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Constraints on the

Spin Evolution of Young

Planetary

Mass Companions

Marta

L. Bryan

Bjö

rn Benneke

Heather

A. Knutson

, Konstantin Batygin

Brendan P.

Bowler

Cahill Ce

nter for

Astronomy and Astrophysics, California Institute of Technology, 1200 East

California Boulevard, MC 249

17, Pasadena, CA 91125, USA

Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena,

CA 91125, USA

McDonald

Observatory and Department of Astronomy, University of Texas at Austin, Austin,

TX 78712, USA

Surveys of young star

forming regions have

discovered

a growing

population

of planetary

mass (<

13 M

Jup

)

companions

around

young

stars

There is an ongoing deb

ate as to

whether these comp

anions formed like planets (that is

, from the circumstellar disk)

, or if

they

represent the low

mass tail of the star formation process

In this study

utilize

high

resolution spectroscopy to measure

rotation rate

of three

young (2

300 Myr)

planetary

mass companions

and combine these measurements with published rotation

rate

s for two additional companions

to provide a look at the spin distribution of these

objects

compare this distribution to complementary

rotation r

ate

measurements for

six

brown dwarfs

with masses

20 M

Jup

, and

show

that these distributions are

indistinguishable

. This

suggests that

either that these two populations formed via

the same

mechanism, or that

processes

regulating

rotation rates

are

independent of formation

mechanism.

find

that rotation rates

for both populations are well below their break

velocities and

do not evolve significantly

during

the first few

hundred

million years

after

the end of accretion.

This suggests

that

rotation rates

are set during

late stages of accretion,

possibly by interactions with

a circumplanetary disk.

This result has important

implications for our understanding of

the

processes

regulating

the

angular momentum

evolution of young planetary

mass o

bjects,

and

the physics of gas accretion and

disk

coupling in the planetary

mass regime

Previous studies have sought to constrain t

he origin of planetary

mass companions around young

stars

by characterizing

the

mass and semi

major axis distribution

s, but

this approach is limited

the relatively s

mall size of the current sample

Here

we propose a

different

approach,

which

measure

rotation r

ates for

planetary

mass companions

to probe their accretion histories

and subsequent angular momentum evolution.

n the absence of any braking mechanism,

actively accreting gas giant planet embedded in a circumstellar disk should spin up to

rotation

rates approaching break

up (that is,

the

maximum physically allowed) velocity. However,

observations of the solar system gas giants indicate that they are rotating 3

4 times more slowly

than their primordial break

up velocitie

This may be due to

magnetic coupling with a

circumplanetary gas ac

cretion disk

, which could

provide a channel for young planets to

shed

their angular momentum

. After the dispersal of the circumstellar and circumplanetary gas disks,

late giant collisions or gravitational tides can further alter the

rotation rates of som

e planets

currently

have a much better understanding of the angular momentum evolution of stars,

whose rotation rates have

been

well characterized

by large surveys of star

forming regions

imilar to

the

general picture for

gas giant planets

, stars spin

as they accrete

material from a

circumstellar gas

disk

. Unlike

planets

stars have several

known

mechanisms for regulating

this

angular momentum, including

interactions between the star and its

gas

disk

and angular

momentum loss

stellar winds

Extending into the substellar mass regime,

surveys of mid

to high

mass brown dwarfs (

–

80 M

Jup

)

have shown

that the

se objects tend to rotate faster

and spin down more gradually than

stars

indicating

that the

processes that allow stars to shed

angular momentum become less ef

ficient in this mass range

However,

these studies are

generally limited to brown dwarfs in nearby young clusters and star

forming region

, as

most

field

brown dwarfs

with measured ro

tation rates

have

poorly constrained ages and

correspondingly uncertain mass constraints

As a result,

only a handful of rotation rates have

been measured for brown dwarfs

with well

constrained masses less than 20 M

Jup

14,15

, and

there

are no published

studies of the rotation rate distribution

and angular momentum evolution

in this

mass range

Here,

use

the near

infrared spectrograph

NIRSPEC at the Keck II 10

m telescope to

measure

rotational line broadening for

three

young

planetary

mass

companions

with

wide

projected

orbital separations

ROXs 42B b

, GSC 6214

210 b

, and VHS 1256

1257 b

e also

observe

five

isolated

brown dwarfs

that

were chosen to have ages and spectral types comparable

to t

hose of the

sample of planetary

mass

companions:

