and GJ 3470b ($4.57$R⊕, $0.0355$ AU).
We measured K2-25’s Ly$\alpha$ and X-ray flux and inferred the EUV flux in
order to determine the planetary irradiation. Although all three planets
experience EUV irradiation on the same order of magnitude, K2-25b experiences
enhanced X-ray irradiation compared to the two other hot Neptunes. This is a
result of K2-25’s young age, and the slower decline of EUV with age compared
to X-rays (Peacock et al., 2020). The large XUV irradiation results in an
estimated maximum mass loss rate ($10.6^{+15.2}_{-6.13}\eta\times 10^{10}$ g
s-1) that is five times larger than for Gl 436b ($2.20\eta\times 10^{10}$ g/s;
Bourrier et al., 2016). The larger mass loss rate for K2-25b means that if the
same atmospheric escape process translates across all three planets, then we
should be able to observe an escaping neutral hydrogen exosphere surrounding
K2-25b.

Our analysis of two HST/STIS observations of K2-25 in Ly$\alpha$ yielded a
non-detection of an extended neutral hydrogen envelope around K2-25b despite
the large mass loss rate. This suggests that K2-25b is either not experiencing
significant atmospheric escape, or other factors result in mass loss that is
unobservable in our Ly$\alpha$ observations. The latter could be due to
properties of the system itself. While adopting the $24.5$M⊕ mass in place of
the value from mass-radius relations results in a factor of three lower mass-
loss rate, this reduction is not sufficient to imply a non-detection of an
evaporating atmosphere, and we therefore explore the latter possibility.

###  5.2 Astrophysical causes for a non-detection

A possible explanation for the suppression of a neutral hydrogen exosphere is
a denser, non-hydrogen dominated atmosphere. A higher mean molecular weight
would decrease the atmospheric scale height, reducing the likelihood of
hydrodynamic escape. This could be possible if K2-25b lost its primordial
atmosphere when it was younger, which would be consistent with the potentially
short $\sim 100$ Myr timescale of photoevaporation and inconsistent with the
longer Gyr timescale proposed by King & Wheatley (2021). García Muñoz et al.
(2020) suggest a transition around $\rho_{\text{p}}\sim 2-3$ g cm-3: planets
with bulk densities below 2 g cm-3 have neutral hydrogen-dominated atmospheres
that allow the detection of an extended atmosphere in Ly$\alpha$. Planets with
bulk densities above 3 g cm-3 could have atmospheres dominated by heavier
species, resulting in no detection of atmospheric absorption in Ly$\alpha$
regardless of mass loss rate. With the planetary mass estimate based on mass-
radius relations ($7^{+10}_{-4}$M⊕), K2-25b has a bulk density
($\rho_{\text{p}}=0.94^{+1.35}_{-0.54}$ g cm-3) that places it within the
regime of H-dominated atmospheres that should result in Ly$\alpha$ absorption
if the atmosphere is escaping. However, the measurement from Stefansson et al.
(2020) results in a bulk density that agrees with either a H-dominated or
water-dominated atmosphere ($\rho_{\text{p}}=3.28\pm 0.8$ g cm-3). In the
framework of García Muñoz et al. (2020), K2-25b could potentially then fall
into the regime where planets typically do not exhibit Ly$\alpha$ absorption
even with high mass loss rates.

K2-25’s youth could also impact the detectability of the planet’s Ly$\alpha$
transit. The estimated lifetime before ionization of an escaping neutral
hydrogen atom in K2-25b’s exosphere is short ($2.86$ hours) compared to Gl
436b ($\sim$12 hours). The short lifetime could indicate that enough neutral
atoms are ionized before they are able to travel far enough to form a large
neutral hydrogen exosphere. The result may be a smaller exosphere more similar
to that of GJ 3470. Our observational strategy was set prior to the detection
of GJ 3470, and was optimized for the detection of a more extended exosphere.

