1
Magnetospherically Driven Optical and Radio Aurorae at the
End of the Main Sequence
G. Hallinan, S. P. Littlefair, G. Cotter, S. Bourke, L. K. Harding, J. S. Pineda, R. P.
Butler, A. Golden, G. Basri, J.G. Doyle, M.
M.
Kao, S. V. Berdyugina, A. Kuznetsov
,
M. P. Rupen & A. Antonova
Affiliations
G
.
Hallinan, S
.
Bourke
,
J.
S
.
Pineda & M
.
M.
Kao
California Institute of Technology, 1200 East California Boulevard, Pasadena,
California 91125, USA
S
.
P.
Littlefair
Department of Physics and Astronomy, University of
Sheffield, Sheffield S3 7RH, UK
.
G
.
Cotter
University of Oxford, Department of Astrophysics, Denys Wilkinson Build
ing, Keble
Road, Oxford OX1 3RH
L. K. Harding
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive,
Pasadena,
CA 91109
-
0899, USA
R
. P.
Butler
Centre for Astronomy, National University of Ireland
-
Galway, University Road,
Galway, Ir
eland
2
A
.
Golden
Department of Mathematical Sciences, Yeshiva University, New York, NY 10033, USA
G
. Basri
Astronomy Department, University of California,
Campbell Hall
, Berkeley, CA 94720,
USA
J. G. Doyle
Armagh Observatory, College Hill, Armagh BT61 9DG, N. Ireland
S
.
V.
Berdyugina
Kiepenheuer Institut f
ü
r Sonnenphysik, Sch
ö
neckstrasse 6, D
-
79104
Freiburg, Germany
A
.
Kuznetsov
Institute of Solar
-
Terrestrial Physics, Irkutsk664033, Russia
M. P. Rupen
National Radio Astronomy Observatory, P.O. Box O, Socorro, NM 87801
A. Antonova
Department of Astronomy, Faculty of Physics, St Kliment Ohr
idski
,
University of Sofia,
5 James Bourchier Boulevard, 1164 Sofia, Bulgaria
3
Aurorae are detected from all the magnetized planets in our Solar System,
including Earth
1
. They are powered by magnetospheric current systems that lead
to the precipitation of energetic
electrons into the high
-
latitude regions of the
upper atmosphere. In the case of the gas giant planets, these aurorae include
highly polarized radio emission at kHz and MHz frequencies produced by the
precipitating electrons
2
, and a myriad of continuum an
d line emission in the
infrared, optical, ultraviolet and X
-
rays associated with the collisional excitation
and heating of the hydrogen
-
dominated atmosphere
3
. Here we present
simultaneous radio and optical spectroscopic observations of an object
at the end
of the main sequence,
located ri
ght at the boundary between stars and brown
dwarfs, from which we have detected radio and optical auroral emissions both
powered by magnetospheric currents.
Whereas the magnetic activity of stars
like
our Sun
is powered by processes
that occur
in their low
er atmospheres,
these
aurorae are powered by processes
originating
much further out in the
magnetosphere of the dwarf that co
uple energy into the
lower
atmosphere.
The
dissipated power is at least 10
4
times larger than produced in the Jovian
magnetosphere
, revealing aurorae to be a potentially ubiquitous signature of large
-
scale magnetospheres that can scale to luminosities far greater than observed in
our Solar System. T
hese magnetospheric current systems may
also
play a causal
role in some of the reporte
d weather phenomena on brown dwarfs.
LSR J1835+3259 (hereafter LSR J1835) is a dwarf of spectral type M8.5 with a
bolometric luminosi
ty
10
-
3.4
times
that of the Sun
, located at a distance of 5.67 +/
-
0.02
pc
4
. It is positioned close to a transition in
magnetic activity near the end of the main
sequence, where the fractional X
-
ray luminosity (
L
x
/L
bol
), indicative of the presence of a
4
magnetically heated corona, drops by two orders of magnitude over a small range in
spectral type
5
. Simultaneously, rapid r
otation becomes ubiquitous, indicating a dearth of
stellar wind assisted magnetic braking
6
. Together, these results suggest that the coolest
stars and brown dwarfs possess a comparatively cool and neutral outer atmosphere
relative to earlier type dwarf sta
rs. Consistent with this picture, LSR J1835 is a rapid
rotator with a period of rotation of just 2.84 hours, and previous deep
Chandra
observations have failed to detect any X
-
ray emission associated with the presence of a
magnetically heated corona
7
.