OPH 90

, USco J1608

2315

, PSO

J318.5

, 2M0355

+1133

, and KPNO Tau 4

We reduce the data and measure rotation

rates as described in

the

Methods

section

(Fig 1 and

Table 1)

also search for

brown dwarfs

in the literature with spectra

l types later than M6, well

constrained ages

typically

less than

Myr, and measured rotation rates

We use the published magnitudes, spectral types, distances,

and ages to derive new mass estimate

s for both these objects and the

NIRSPEC sample of low

mass brown dwarfs

in a uniform manner

(see Methods

), rather than relying on the relatively

heterogeneous approaches from the literature.

select

our comparison sample of low

mass

brown dwarfs

using a cutoff of 20 M

Jup

, which yields six objects with measured rotation rates

including three from our survey (OPH 90,

USco J1608

2315, PSO J318.5

22), and three from the

literature (2M1207

3932, GY 141, KPNO Tau 12)

14,15

. Although this mass range includes

some objects abov

e 13 M

Jup

, we note that 1

uncertainties on the mass estimates for some of the

planetary

mass companions approach 20 M

Jup

, and this mass distribution is therefore

consistent

with that of

our bound companion sample.

compare

rotation

al velocities

for

our

sample of

planetary

mass companions

to those of the

low

mass

brown dwarf

. Because these brown dwarfs

likely formed via direct fragmenta

tion of

molecular cloud

systematic differences in the observed rotation rates between the two

populations

would

suggest

differing formation histories.

For this analysis we also

include

pub

lished spin measurements for t

additional

planetary

mass

companions,

Pic b and

2M1207

3932

for

a total sample size of

five

planetary

mass companions

and

six

low

mass

20 M

Jup

)

brown dwarf

We note that the bound

brown dwarf companions GQ Lup B and HN

Peg B

also have measured rotation rates

, but

they were excluded from our sample because of

their

higher masses

For the objects in our observed sample we take the posterior distribu

tions

from the

Markov chain Monte Carlo (

MCMC

)

fits and divide by the probability distribution for

sin

where

we used an inclination distribution uniform in cos

Here

is the inclination of the

object’s

rotational axis with respect to our line of sight.

For the objects observed by previous

surveys, we produce a Gaussian distribution for each rotation rate

centered on the measured

sin

values,

and

for those with measured

sin

values

divide that by

the probability distribution

for sin

. This left us with a distribution of rotation rates for each object where we took into

account the unknown inclination

. We then compare the resulting set of velocity distributions to

models in which the rotational

velocities of both populations are either drawn from a single

Gaussian or from two distinct Gaussians using the Bayesian Information Criteria (BIC).

The two

Gaussian model BIC differs from the single Gaussian model BIC by >10

, indicating the single

Gaussian model is strongly preferred.

We also calculate the Akaike information criterion

(AIC)

and find that the single Gaussian model is also strongly preferred,

with

AIC > 10

Finally, we

calculate

the

evidence ratio

the tw

o models, and again f

nd that the single Gaussian model

favored

We conclude that at the level of our observations, there is no evidence for a systematic difference

in the measured rotation rates between the sample of planetary

mass

companions and brown

dwarfs with comparable masses.

This suggests that either the

planetary

mass companions

formed via the same mechanism

as the brown dwarfs

(that is,

turbulent fragmentation), or that

the processes that regulate spin are independent of forma

tion mechanism at the level probed by

our observations.

This is consistent with a picture

in which

spin is regulated via interactions

with the circumplanet

ary disk, as

planetary

mass brown dwarfs should also host circumplanetary

disks early in their lifetimes.

However,

it has been suggested that the

properties

of these disks

might vary depending on the formation channel

, and dis

ks around isolated objects likely

evolve

differently than those embedded in a circumstellar disk.

spin is

indeed

regulated via

interactions with a circumplanetary disk, our findings imply that both classes of objects should

have broadly similar disk properties.

We note that while

there are

other mechanisms such as

planet

scattering,

collisions

disk

migration,

and

tides imposed by exomoons

that

could in

theory

alter the rotation rates of

our bound companions

we do not expect any of these to affect

the angular momentum evolution of these objects at the level measured by our observations

compare the

rotation rate for each object

its

corresponding break

velocity

, taking

into account uncertainties in

the measured rotational line broadening

, unknown inclination

angle

estimated

masses,

radii

, and ages

for the objects in the

sample

(Fig 2)

In the absence of

any

braking

mechanism

we would expect actively accreting objects to spin up until they reac

this critical rotation rate

. The ratio of the observed rotation rate to the predicted break

velocity therefore provides a useful

measure of the relative efficiency of angular momentum loss

mechanisms both during and after the end of accretion.

