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Bond albedo.
2. We ﬁrmly rule out the “null hypothesis”, whereby all
transiting planets can be ﬁt by a single ABandε. It
is not immediately clear whether this stems from diﬀer-
ences in Bond albedo, recirculation eﬃciency, or both.
3. In the few cases where it is possible to unambiguously
infer an albedo based on optical eclipse depths, they are
extremely low, implying correspondingly low Bond albe-
dos (<10%). If one adopts such low albedos for all
the planets in our sample, the discrepancies in day-side
eﬀective temperature must be due to diﬀerences in recir-
culation eﬃciency.
4. These diﬀerences in recirculation eﬃciency do not
appear to be correlated with the presence or absence of
a stratospheric inversion.
5. Planets cooler than Tε=0= 2400 K exhibit a wide va-
riety of circulation eﬃciencies that do not appear to be
correlated with equilibrium temperature. Alternatively,
theseplanetsmayhavediﬀerent (but generallylow)albe-
dos. Planets hotter than Tε=0= 2400 K have uniformly
low redistribution eﬃciencies and albedos.
The apparent decrease in advective eﬃciency with
increasing planetary temperature remains unexplained.
One hypothesis, mentioned earlier, is that TiO and VO
would provide additional optical opacity in atmospheres
hotter than T∼1700 K, leading to temperature in-
versions and reduced heat recirculation on these plan-
ets (Fortney et al. 2008). But if our sample shows any
sharp change it behavior it occurs near 2400 K, rather
than 1700K. One couldinvokeanotheroptical absorber,
but in any case the lack of correlation —pointed out in
thisworkandelsewhere—betweenthepresenceofatem-
perature inversionand the eﬃciency of heat recirculation
makes this explanation suspect. Another possible expla-
nation for the observed trend is that the hottest planets
have the most ionized atmospheres and may suﬀer the
most severe magnetic drag (Perna et al. 2010).
The simplest explanation for this trend is simply that
the radiative time is a steeper function of temperature
than the advective time: advective eﬃciency is given
roughly by the ratio of the radiative and advective times
(eg: Cowan & Agol 2010). In the limit of Newtonian
cooling, the radiative time scales as τrad∝T−3. If one
assumes the wind speed to be of order the local sound
speed, then the advective time scales as τadv∝T−0.5.
One might therefore naively expect the advective eﬃ-
ciency to scale as T−2.5. Such an explanation would notAlbedo and Heat Recirculation on Hot Exoplanets 9
explain the apparent sharp transition seen at 2400 K,
however.
The combination of optical observations of secondary
eclipses and thermal observations of phase variations is
the best way to constrain planetary albedo and circu-
lation. The optical observations should be taken near
the star’s blackbody peak, both to maximize signal-to-
noise, and to avoidcontaminationfrom the planet’s ther-
mal emission, but this separationmay not be possible for
the hottest transiting planets. The thermal observations,
likewise, should be near the planet’s blackbody peak to
better constrain its bolometric ﬂux. Note that this wave-
length is shortwardof the ideal contrastratio, which typ-
ically falls on the planet’s Rayleigh-Jeans tail. Further-
more, the thermal phase observations should span a full
planetaryorbit: thelightcurveminimumisthemostsen-
sitive measure of ε, and should occur nearly half an orbit
apart from the light curve maximum, despite skewed di-
urnal heatingpatterns (Cowan & Agol 2008, 2010). This
means that observing campaigns that only cover a little
more than half an orbit (transit →eclipse) are probably
underestimating the real peak-trough phase amplitude.A possible improvement to this study would be to per-
form a uniform data reduction for all the Spitzerexo-
planet observations of hot Jupiters. These data make up
the majority of the constraints presented in our study
and most are publicly available. And while the pub-
lished observations were analyzed in disparate ways, a
consensus approach to correcting detector systematics is
beginning to emerge.
N.B.C. acknowledges useful discussions of aspects of
this work with T. Robinson, M.S. Marley, J.J. Fort-
ney, T.S. Barman and D.S. Spiegel. Thanks to our
referee B.M.S. Hansen for insightful feedback, and to
E.D. Feigelson for suggestions about statistical methods.
N.B.C. was supported by the Natural Sciences and Engi-
neering Research Council of Canada. E.A. is supported
by a National Science Foundation Career Grant. Sup-
port for this work was provided by NASA through an
award issued by JPL/Caltech. This research has made
use of the Exoplanet Orbit Database and the Exoplanet
Data Explorer at exoplanets.org.
REFERENCES
Agol, E., Cowan, N. B., Knutson, H. A., Deming, D., Steﬀen,
J. H., Henry, G. W., & Charbonneau, D. 2010, ApJ, 721, 1861
Alonso, R. et al. 2009a, A&A, 506, 353
Alonso, R., Deeg, H. J., Kabath, P., & Rabus, M. 2010, AJ, 139,
1481
Alonso, R., Guillot, T., Mazeh, T., Aigrain, S., Alapini, A. ,
Barge, P., Hatzes, A., & Pont, F. 2009b, A&A, 501, L23
Anderson, D. R. et al. 2010, A&A, 513, L3+
Barman, T. S. 2008, ApJ, 676, L61
Barnes, J. R., Barman, T. S., Prato, L., Segransan, D., Jones ,
H. R. A., Leigh, C. J., Collier Cameron, A., & Pinﬁeld, D. J.