LS
R J1835 has previously been identified to produce highly circularly polarized radio
emission, periodically pulsed on the rotation period of 2.84 hours
8
. Since their initial
detection as a new population of radio source
s
9
, similar behaviour has been observ
ed for
a number of very low mass stars and brown dwarfs spanning the spectral range M8
-
T6.5
10,11
. In some cases, periodic variability has also been detected in broadband optical
photometric bands and the Hα line
12,13,14
. Together these characteristics ar
e unlike
anything observed for earlier type stars
15
.
We pursued spectroscopic data in radio and optical bands to investigate a possible
relationship between the periodic radio, broadband optical and Balmer line emission
.
We used the
Karl G. Jansky Very La
rge Array
(VLA) radio telescope to produce a
dynamic spectrum of the periodic radio emission from LSR J1835. Simultaneously we
conducted time resolved optical spectrophotometry using Double Spectrograph (DBSP)
on the 5.1m Hale telescope at the Palomar Obse
rvatory. Follow
-
up observations,
involving an additional 7 hours of more sensitive time resolved optical
5
spectrophotometry, were carried out using the
Low Resolution Imaging Spectrometer
(LRIS) on the 10 m Keck telescope
.
The broadband (
δν
/
ν
~ 1) dynamic r
adio spectrum produced with the VLA reveals a
number of distinct components periodically repeating on the 2.84 hour rotation period
(Figure 1). The observed periodic features can exhibit a cut
-
off in frequency, are 100%
circularly polarized and are very sh
ort duration relative to the rotation period,
the latter
implying sharp beaming. These properties are consistent with electron cyclotron maser
emission produced near the electron cyclotron frequency
at the source of the radio
emission (
ν
MHz
≈ 2.8 x
B
Gauss
), a coherent emission proces
s responsible for
planetary
auroral
radio emission
2
. From the dynamic spectrum
of
the radio emission from
LSR
J1835, we can infer magnetic field strengths in the source region of the emission
ranging from 1550 to
at least
2850
Gauss
, close to the maximum photospheric magnetic
field strengths found in late M dwarfs
16
.
The proximity of the radio source close to the
photosphere, t
ogether with the persistent nature of the periodic radio emissio
n, requires
a
current system driving a
continuously propagating electron beam
in the lower
atmosphere of the dwarf
.
The simultaneous optical spectroscopic data collected with the Hale telescope are also
modulated on the 2.84 hour rotational period (Fig. 1). This behaviour is confirmed with
the
follow
-
up higher signal
-
to
-
noise Keck data, where the same periodic modulation is
observed at the same amplitude. This modulation is clearly present in both spectral line
emission, including the Balmer lines, and the broadband continuum optical emission of
the dwarf. Most notably, the Balmer line emission and nearby continuum vary in phase
(Fig. 2), indicating a co
-
located region of origin.
6
We find that the surface feature responsible for the periodic variability in the optical
spectrum can be modelled as
a single component approximated as a blackbody of
temperature T ~ 2200
K
, with surface coverage of < 1% (Fig. 2). We attribute this
blackbody
-
like spectrum to an optically thick region with the dominant opacity
contributed by the negative hydrogen ion (H
-
)
. H
-
is the dominant source of solar
optical continuum opacity, but is superseded by b
ound
-
bound molecular line opacities
,
such as TiO, for cool M dwarfs due to the scarcity of free electrons available to form the