Taking the error

weighted average o

ver

our

sample, we find that

the

five planetary

mass companions

that we observed

are rotating at

0.13

058

of their break

up velocity

Our

sample

of low

mass brown dwarfs has a

similar

average rot

ation rate of 0.1

0.0

of their break

up velocity

If we combine both samples

together, we find an

average rotation rate

0.126

0.0

times the break

up vel

ocity, suggesting

that both

populations

have shed an appreciable fraction of the angular momentum acquired

during accretion

Previous studies of young stars and higher mass brown dwarfs indicate that there is a correlation

between their rotation rates a

nd masses, with lower mass objects

rotating faster on

ave

rage

consider

whether

this correlation

extends down into the planetary

mass regime

, as has

been suggested

by previous studies

As before, we

include

published rotation rates for

Pic b

and

2M1207

3932

and

show

Jupiter and Saturn

for reference

(

Supplementary

Fig. 5

)

exclude t

he terrestrial

and ice giant

solar system planets

their masses and spins are dominated

by the accretion of solids rather than hydrogen and helium,

and

in some cases have

been further

altered by giant

impacts and/or tidal evolution

We find no evidence

(Pearson correlation

coefficient of

0.0788)

for

any

correlation between rotation rate and mass for

our

sample of

planetary

mass companions, brown

dwarfs with masses below 20 M

Jup

, and

the

solar system gas

giants

Jupiter and Saturn

This suggests that the mechanisms for shedding angular momentum

are effectively independent of mass in the 1

20 M

Jup

range.

We next

investigate

how

the observed rotatio

n rates for

our

sample of

planetary

mass

companions and brown dwarfs

evolve

during

the first several hundred Myr (

Fig

)

We find that

the rotation rates for

both

populations

appear to remain constant with respect to their break

velocities

for ages between

00 Myr

Furthermore,

the rotation rates

for these objects

are also

similar to the present

day rotation rates of Jupiter and Saturn

; given the lack of an observed

correlation between planet mass and rotation rate, this

suggest

that th

ere is also

no significant

spin evolution

on timescales of billions of years

This

suggests that

the observed angular

velocities

planetary

mass objects

are set very early in the

evolutionary lifetimes,

perhaps

through exchange of angular momentum bet

ween the

object

and

its

circumplanetary

gas

disk

Although the mechanism

that

mediates angular momentum transfer in

planetary

mass

companions

is currently unknown, we

use the observations presented here to estimate

its

efficiency.

In the

Methods section

, we present a

calculation that

approximates

the angular

momentum evolution of a newly formed

Jup

object

surrounded

by a circumplanetary disk.

Accounting for spin

up due to gravitational contraction and accretion of disk material, we find

that the spin

down mechanism must extract angular momentum from the planetary

mass object

at a characteristic rate of dL/dt~10

kg m

during the disk

bearing epoch in order to reproduce

the obser

ved rotation rates in the sample.

Understanding and modeling the physical nature of this

mechanism

represents

an intriguing problem, worthy of future

exploration.

The observations presented here

provide constraints on the

primordial rotation rates

and angular

momentum evolution of young

planetary

mass companions

and

brown dwarfs

with comparable

masse

The degree of similarity between these two classes of objects suggests that i

rrespective

of the

formation mechanism

the physical processes

that

regulate angular momentum are likely to

be the same for gas giant planets as they are for planetary

mass br

own dwarfs. As a

consequence, these observations

lay the foundation for

new

theoretical

investigations into

the

mechanism

s that regulate

gas

accretion onto growing

planetary

mass

objects

Looking ahead,

hese results pave

the way for future studies

of gas giant planets

using instruments

on the

upcoming generation of thirty

meter class telescopes such as the Giant Magellan Telescope’s

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Acknowledgments

The data presented herein were obtained at the W. M. Keck Observatory,

which is operated as a scientific partnership among the California Institute of Technology, the

University of California and the National Aeronautics and Space Administration. The

Obse

rvatory was made possible by the generous financial support of the W. M. Keck

Foundation. We acknowledge the efforts of the Keck Observatory staff. The authors wish to

recognize and acknowledge the very significant cultural role and reverence that the su

mmit of

Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate

to have the opportunity to conduct observations from this mountain. HAK acknowledges

support from the Sloan Fellowship Program. Support for this work was pr

ovided by NASA

through Hubble Fellowship grant HST

HF2

51369.001

A awarded by the Space Telescope

Science Institute, which is operated by the Association of Universities for Research in

Astronomy, Inc., for NASA, under contract NAS5

26555.