Bond albedo.

2. We ﬁrmly rule out the “null hypothesis”, whereby all

transiting planets can be ﬁt by a single ABandε. It

is not immediately clear whether this stems from diﬀer-

ences in Bond albedo, recirculation eﬃciency, or both.

3. In the few cases where it is possible to unambiguously

infer an albedo based on optical eclipse depths, they are

extremely low, implying correspondingly low Bond albe-

dos (<10%). If one adopts such low albedos for all

the planets in our sample, the discrepancies in day-side

eﬀective temperature must be due to diﬀerences in recir-

culation eﬃciency.

4. These diﬀerences in recirculation eﬃciency do not

appear to be correlated with the presence or absence of

a stratospheric inversion.

5. Planets cooler than Tε=0= 2400 K exhibit a wide va-

riety of circulation eﬃciencies that do not appear to be

correlated with equilibrium temperature. Alternatively,

theseplanetsmayhavediﬀerent (but generallylow)albe-

dos. Planets hotter than Tε=0= 2400 K have uniformly

low redistribution eﬃciencies and albedos.

The apparent decrease in advective eﬃciency with

increasing planetary temperature remains unexplained.

One hypothesis, mentioned earlier, is that TiO and VO

would provide additional optical opacity in atmospheres

hotter than T∼1700 K, leading to temperature in-

versions and reduced heat recirculation on these plan-

ets (Fortney et al. 2008). But if our sample shows any

sharp change it behavior it occurs near 2400 K, rather

than 1700K. One couldinvokeanotheroptical absorber,

but in any case the lack of correlation —pointed out in

thisworkandelsewhere—betweenthepresenceofatem-

perature inversionand the eﬃciency of heat recirculation

makes this explanation suspect. Another possible expla-

nation for the observed trend is that the hottest planets

have the most ionized atmospheres and may suﬀer the

most severe magnetic drag (Perna et al. 2010).

The simplest explanation for this trend is simply that

the radiative time is a steeper function of temperature

than the advective time: advective eﬃciency is given

roughly by the ratio of the radiative and advective times

(eg: Cowan & Agol 2010). In the limit of Newtonian

cooling, the radiative time scales as τrad∝T−3. If one

assumes the wind speed to be of order the local sound

speed, then the advective time scales as τadv∝T−0.5.

One might therefore naively expect the advective eﬃ-

ciency to scale as T−2.5. Such an explanation would notAlbedo and Heat Recirculation on Hot Exoplanets 9

explain the apparent sharp transition seen at 2400 K,

however.

The combination of optical observations of secondary

eclipses and thermal observations of phase variations is

the best way to constrain planetary albedo and circu-

lation. The optical observations should be taken near

the star’s blackbody peak, both to maximize signal-to-

noise, and to avoidcontaminationfrom the planet’s ther-

mal emission, but this separationmay not be possible for

the hottest transiting planets. The thermal observations,

likewise, should be near the planet’s blackbody peak to

better constrain its bolometric ﬂux. Note that this wave-

length is shortwardof the ideal contrastratio, which typ-

ically falls on the planet’s Rayleigh-Jeans tail. Further-

more, the thermal phase observations should span a full

planetaryorbit: thelightcurveminimumisthemostsen-

sitive measure of ε, and should occur nearly half an orbit

apart from the light curve maximum, despite skewed di-

urnal heatingpatterns (Cowan & Agol 2008, 2010). This

means that observing campaigns that only cover a little

more than half an orbit (transit →eclipse) are probably

underestimating the real peak-trough phase amplitude.A possible improvement to this study would be to per-

form a uniform data reduction for all the Spitzerexo-

planet observations of hot Jupiters. These data make up

the majority of the constraints presented in our study

and most are publicly available. And while the pub-

lished observations were analyzed in disparate ways, a

consensus approach to correcting detector systematics is

beginning to emerge.

N.B.C. acknowledges useful discussions of aspects of

this work with T. Robinson, M.S. Marley, J.J. Fort-

ney, T.S. Barman and D.S. Spiegel. Thanks to our

referee B.M.S. Hansen for insightful feedback, and to

E.D. Feigelson for suggestions about statistical methods.

N.B.C. was supported by the Natural Sciences and Engi-

neering Research Council of Canada. E.A. is supported

by a National Science Foundation Career Grant. Sup-

port for this work was provided by NASA through an

award issued by JPL/Caltech. This research has made

use of the Exoplanet Orbit Database and the Exoplanet

Data Explorer at exoplanets.org.

REFERENCES

Agol, E., Cowan, N. B., Knutson, H. A., Deming, D., Steﬀen,

J. H., Henry, G. W., & Charbonneau, D. 2010, ApJ, 721, 1861

Alonso, R. et al. 2009a, A&A, 506, 353

Alonso, R., Deeg, H. J., Kabath, P., & Rabus, M. 2010, AJ, 139,

1481

Alonso, R., Guillot, T., Mazeh, T., Aigrain, S., Alapini, A. ,

Barge, P., Hatzes, A., & Pont, F. 2009b, A&A, 501, L23

Anderson, D. R. et al. 2010, A&A, 513, L3+

Barman, T. S. 2008, ApJ, 676, L61

Barnes, J. R., Barman, T. S., Prato, L., Segransan, D., Jones ,

H. R. A., Leigh, C. J., Collier Cameron, A., & Pinﬁeld, D. J.