H
-
ion
1
7
. In the solar case, ionized meta
ls fulfil the role of the electron donors necessary
to sustain a H
-
population. Since there are essentially no electron donors in a thermal
gas at T
=
22
0
0 K, another population of electron donors is required.
The periodically variable radio, Balmer line
and optical continuum emission detected
from LSR J1835 can be explained by a single phenomenon, specifically a propagating
electron beam impacting the atmosphere, powered by auroral currents. Integrating over
time and frequency, we can determine that the h
ighly circularly polarized radio
emission
from LSR J1835 contributes at least
10
15
W
of power, requiring
10
17
–
10
19
W
of power available in the electron beam for dissipation in the atmosphere, assuming an
efficiency of 10
-
2
–
10
-
4
for the radio emission
2
. We note that this amounts to
~
10
-
6
–
10
-
4
of the bolometric luminosity of the dwarf. Collisional excitation of the neutral
hydrogen atmosphere by the precipitating electrons leads to subsequent radiative de
-
excitation, resulting in Balmer line emission w
ith an average power of 2.5 x
10
17
W
,
consistent with the energy budget of the precipitating electron beam. This is similar to
the main Jovian auroral oval, where, radio emission from electron beams contributes
only 10
-
4
of the auroral power, with the bulk
of the power produced in the infrared
(H3+; thermal), far ultraviolet (Lyman and Werner band H
2
emission) and optical
7
(Balmer line) due to dissipation of the electron beam energy in the atmosphere
3
.
Similar ratios are also observed in Io’s magnetic footp
rint in the Jovian atmosphere
18
.
We propose that this same electron beam is also causally responsible for the co
-
located
optical broadband variability. In this model the associated increased electron number
density contributes excess free electrons leading
to an increase in the H
-
population, in a
process that has previously been invoked for white light solar flares
1
9
. This results in an
optically thick layer at a higher altitude, and thus lower temperature, than the
photosphere. Despite the lower temperatu
re, the absence of deep absorption bands in the
spectrum results in a bright feature in optical bands, responsible for the broadband
optical variability. However, in regions of the dwarf spectrum devoid of absorption
bands, particularly towards the redder
end of the spectrum, there is a reversal, with the
auroral feature appearing dimmer than the surrounding photosphere. This results in
some optical wavelengths displaying lightcurves which are in anti
-
phase to other
wavelengths (Fig 2). This phenomenon also
accounts for the anti
-
phased lighturves seen
in multi
-
band photometry of an M9 dwarf
1
3
.
Our observations point to a unified model involving global auroral current systems to
explain the periodic radio, broadband optical and Balmer line emission detected
from
LSR J1835, as well as other low mass stars and brown dwarfs.
Extending to cooler
objects, it is notable that radio emission has now been detected from
a brown dwarf
of
spectral class T6.5 (~900 K)
11
. The brown dwarf in question
is also notable for hos
ting
weak Hα emission, one of only 3 such T dwarfs confirmed to emit Balmer line
emission
20
.
The similarities with
LSR J1835
suggest that the
auroral mechanism
robustly operates well into the regime
occupied by the coolest brown dwarfs of spectral
8
types l
ate L and T. It is also notable that a
large degree of variability
has been
observed
in the infrared
in this spectral regime, partic
ularly
near the transition between
the L and
T
spectral classes
21
,2
2
. This variability
explicitly requires variation in temp
erature or
photospheric opacity across the surface of these brown dwarfs
2
3
, which
has been
attributed to the spatially inhomogeneous distribution of condensate clouds in their
atmospheres, effectively a manifestation of weather.
This is supported by the mapping
of such cloud patterns on the surface of the recently discovered Luhman 16B
2
4
. We
speculate that the magnetospheric currents powering aurorae in brown dwarfs may also
play a causal
role in driving some of the more extreme examples of the weather
phenomenon in brown dwarfs; s
pecifically,
via
downward propagating electron beams
modif
ying
atmospheric
temperature and opacity in the same fashion as has been shown
for LSR J1835.
The natur
e of the electrodynamic engine powering brown dwarf aurorae remains an
outstanding open question. For solar system planets, this electrodynamic engine can be
a) magnetic reconnection between the planetary magnetic field and the magnetic field
carried by t
he solar wind (e.g. Earth and Saturn)
2
5
b) the departure from co
-
rotation with
a plasma sheet residing in the planetary magnetosphere (e.g. Jovian main auroral
oval)
26,27
or c) interaction between the planetary magnetic field and orbiting moons (e.g.