Author Contributions.

M. L. B. led the observational program, analyzed the resulting data, and

wrote the paper.

B. helped to design and execute the observations and provided advice on the

analysis as well as atmosphere models for each object.

H. A. K

. provided advice and guidance

throughout the process.

K. B. calculated the approximate angular momentum evolution of a

newly formed

10 M

Jup

object surrounded by a circumplanetary disk.

B. P. B. helped to identify

and characterize suitable targets, inclu

ding calculating new mass estimates for all

of the

brown

dwarfs included in this study.

Author Information

Reprints and permissions information is available at

www.nature.com/reprints

. The authors have no

competing financial interests to report.

Correspondence and requests for materials should be addressed to

mlbryan@astro.caltech.edu

Fig. 1

Rotational broadening in the ROXs 42B b spectrum.

ross

correlation between the

ROXs 42B

b spectrum and a model atmosphere broadened to the instrumental resolution

(

black

points)

with 1

σ

uncertainties

from a

jackknife resampling technique (see Methods).

The cross

correlation functions between a model atmospher

e broadened to the instrumental resolution and

that same model additionally broadened by a range of rotation rates (5, 10, 15, 20, 25, 30 km/s)

are

overplotted

in color

The autocorrelation

for a model with no rotational line broadening

shown as

dash

ed pink line.

Fig. 2.

Distributions of

observed

rotation rates as

a fraction

of the corresponding break

velocity for each

object

The distributions for the

planetary

mass companions

are shown in the

left panel and the distributions for

brown dwarfs

with masses less than

20 M

Jup

are shown in the

right panel. Note that these distributions take into account the uncertainties

in the

object’s

mass,

age, and radius, as well as the unknown inclination of its rotation axis with respect to our line of

sight

The uncertainties on the break

up velocities dominate the spread of these distributions.

Fig 3.

Angular momentum evolution of planetary

mass objects.

bserved

rotation rates as

fractions of break

up velocities

are plotted

for

our

sample of five

planetary

mass companions

(blue squares)

as well as a comparison sample of six

isolated

brown dwarfs

with masses

less

than

20 M

Jup

(red triangles)

, and

Jupiter

and Saturn

(

purple

squares)

For comparison w

e also

plot published rotation rates for all

brow

n dwarfs

with well

constrained ages

typically less than

20 Myr

, and spectral types later than M6

(filled circles)

, where

the

shade of grey

corresponds to

our new estimates of the

brown dwarf mass

determined using

the published magnitudes,

spectral types,

distances, and ages of these objects.

We show

σ

uncertainties

for all objects;

these

are dominated by uncertainties in the estimated break

up velocity for each object, with an

additional contribution from the measured rotation rate and unknown inclination with respect to

our line of sight.

Table 1.

Measured rotation rates for our

sample of three new planetary

mass companions.

Planet

ary

Mass

Companion

Mass [M

Jup

]

Age [Myr]

Ref.

sin

[km/s]

ROXs 42B b

10 +/

3 +/

(

), (

)

9.5 (+2.1

2.3)

GSC 6214

210 b

–

11 +/

(

), (

)

6.1 (+4.9

3.8)

VHS 1256

1257 b

–

150

–

300

(

), (

)

13.5 (+3.6

4.1)

Methods

NIRSPEC O

bservations

We observed our targets in K band

(2.03

–

2.38 um)

using the near

infrared spectrograph

NIRSPEC at the Keck II 10 m telescope, which has a resolution of approximately 25,000. We

used the 0.041x2.26 arcsec slit for our

adaptive optics (

)

observations and the 0.432x24

arcsec slit for natural seeing observ

ations, and obtained our data with a standard ABBA nod

pattern. We observed the planetary

mass companions ROXs 42B b and GSC 6214

210 b (1.2"

and 2.2" separations, respectively) in AO mode in o

der to minimize blending with their host

stars; all other tar

gets were observed in natural seeing mode, which has a much higher (~10x

greater) throughput. For ROXs 42B and VHS 1256

1257 we were able to observe b

oth the host

star and planetary

mass companion simultaneously, which made it easier to calculate a

wavele

ngth solution and telluric correction for the much fainter companions in these systems

(see

Methods section

2). We could not do this for GSC 6214

210 b because the planetary

mass

companion was located at a separation of 2.2”, which was comparable to the s

lit length. For this

object we obtained a separate spectrum for the star after completing our observations of the

companion. See

Supplementary

Table

for observation details.