The exosphere could be interacting with high velocity stellar winds from its
host (e.g., Cohen et al., 2015; Kislyakova et al., 2014; Bourrier & Lecavelier
des Etangs, 2013). Via charge exchange, fast-traveling protons from the
stellar wind may receive an electron from the neutral hydrogen in the
transiting planet’s atmosphere, suffering little deflection or change in
kinetic energy from the collision. Therefore, if the stellar wind is high
speed, the now-neutral stellar wind could result in Ly$\alpha$ absorption
beyond the Ly$\alpha$ emission feature–where, for K2-25, there is little
signal (cf. Bourrier et al., 2016).

Carolan et al. (2021) explored the effects of varying stellar wind strengths
on the Ly$\alpha$ transits of close-in planets with 3D hydrodynamic
simulations (see also Carolan et al. (2020) and Vidotto & Cleary (2020)). They
showed that stellar winds provide an external pressure that can confine the
atmosphere of an exoplanet and decrease the exosphere’s transit depth by a
factor of about two. We make a rough estimate of the stellar mass loss rate
for K2-25 using its X-ray flux ($3.17\times 10^{6}$ erg s-1 cm-2) and the
observed relationship between X-ray flux and the mass loss rates of main
sequence stars (Wood et al., 2005, c.f. Fig 3). This gives $\sim
0.8\dot{\text{M}}_{\odot}$. When making this estimate, we assume that the mass
loss rate per unit surface area reaches an asymptote at $\sim
9\dot{\text{M}}_{\odot}/\text{A}_{\odot}$ (units of solar mass loss rate per
solar surface area) for X-ray fluxes greater than $10^{6}$ erg s-1 cm-2. The
inferred mass loss rate for K2-25 is significantly lower than the wind
strengths from Carolan et al. (2021) and so we do not a priori expect that the
wind would inhibit the detection of an escaping exosphere. Our stellar mass
loss estimate is highly uncertain, however, as there are no observational
constraints on K2-25’s stellar wind strength and current
$\dot{\text{M}_{\star}}\propto F_{X}$ relations do not extend to the high
X-ray fluxes exhibited by young M dwarfs.

The stellar and planetary properties that concern the properties of
evaporating atmospheres are far from clear, and K2-25b is currently the only
young Neptune in the literature with an analysis of its Ly$\alpha$ transit.
Our non-detection of an extended atmosphere is inconsistent with the extended,
cometary-tail seen around Gl 436b, but permits the presence of smaller
exospheres more similar to that of GJ 3470b. We considered the scenario where
we obtained a total of 3 HST visits similar in quality to our Visit 2. We can
only detect an exosphere of $>0.5$R⋆ in the Ly$\alpha$ blue-wing, which does
not enable us to probe atmospheres as small as could be expected. K2-25b needs
further modeling to learn about the expected size and shape of a potential
exosphere, which could then guide future observations.

We sincerely thank Dr. Hans R. Müller and Dr. Brian Chaboyer for helpful
discussions and thoughtful comments throughout the progress of this work. The
authors appreciate the support, care, and community provided by the astronomy
graduate students and post-docs within Dartmouth College’s Department of
Physics & Astronomy. We would like to express our appreciation for Nova,
Charlie, Edmund, and Nessie who have gifted us with smiles during many months
of remote work. This research is based on observations made with the NASA/ESA
Hubble Space Telescope obtained from the Space Telescope Science Institute,
which is operated by the Association of Universities for Research in
Astronomy, Inc., under NASA contract NAS 5–26555. These observations are
associated with HST-GO-14615. Support for program HST-GO-14615 was provided by
NASA through a grant from the STScI. This work is based on observations
obtained with XMM-Newton, an ESA science mission with instruments and
contributions directly funded by ESA Member States and NASA. These
observations are associated with OBSID 0782061001. This material is based upon
work supported by the National Science Foundation under Grant No. 2008066.
This work benefited from the Exoplanet Summer Program in the Other Worlds
Laboratory (OWL) at the University of California, Santa Cruz, a program funded
by the Heising-Simons Foundation. This project has received funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (project Four Aces; grant agreement No
724427; project Spice Dune; grant agreement No 947634). V.B. acknowledges
support by the Swiss National Science Foundation (SNSF) in the frame of the
National Centre for Competence in Research “PlanetS”. HST/STIS, XMM-Newton

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