Jupit
er
-
Io current system)
28
. Of these models, the sub
-
corotation of magnetospheric
plasma on closed field lines, in turn powering
magnetosphere
–
ionosphere
coupling
currents, has been put forward as a plausible model that can be extrapolated from the
Jovian ca
se to the brown dwarf regime
2
9
,
30
. This latter model requires a continuously
replenished body of plasma within the magnetosphere. This mass
-
loading can be
achieved by multiple avenues, including interaction with the ISM, the sputtering of the
9
dwarf atmosp
here by auroral currents, a volcanically active orbiting planet or magnetic
reconnection at the photosphere. Alternatively, considering the case of an orbiting
planetary body embedded within the magnetosphere of LSR J1835, a simple scaling
from the Jupiter
-
Io system indicates that an Earth
-
sized planet (magnetized or
unmagnetized) orbiting within 20 radii (<30 hour orbital period) of LSR J1835 will
generate a current sufficient to power the observed aurorae. However, we note that the
observed rotational
mod
ulation of the radio emission would require a substantially
asymmetric magnetic field configuration for LSR J1835.
Indeed, the aurorae would
display modulation on both the rotational and orbital period, which may be consistent
with the large degree of vari
ability reported for the radio pulsed emission from these
objects.
A possible avenue to resolving the nature of the electrodynamic engine lies with the
strong rotational modulation of the Balmer line emission of LSR J1835, which implies
the auroral featu
re is not axisymmetric relative to the rotation axis of the dwarf. This
should result in a variation in the width, intensity and velocity structure of line profiles
with rotation that can be used to help map the aurora
(e.g. auroral oval vs polar cap)
,
an
alogous to Doppler imaging, which in turn will inform on the location of the
electrodynamic engine.
Our results imply that the available power for generating aurorae on brown dwarfs is
dependent on magnetic dipole moment and rotation, and may be weakly co
upled to
other physical characteristics, such as bolometric luminosity. This accounts for the
continuous presence of auroral radio emission at similar luminosity from spectral type
M8 through to T6.5, despite a 2 orders of magnitude decrease in bolometric
luminosity
10
over the same spectral range
8
. This suggests aurorae may be present at detectable levels
on even the faintest T and Y dwarfs and
bodes well for searches for similar emission
from
ex
oplanets
.
Online Content
Methods, along with any additional Ext
ended Data display items and
Source Data, are available in the online version of the paper; references unique to these
sections appear only in the online paper.
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Supplementary Information
is linked to the
online version of the paper at
www.nature.com/nature.
Acknowledgements
We are grateful to T. Readhead, S. Kulkarni and J. McMullin for
working to ensure that simultaneous Palomar and VLA observations could occur. We
thank the staff of the Palomar Observat
ory, the W.M. Keck Observatory and the
National Radio Astronomy Observatory for their support of this project. The W.M.
Keck Observatory is operated as a scientific partnership among the California Institute
14
of Technology, the University of California and
the National Aeronautics and Space
Administration. The Observatory was made possible by the generous financial support
of the W.M. Keck Foundation. The VLA is operated by the National Radio Astronomy
Observatory, a facility of the National Science Foundati
on operated under cooperative
agreement by Associated Universities, Inc. Armagh Observatory is grant
-
aided by the
N. Ireland Dept. of Culture, Arts & Leisure. G.H. acknowledges the generous support
of D. Castleman and H. Rosen
.
This material is based upon
work supported by the
National Science Foundation under grant no.
AST
-
1212226
/DGE
-
1144469
. We thank
Jeff Linsky and Peter Goldreich for helpful comments on this manuscript.
Contributions
G.H., S.B., M.R., A.A., A.G., A.K., M.
M.
K. and
J.
G.D. proposed, plann
ed and
conducted the radio observations.
G.H. and S.B. reduced the VLA data and the dynamic
spectrum was outputted by S.B.
G.H. interpreted the dynamic radio spectra. G.H.,
S.
P.
L., G.C., R.
P.
B., S.P. and L.
K.
H. proposed and conducted the Keck observations.
G.C. carried out the Palomar observations and reduced the publication data. S.
P.
L. and
G.C. reduced the Keck spectroscopic data, with the final publication
data delivered by
S.
P.
L. G.H. G.C.
and S.B.
G.H.
,
S.
P.
L.
and J.S.P
developed the interpretation of
the
optical data. S.
P.
L. carried out the detailed model fitting of the Keck spectra. G.B.
analyzed high resolution archival spectra and provided insight on interpretation of the
optical data. S.V.B. coordinated contemporaneous spectropolarimetry with the
observations presented in this paper. S.
P.