2. 1D Wavelength Calibrated Spectrum Extraction

We extracted 1D spectra from our images using a Python pipeline modeled after

ref

. After flat

fielding, dark subtracting, and then differencing each nodded AB pair, we stacked and aligned

the set of differenced images and combined them

into a single image. We then fit the spectral

trace for each order with a third order polynomial in order to align the modestly curved 2D

spectrum along the

(dispersion) axis. For our sample of planetary

mass companions we fit the

trace of the host sta

r and used this fit to rectify the 2D spectra of both the star and the

companion; this leveraged the high signal

noise of the stellar trace in order to provide better

constraints on the shape of the fainter companion trace. Although we were not able to

place

GSC 6214

210 A and its companion in the slit simultaneously, we found that the shape of the

spectral trace changed very little during our relatively modest 2.2” nod from the host star to the

companion and therefore utilized the same approach with th

is data set. For both ROXs 42B b

and GSC 6214

210 b, the initial solutions obtained from the stellar trace had a slope that differed

by 2

3 pixels from beginning to end when applied to the companion trace. We corrected for this

effect by rectifying the s

pectra of these two companions a second time using a linear function.

Supplementary

Figure

1 shows an example 2D rectified spectrum for VHS 1256

1257 A and

VHS 1256

1257 b.

We note that the NIRSPEC detector occasionally exhibits a behavior, likely due to

variations in

the bias voltages, in which one or more sets of every eight rows will be offset by a constant value

for individual quadrants located on the left side of the detector. Our GSC 6214

210 b

observations were the only one

that

appeared to exhib

it this effect, which produced a distinctive

striped pattern in the two left

hand quadrants. We corrected for this effect by calculating the

median value of the unaffected rows and then adding or subtracting a constant value from the

bad rows in order to

match this median pixel value. While the left side of the detector remained

slightly noisier than the right in our GSC 6214

210 b data set, this noise was not high enough to

preclude its use in our analysis.

After producing combined, rectified 2D spectr

a for each order, we extracted 1D spectra in pixel

space for each positive and negative trace. We calculated an empirical PSF profile along the

(cross

dispersion)

axis of the 2D rectified order, and used this profile to combine the flux along

each colum

n to produce a 1D spectrum. For the ROXs 42B and VHS 1256

1257 datasets, which

include both the star and the planet in the slit simultaneously, we plotted this empirical PSF

profile and confirmed that the stellar and companion traces were well

separated i

n the cross

dispersion direction. We identified the range of

(cross

dispersion) positions containing the

stellar PSF and set these to zero before extracting the companion spectrum. When extracting the

host star spectrum, we similarly set the region con

taining the companion trace to zero.

We next calculated a wavelength solution for each spectral order. Because we maintained the

same instrument configuration (filter, rotator angle, etc.) throughout the night, the wavelength

solution should remain effec

tively constant aside from a linear offset due to differences in the

placement of the target within the slit. As with the 2D traces, we leverage the increased SNR of

the host star spectra to obtain a more precise solution for our sample of planetary

mass

companions. We fit the positions of tellu

ric lines in each order with a third

order polynomial

wavelength solution of the form:

= ax

+ bx

+ cx + d

, where x is pixel number. We then apply

this solution to the companion spectrum using a linear offset c

alculated by cross

correlating the

companion spectrum with a telluric model spectrum.

For our brown dwarf observations we

found that the SNR of the spectra was typically not high enough to obtain a reliable wavelength

solution using telluric lines, and therefore determined this solution by fitting higher SNR

standard star observations obt

ained at a similar airmass immediately before or after each

observation and applying a linear offset (i.e., the same approach as for the companion spectra).

We next remove telluric lines by simultaneously fitting a telluric model and an instrumental

profile to each order in the extracted spectra. For the instrumental profile, we use a Gaussian

function where we allow the width to vary as a free parameter. Although we also considered an

instrumental profile with a central Gaussian and four satellite

Gaussians on either side

, we

found that our choice of instrumental profile had a negligible effect on our final rotational

broadening measurement for ROXs 42B b and therefore elected to use the simpler single

Gaussian model in our subsequent analyses.

e determine the

best

fit

telluric

models

for our

planetary

mass

companion

and low

mass brown dwarf

spectr

by fitting the spectrum of

either

the host star

or the standard star, respectively,

and then applying a linear offset

before

dividing

this model from

the data

Supplementary Figure 2 shows an example 1D wavelength calibrated

and telluric corrected

spectrum for

2M0355+

1133

3. MCMC Fits to Determine Rotational Line Broadening

We measure th

e rotational line broadening

sin

, where

is the rotational

velocity and

is the

unknown inclination, and radial velocity offset for each object by calculating the cross

correlation function (CCF) between the first two orders of each object’s spectrum

(

= 2.27

2.38 um) and a model atmosphere, where the model h

as first been broadened by the measured

instrumental profile (R

25,000). We utilize these two orders because they contain absorption

lines from both water and CO, including two strong CO bandheads, and because they

have

the

most accurate telluric

correction

and wavelength solution

. We generate atmospheric models

for both our

samples of

planetary

mass companions and low

mass brown dwar

fs using the

SCARLET code

, with the parameters used for each object listed

Supplementary

Table 2

We next see

k to match the shape of the measured CCF for each object by comparing it to the

CCF between a model atmosphere with instrumental broadening and the same atmosphere model

with both a radial velocity offset and additional rotational line broadening.

We rota

tionally

broaden the atmos

pheric model using a wavelength

dependent broadening kernel calculated

using E

quation 18.11 taken from ref.

for a quadratic limb darkening law.

The shape of the

rotati

onally

broadened line profile depends on the planet's limb

darkening, which varies

smoothly across the covered wavelength range and between line centers and line wings. We

therefore calculate limb

darkening coefficients for each of the 2048 individual wavelength bins

in our spectrum using the SCARLET model. We fir

st compute the thermal emission intensity

from the planet's atmosphere across a range of different zenith angles. From those intensities we

then generate model intensity profiles at each wavelength, which we fit with quadratic limb

darkening coefficients.

Finally, we use the resulting limb

darkening coefficients to calculate the

appropriate rotational broadening kernel at that wavelength position.

We fit for th

e rotational line broadening

sin

and radial velocity offset of each object using a

MCMC

techni

que. We assume uniform priors on both parameters, and calculate the log

likelihood function as

−

0

.

5

!

!

!

!

!

!

!

!

!

!

!

!

, where

is the CCF between the data and the model

spectrum with instrumental broadening only and

is the CCF of this model and the sa

me model

with additional rotational line broadening and a velocity offset applied. We calculate the

uncertainties

휎

on the CCF of the model with the data using a jackknife resampling technique:

(1)

휎

!"#$$%&'(

!

=

(

!

!

!

)

!

(

푥

!

−

푥

)

!

!

!

!

!

where

is the total number of samples (defined here as the number of individual AB nod pairs),

is the cross

correlation function calculated utilizing all but the

th AB nod pair, and

is the

cross

correlation function calculated using all AB nod pairs

. The number of individual nod pairs

for each target ranged between four and nine; see

Supplementary

Table 1

for more details.

In addition to the measurement uncertainties on our extracted spectra, we also accounted for the

uncertainty on the instrument

al profile in our fits to the CCF. We did this by first fitting for the

instrumental profiles in individual AB nod pairs using telluric lines in our high SNR stellar

spectra (either host star or standard star). We then calculated the median resolution fo

r each

night and set the corresponding uncertainty on this value to the standard deviation of all

resolution values divided by the square root of the number of AB nod pairs utilized. When

itting the CCF functions of the

planetary

mass companions and brow

n dwarfs

that we observed

we drew a new resolution from a Gaussian distribution with a peak located at the median

resolution and width equal to the calculated uncertainty on that resolution at each step in the

MCMC chain. We plot both the measured CCFs a

nd the best

fit model CCFs for each object in

our sample in

Supplementary

Figure 3

, and report the best

fit values and corresponding

uncertainties in Table 1 and

Supplementary

Table 2

We next considered whether our measured rotational broadening values

might be inflated by

small offsets in the relative positions of individual spectra within our sequence of AB nod pairs.