L and G.H. wrote the Supplementary
Information. All authors discussed the result and commented on the manuscript.
15
Author Information
Reprints and permissions information is available at www.nature.com/reprints.
The
authors declare no competing financial interests. Correspondence and requests for
materials should be addressed to G.H. (gh@astro.caltech.edu).
Figure 1 |
Simultaneous optical and radio
periodic variability
of LSR J1835
.
a.
Balmer line emission
extr
acted from spectra detected with
the Hale telescope.
b
.
Dynamic spectra of the
right circularly polarized
radio emission
detected
from LSR
J1835
with the VLA,
with the y axis truncated to remove the large gap between
observing bands
(see Methods for detail
s)
.
The
offset in phase of the radio features
relative to the optical peak
can be accounted for by the complex beaming of the radio
emission
.
16
Figure 2 | Modelling the o
ptical
variability
of
LSR J1835.
a.
Adopted models for the
surface brightness of the pristine photosphere (green) and auroral feature (blue)
.
At
certain wavelengths the auroral feature is brighter, whereas the pristine photosphere is
brighter at other wavelengths.
b.
L
ightcurves constructed
f
rom the Keck
spectrophotometry for three regions of the spectrum highlighted in panel
a
.
The
lightcurve produced for
Hα emission is tightly correlated
with the lightcurve of the
nearby continuum
, confirming the Balmer line emission and excess continuum e
mission
to be approximately co
-
located
. Meanwhile,
redder wavelengths are
in anti
-
correlation
as expected for our model.
c.
The amplitude of variability as a function of wavelength
(black) with 2σ uncertainties (shaded gray region)
,
derived from the ‘low’
and ‘high’
states for the auroral emission defined as the red and blue shaded regions respectively in
panel
b
. The variability predicted by the model presented in panel
a
, is shown in red
(see Methods for details).
We note that the line emission is not
represented in the model.
17
Methods
Radio Data Reduction
D
ata were reduced using the Common Astronomy Software Applications (CASA
Release 4.1.0
)
and
Astronomical Image Processing System (AIPS) packag
es. The
amplitude and phase of the
data were calibrated usi
ng short observations of the
quasars
QSO J1850+284 and
3C286 that were interspersed throughout the 6 hour
observation of
LSR J1835. Bad
data, particularly those contaminated by radio freq
uency interference
(RFI), were
flagged. The tasks
fixvis/UVFIX
were u
sed to shift the source to
the phase
center and
the tasks
clean/IMAGR
were used to image the data. The tasks
uvsub/UVSUB
were used to subtract the source models for nearby background sources from the
visibility
data. The real part of the complex visibiliti
es as a function of time and
frequency
for
each polarization were then exported from CASA and AIPS and plotted to
produce the dynamic spectrum of LSR J1835 shown in the main paper.
We observed LSR J1835 using 2 x 1 GHz sub
-
bands spanning frequencies of 4
.3
-
5.3
GHz and 6.9
-
7.9 GHz. Two full rotation perio
ds were captured during the 5.7
-
hour
observation. We
show
the dynamic spectrum detected by
the
right circularly polarised
feed
s of the VLA antennas
. The original data has a time and frequency resolution of
1
second and 2 MHz respectively, but is binned and smoothe
d to produce the data shown
in F
igure 1
,
with the y axis truncated to remove the large gap between observing bands
.
Periodic features of 100% circularly p
olarized radio emission occupy
~3% of the
d
ynamic spectrum, allowing us to infer that it is beamed from the source in an angular
emission pattern that occupies a similar fraction of a 4π steradian sphere. Studies of
electron cyclotron maser emission from planetary magnetospheres have revealed the
18
e
mission to be beamed in a hollow cone with walls a few degrees thick and a large
opening angle (typically ~ 70° to 90°) relative to the local magnetic field
2
. This is
consistent with our data and accounts for the offset in phase of the radio features
relat
ive to the optical peak. Integrating over time and frequency and inferred beaming
pattern, we determine the auroral power contributed by the polarized radio emission to
be ~10
15
W
.
We detect at least 6 distinct components in the dynamic spectrum for LSR J1835, each
of which is likely powered by a distinct local field
-
aligned current. The relationship
between these individual current systems within the large scale magnetosphere will
l
ikely remain uncertain until the nature of the electrodynamic engine is established.
Although a number of these components exhibit a cut
-
off in emission rising to higher
frequencies, there are still components present all the way to the top of the band,
im
plying that the true cut
-
off in emission, associated with the largest strength magnetic
fields near the surface
of the dwarf
,
was not captured
.
Hale Optical Data Reduction
The data from the Double Spectrograph (DBSP) on the Hale telescope were reduced
us
ing standard techniques with the aid of the IRAF software suite. First, bias level was
subtracted from the raw frames. Then pixel
-
to
-
pixel gain variations in the CCD were
corrected by normalising against exposures taken with the DBSP internal broadband
lam
p and the illumination function of the long slit was corrected by normalising against
twilight sky exposures. Next the {x,y} pixels of the CCD were transformed to a
rectilinear {wavelength,
sky position} solution using the DBSP internal arc lamps.
19
Finally
the night sky emission was subtracted by fitting a fourth
-
order polynomial to
each column along the sky direction, with the stars in the frame masked out, and cosmic
rays rejected via sigma
-
clipping. A tramline extraction was then used to make 1
-
D
spectra
of the target and reference stars.
K
eck Optical Data Reduction
Data
were
reduced
using
the
LOW
-
REDUX
pipeline
(
http://www.ucolick.org/~xavier/LowRedux
). Bias frames were constructed by median
stackin
g 5 individual bias frames. Non
-
uniform pixel response was removed by dividing
by a dome flat field produced by stacking 7 individual flat fields together. Individual
objects were located on the slit, and an optimal extraction routine
31
wa
s used to extract
the object spectra on each frame. Wavelength calibration was performed using fits to arc
line spectra taken at the start of the night, which gave a dispersion
of 2.4
AA/pix (red)
and 3.2 AA/pix (blue), and RMS values of 0.7 pix (red) and 0.4 pix (blue).
Each object
spectrum was corrected for flexure using fits to night sky lines. Flexure corrections
ranged from
-
2 to +2 pixels. Flux calibration was performed via a high
-
order
polynomial fit to the flux standard Feige
-
110, with the Balmer lines masked out.
The data were corrected
for light falling outside of the spectrograph slit
using the
additional comparison stars observed. Since the wavelength coverage of each object
differs slightly due to the location of slits in the object mask, slit loss correction
s were
determined as follows. Each comparison star was divided by an average of all the
frames, to remove the spectral shape. The resulting spectrum was fit with a 1
st
order
polynomial to give a series of wavelength
-
dependent slit loss corrections for each
frame. This polynomial was then re
-
binned onto the same wavelength scale as the target
20
spectrum. Not all comparison star spectra were used to correct for slit losses in the
target spectrum. Instead, a master slit loss correction was produced via a straigh
t mean
of slit loss corrections for selected comparison stars, with the quality of slit loss
correction being judged by
-
eye. Comparison stars were removed from the slit loss
correction calculations because they were either extremely blue, or not well align
ed on
their slits in the slit mask. LSR 1835 was slit loss corrected using the two reddest
comparison stars.
Wavelength dependence of optical variability
The amplitude of variability as a function of wavelength (which we term the difference
spectrum) was
estimated as follows.
A rotational phase was assigned to each spectrum using the ephemeris of LSR J1835
.
We created a spectrum representing the “high state” and the “low state” of LSR J1835
by averaging all spectra with phases between
0.95
-
1.00 & 0.00
-
0.3
5
and 0.45
-
0.85
respectively. The
difference spectrum is then simply
the difference between the high
state and the low state spectra.
Statistical uncertainties on the difference spectrum are negligible compared to
systematic errors which arise from imper
fect correction of slit losses, sky subtraction
and removal of telluric absorption.
To
estimate these systematic errors
, we produced a
difference
spectrum using an independent method. In this method
, lightcurves were
produced from a series of 5Å bins, and
a sinusoid of fixed phase and period was fitted to
the lightcurves. A
difference
spectrum was produced using the amplitude of the sinusoid
fit at each wavelength. The two methods yield very similar spectra, except in the range
21
between 7600 and 7650 AA, wh
ere the spectra are affected by telluric absorption.
Subtracting the two difference spectra gave an estimate of the uncertainty, which is
shown in Fig. 2.