We tested for this by calculating a CCF for each individual AB nod pair in our ROXs 42B b

observations, where we treat the positive an

d negative traces separately, and measuring the

location of the CCF peak in wavelength space. We found that within the set of individual

positive trace spectra (A nods) and negative trace spectra (B nods) the observed wavelength

shifts were minimal, typic

ally less than 1 km/s. However, the difference between the median

positive and negative trace offsets could be as large as 5 km/s. As a result we opted to fit the

positive and negative trace spectra separately, resulting in two independent estimates of t

rotational broadening and velocity offset for each object. We find that in all cases these two

values are consistent within the errors, and report their

error

weighted average in Table

1 and

Supplementary

Table 2

We also plot these values as a functi

on of time in Supplementary Figure

As an additional check, we

also

used the extracted spectra for the host star ROXs 42B, which

have a much higher SNR in individual exposures, to determine the spin rate for each individual

AB pair. We found that our measured rotational broadening also remained consistent across the

full

set of AB pairs, and agreed with the value calculated directly from the composite stellar

spectrum (i.e.,

including all AB nod pairs) to <0.5

We next compare the measured radial velocity offsets for our sample of bound planetary

mass

companions to thos

e of their host stars. For the host stars, we used Phoenix spectra

to model

their spectra, where we select the model with log(g) and T

eff

values closest to those reported in

the literature for each system. We determined the rotation rates of ROXs 42B, G

SC 6214

210

and VHS 1256

1257

to be 43.6

0.2 km/s, 28.8+/

2.5 km/s, and 75.2

(+2.7

2.3)

km/s

respectively, and

measured

velocity offsets of 1.8

0.2 km/

12.6 (+2.0

2.2)

km/s, and 1.5

(+2.0

2.2)

km/s respectively. We would expect both star and

planetary

mass companion to

share the same RV offset, as the predicted orbital velocities of these relatively wide separation

companions should be much smaller than the precision of our measurements. For our lower S/N

spectra (ROXs 42Bb and GSC 6214

210b)

, we find that the reported RV values for the planets

differ from those of their host stars by 4.1 km/s and 5.3 km/s respectively. If we take the formal

RV errors of 0.7 km/s and 1.3 km/s from our MCMC analysis at face value, this would

correspond to RV o

ffsets of 7.7

and 2.2

respectively. However, we note that for these two

relative

low SNR targets

the telluric lines we use for calibrating the linear offset in the planet's

wavelength solution becomes the dominant source of uncertainty in our measure

ment of the

planet’s RV offset

;

this is not accounted for in our formal jackknife

error

analysis

, which

assumes an error

free wavelength solution

. We therefore adopt a systematic noise floor of 4.0

km/s for the reported RV values for these two relatively

low SNR targets. We also test the

possible effects of 4

5 km/s errors in our wavelength solutions for these two planets by setting

ROXs 42Bb's radial velocity equal to that of its host star (1.8 km/s vs

2.3 km/s) and re

running

our MCMC analysis. We fin

d that the measured rotation rate for the companion in this fit is 9.8

(+2.0

2.1) km/s, consistent wi

th our original measurement at

0.1

We also investigate whether or not night

night variations in the instrumental broadening

profile might affect our

estimated values for rotational line broadening. We test this by fitting

for the rotational line broadening of the host star ROXs 42B, which was observed

along with its

planetary

mass companion

on two separate nights with an estimated in

strumental resolu

tion of

R~30,000 and R~

26,000, respectively. We found that the measured spins for the first and

second

night differed by ~1

, indicating that our method for determining the instrumental broadening

profile using telluric lines is providing a reliable characterization of this parameter.

On the modeling side, we also check whether variations in the C/O ratio of the atmospheric

odel used in the cross

correlation might affect our measured spin rates. We test this by

repeating our CCF analysis of the ROXs 42B b spectrum using models with C/O ratios of 0.8,

0.54 (solar), and 0.35. We find that the measured spin rates for the low a

nd high C/O models are

consistent

with our solar C/O model at <0.5

. We also consider the possibility that pressure

broadening might cause us to over

estimate the amount of rotational line broadening in these

objects. Our fiducial solar metallicity model

s were generated using opacities

from the ExoMol

database

, which does not include pressure broadening, but has line locations that better match

our observed spectra. Alternative opac

ity tables such as HiTemp

do include pressure

broadening, but do not

match the line locations in our spectra as well as the ExoMol database.

We test the potential effects of pressure broadening on our estimate of the spin rate by generating

a new version of our atmosphere model for ROXs 42B b using HiTemp molecular opaci

ties

and

comparing the resulting

sin

value to the one measured using our original ExoMol models.