Modelling the variability
We construct a two
-
phase model of the optical emission from LSR J1835, wit
h
emission from a ‘pristine’ photosphere P, and a surface feature S. We assume that the
relative contribution from these two phases varies as the dwarf rotates. If the surface
feature covers a fraction
f
h
during the high state and a fraction
f
l
during the
low state,
then the difference between high and low states can be written as
∆
=
푓
ℎ
푆
+
(
1
−
푓
ℎ
)
푃
−
푓
푙
푆
−
(
1
−
푓
푙
)
푃
,
which can be simplified to
∆
=
(
푓
ℎ
−
푓
푙
)
(
푆
−
푃
)
=
휖
(
푆
−
푃
)
.
We model the photosphere, P, of LSR J1835 using an M8 template from a SDSS library
of
composite M
-
dwarf spectra
32
. To ensure the absolute surface flux of the template is
correct, we scale the template so that the bolometric flux matches that of a DUSTY
model atmosphere
33
with
surface gravity of
log g=5.
0 and an
effective temperature
2600K.
We model the surface feature, S, as a black body with temperature
T
b
. Since our
models for S and P give the flux crossing unit surface of the star, to match them to our
data they need to be multiplied by a factor
N =
(R/d)
2
, where R is the radius of LSR
J1
835 and d is the distance
to LSR J1835
. Since this is a simple multiplication of the
model, this factor can be combined with the parameter
휖
. The two free parameters of
our model are therefore Tb, and the scaling constant,
휖
.
22
We draw samples from the po
sterior distributions of our parameters
by a Markov
-
Chain
Monte Carlo (MCMC) procedure.
Because our difference spectrum has unknown
uncertainty
, a nuisance parameter σ is added.
The uncertainty on the difference
spectrum is set to
σ
everywhere, except at w
avelengths corresponding to emission lines
where the uncertainty is set to an arbitrary large value.
P
osterior probability distributions of Tb,
휖
and σ
are estimated
using an
affine
-
invariant
ensemble sampler
3
4
. Uninformative priors were used for all parameters, with the
exception that
휖
was forced to be positive.
The MCMC chains consist of a total of
48,000 steps of which 24,000 were discarded as burn
-
in, giving 560, 770 and 860
independent samples of
Tb,
휖
a
nd σ
, respectively. The posterior probability distributions
of our parameters are shown in Extended Data Fig. 1. Chisq of the most probable model
was 860, with 920 degrees of freedom, showing that the model is an excellent fit to the
data. The only wavelen
gth regions where the model fails to reproduce our data is in the
emission lines. The emission lines are likely caused by collisional excitation of the
neutral atmosphere by the precipitating electrons leads to subsequent radiative de
-
excitation; a process
not captured by our simple two
-
phase model.
The best fitting parameters are Tb = 2180±10K,
휖
=
(
1
.
64
±
0
.
02
)
×
10
−
21
and
휎
=
0
.
018
±
0
.
0004
mJy
.
Using a model
-
based estimate for the radius and the measured
distance for
LSRJ1835 to correct
휖
for the (R/d)
2
fact
or
,
we find that the difference in
covering fractions between the high state and low state is between 0.5 and 1%.
These
error bars do not take into account systematic uncertainties. For example, the
photospheric temperature of LSR J1835 is not determined t
o 10K; adopting a different
23
template, of M9 spectral type, for the pristine photosphere can alter Tb by
approximately 50K. Similarly, systematic errors in the scaling factor
휖
are probably
around five times higher than the statistical errors quoted above.
Modelling the high state
By fitting the difference spectrum we are able to constrain the auroral surface feature’s
spectrum with a minimum of assumptions. Nevertheless, to give confidence in our
modelling, one might wish to compare our observed high stat
e spectrum with the
predictions of our model. This requires a few additional assumptions. The high state can
be written as
퐻
=
푁
[
푓
ℎ
푆
+
(
1
−
푓
ℎ
)
푃
]
.
Assuming the same pristine spectrum P and surface feature spectrum S that gave the
best fit to our difference sp
ectrum, we use an identical MCMC procedure to that
outlined above (including a
similar
nuisance parameter
휎
for the uncertainties) to draw
posterior samples of N and
푓
ℎ
. We find
푓
ℎ
=
0
.
024
±
0
.
004
,
푁
=
(
8
.