Depending on our choice of pressure

temperature profile, we found that the measured rotation

rate calculated using the HiTemp models was 2

5 km/s (0.7

1.8

) lower than the rotation rate

using our original ExoMol models.

utilize

the ExoMol opacities

in our final rotation rate

analysis

for three reasons: (1) the pressure

temperature profiles for these young planetary

mass

objects are poorly constrained by

current observations, (2) the pressure

broadened profiles for

many molecules at high temperatures are not currently well understood, and (3) the ExoMol line

locations are a better match for our spectra than the HiTemp line locations. We note that

includi

ng pressure

broadening in our models would likely decrease our estimated rotation rates

by several km/s, corresponding to a change of approximately 1

for most of the objects in our

sample. However, this would not affect our conclusion that young planetar

mass objects appear

to be rotating at much less than their break

up velocities, and that their rotation rates do not

evolve significantly in time.

Finally, we test whether uncertainties in assumed effective temperatures and surface gravities

could impac

t our measured rotation rates. We generate atmospheric models for PSO J318.5

using T

eff

and log(g) values determined in the forward model analysis of Allers et al 2016

, T

eff

= 1325 (+350

12) K and log(g) = 3.7 (+1.1

0.1). We recalculate rotation rates

and velocity

offsets

using a model with T

eff

= 1313 K and log(g) = 3.6, and another model with T

eff

= 1675 K

and log(g) = 4.8. We find that these

parameters differ from

the

original value

by less than

0.3

4. Calculating the Break

up Velocity

To calculate break

up velocities, we need estimates of the masses and radii of the objects in

our

sample. For our sample of bound planetary

mass companions we utilize mass

estimates from the

literature. For our sample of low

mass brown dwarfs, we derive new mass estimates in a

homogeneous manner rather than relying on the heterogeneous approaches from the literature

(Supplementary Table 3)

. We first calculate bolometric lu

minosities for these objects using the

band bolometric correction for young ultracool dwarfs fr

ref

together with their distances

and spectral types. We then calculate masses using their ages and luminosities together with a

fin

ely interpolated grid of hot

start evolutionary models from

ref

. We incorporate uncertainties

in distance, spectral type, apparent K

band magnitude, and age in a Monte Carlo fashion by

randomly drawing these values from normal distribu

tions for a large number of trials. We note

that estimating masses using the bolometric luminosity is more robust than using absolute

magnitudes as the former is less reliant on the detailed accuracy of atmospheric models. This is

especially true in the

optical where strong molecular opacities are generally more difficult to

reproduce in synthetic spectra compared, for example, to near

infrared wavelengths.

Once we ha

a mas

s estimate, we used COND models

to estimate the radius of each object.

We note

that

by using COND models, the radii we adopt assume a hot

start formation history.

We calculate the 1

minimum radius using the 1

minimum age and mass, and the 1

maximum

radius using the 1

maximum age and mass. Although we could have propagate

d the

uncertainties in mass and age in quadrature, these are not independent quantities as the mass

estimate depends directly on the age estimate, and we therefore opted for a more conservative

approac

h. We calculate the best

fit break

up velocity for eac

h object and the corresponding 1

uncertainties on this parameter by propagating uncertainties from the mass, radius, and age of the

object.

Supplementary Figure 4 compares the distributions of measured rotation rates and

calculated break

up velocities fo

r each object.

5. Angular Momentum Evolution Calculation

Here we seek to approximately characterize the angular momentum evolution of a giant planet or

low

mass brown dwarf

following the primary phase of assembly

n the absence of more

stringent observ

ational constraints

we utilize a simple parameterized model to estimate the

relevant timescale for angular momentum

evolution

. It is readily apparent that this timescale

will be shorter than both the disk lifetime

and the Kelvin

Helmholtz timescale

, but

such a

parameterized model is nonetheless instructive for i

llustrating the relevant forces

at work in this

problem

We begin by discussing the consequences of gravitational contraction.

5.1. Gravitational Contraction

To approximate the interior struct

e of a newly

formed planetary

mass object, we adopt a

polytropic equation of state with index

휉

=

3

/

2

characteristic of a fully

convective body. The

binding energy of such an object is given by:

(2)

E

=

−

푏

!

!

!

!

where

3/(

10

−

2

휉

) = 3/7

is the gravitational constant,

is the mass of the object and

the radius of the object.

Equating the

gravitational energy loss to the radiative flux at the surface,

we have

(3)

!"

=

−

4

휋

푅

!

휎

푇

!""

!

=

푏

!

!

!

!

!

Adopting an initial condition R |

t=0

, the above expression yields a differential equation for

the evolution of the radius:

(4)

=

−

푅

!

!

!

!

!

!

!""

!

!

!

!

!

!

=

−

!

!

!

!

!

!