47
±
0
.
02
)
×
10
20
and
휎
=
0
.
196
±
0
.
003
mJy.
The resulting fit to the high state spectrum and
residuals are shown in Extended Data Fig. 2.
The constraints on
푁
above, and the
constraint on
휖
from fitting the difference spectrum allow us to estimate
푓
ℎ
−
푓
푙
=
0
.
0194
±
0
.
0002
and hence
푓
푙
=
0
.
005
±
0
.
00
4
.
A couple of pertinent features are visible in Extended Data Fig. 2. The first is that the
quality of our model is limited at blue wavelengths by the signal
-
to
-
noise in the M8
template spectrum we have adopted for the pristine photosphere. Although our n
uisance
param
e
ters can
account for
this to some degree, this is another reason why the statistical
errors quoted on parameters are likely under
-
estimates. The second is that there are
24
features in the residuals which are of similar amplitude to features in
the difference
spectrum. These features arise because the M8 template is not a perfect fit to the pristine
photosphere. However, this does not mean that our fit to the difference spectrum is
unreliable. If we label our adopted photosphere template P, and t
he true photosphere P’,
then the error in the high state spectrum will be
퐻
−
퐻
′
=
푁
(
1
−
푓
ℎ
)
(
푃
−
푃
′
)
,
whereas the error in the difference spectrum will be
∆
−
∆
′
=
푁
(
푓
ℎ
−
푓
푙
)
(
푃
−
푃
′
)
.
Thus,
residuals in the high state spectrum will also be present in the difference
s
pectrum, but reduced in size by a factor of more than 50; this implies the residuals will
be smaller than the value we adopt for our nuisance parameter when fitting the
difference spectrum.
An orbiting exoplanet as an electrodynamic engine
An orbiting pla
netary body embedded within the magnetosphere of LSR J1835 will
have motion relative to this magnetosphere and any associated frozen
-
in plasma. If the
planet is conducting, this motion leads to the generation of an electric field across the
planet that can
power auroral emissions on LSR J1835
28
. The expected power produced
is proportional to the intercepted flux of magnetic energy,
P
∝
v
B
⊥
2
R
obs
2
, where
B
⊥
is
the component of the magnetic field perpendicular to the planet’s orbital motion,
v
is the
planets velocity relative to the local magnetic field and
R
obs
is the size of the obstacle
created by the planet, the latter defined by its ionosphere or magnetosphere depending
on whether the planet is magnetized or unmagnetized
17
. For example, a
simple scaling
from the Jupiter
-
Io system indicates that an unmagnetized Earth
-
sized planet orbiting
25
within 20 radii (<30 hour orbital period) of LSR J1835 will generate a current sufficient
to power the observed aurorae.
However, the resulting auroral em
ission is expected to be strongly modulated by the
orbital period of the planet, whereas in the case of LSR J1835, the observed periodicity
of 2.84 hours is consistent with rotation of the dwarf, as inferred from rotational
broadening of spectral lines. In
deed, a planet orbiting with this period would be within
the Roche limit of LSR J1835 and torn apart by tidal forces. An alternative possibility
arises if the magnetic field at the location of the planet varies substantially as LSR
J1835 rotates, which wil
l be the case if LSR J1835 possesses a non
-
axisymmetric
magnetic field. For example, a tilted dipole would result in as much as 4
-
fold variation
in the current produced across the orbiting planet during each rotation of LSR J1835.
The period of the auroral
emission would then be P
aur
= P
ro
t
(1 + P
rot
/P
orb
) and w
ould
approach the rotation period for large values of P
orb
/P
rot
. In this scenario, the aurorae
would display modulation close to the rotation period as well as on the orbital period.
Code
Availability
T
he code used to model the auroral feature of LSR J1835 is publicly available at
https://github.com/StuartLittlefair/lsr1835
31.
Horne, K.
Optimal Spectrum Extraction and Other CCD
Reduction Techniques
.
Publ. of the Astron. Soc. Pacific
,
98
, 609
-
617 (1986)
32.
Bochanski, J. J., West, A. A., Hawley, S. L. & Covey, K. R.
Low
-
Mass Dwarf
Template Spectra from the Sloan Digital Sky Survey
,
Astron. J.
133,
531
-
544
(2007)