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means that lB≫l2
E/d. For magnetic ﬁeld ∼1T, and
lB∼25nm, using the estimate below Eq. (10) that lE∼
400nm we conclude that the width of the ﬂake should
exceedd >10µm. The magnetoplasmon modes (21) are
∼(lB/lE)2slowerthan electrons. Note that these modesare undamped since single-particle excitations cannot be
induced at frequencies below cyclotron frequency ωB.
Conclusions . Graphene p-njunctions are among the
most simple and promising applications of this material.
Single-electron properties of p-njunctions have been ex-
tensively studied. In the present paper we investigated
their collective excitations both with and without mag-
netic ﬁeld. We anticipate that plasmon modes will be
crucial for the optical response of graphene nanostruc-
tures and realistic samples containing electron-hole pud-
dles. High degree of experimental control should make
them of special interest to nanoplasmonics and electron-
ics. Among the most promising applications of plasmons
inp-njunctions we envisage a possibility of a “plasmon
transistor” [4]. In particular, by simply switching the
direction of electric ﬁeld from across the ﬂake to along
it (and back) the propagation of plasmons can be facil-
itated (or prevented). In addition, as follows from the
above Eqs. (1), (11), the plasmon velocity can be con-
trolled with simple change in the magnitude of electric
ﬁeld. This is in a sharp contrast to plasmons in metal-
lic nanostructures, whose spectra are typically ﬁxed once
devices are fabricated.
Acknowledgments. Useful discussions with M. Raikh
and O. Starykh are gratefully acknowledged. This
work was supported by DOE, Grant No. DE-FG02-
06ER46313. P.G.S. was supported by the SFB TR 12.
[*] Present address: School of Physics, University of Exete r,
EX4 4QL, U.K.
[2] M. Wilson, Phys. Today 59, No. 1, 21 (2006).
[3] A.K. Geim and K.S. Novoselov, Nature Mater. 6, 183
(2007).
[4] H.A. Atwater, Sci. Am. 296, 56 (2007).
[5] S.A. Maier, Plasmonics: Fundamentals and Applications
(Springer, New York, 2007).
[6] F. Stern, Phys. Rev. Lett. 18, 546 (1967).
[7] S. Gangadharaiah, A.M. Farid, and E.G. Mishchenko,
Phys. Rev. Lett. 100, 166802 (2008).
[8] O. Vafek, Phys. Rev. Lett. 97, 266406 (2006).
[9] E.H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418
(2007).
[10] A.H. Castro Neto et al.,Rev. Mod. Phys. 81, 109 (2009).
[11] J. Martin et al.,Nature Physics 4, 144 (2008).
[12] J.R. Williams, L. DiCarlo, and C.M. Marcus, Science
317, 638 (2007).
[13] Rigorous derivation of Eq. (2) is based on the “rela-
tivistic” stress energy-momentum tensor, see L.D. Lan-
dau and E. M. Lifshitz, Fluid Mechanics , Butterworth-
Heinemann, Oxford (1987), Ch. 15; M. Mueller, L. Fritz,
S. Sachdev, and J. Schmalian, arXiv:0810.3657.
[14] In the case of gate controlled junctions the image charg es
induced at the gates should be included into Eq. (4).
[15] T.A. Sedrakyan, E.G. Mishchenko, and M.E. Raikh,
Phys. Rev. B 74, 235423 (2006).
[16] P.G. SilvestrovandK.B. Efetov, Phys.Rev.B 77, 155436
(2008).5
[17] In addition even and odd solutions with n >0 have dif-
ferent singular behavior at \|ξ\| ≪1:δρ(even)∼/radicalbig
\|ξ\|,
δρ(odd)∼sign(ξ)//radicalbig
\|ξ\|. Atξ→ ±1 all solutions diverge
asδρ∼1//radicalbig
1−ξ2.
[18] L.M. Zhang and M.M. Fogler, Phys. Rev. Lett. 100,116804 (2008).
[19] I.L. Aleiner and L.I. Glazman, Phys. Rev. Lett. 72, 2935
(1994).
[20] V.A. Volkov and S. A. Mikhailov, JETP Lett. 42, 556
(1985).
arXiv:1001.0012v2 [astro-ph.EP] 20 Dec 2010Draft version May 20, 2018
Preprint typeset using L ATEX style emulateapj v. 8/13/10
THE STATISTICS OF ALBEDO AND HEAT RECIRCULATION ON HOT EXOPL ANETS
Nicolas B. Cowan1,2, Eric Agol2,
Draft version May 20, 2018
ABSTRACT
If both the day-side and night-side eﬀective temperatures of a pla net can be measured, it is possible
to estimate its Bond albedo, 0 < AB<1, as well as its day–night heat redistribution eﬃciency,
0< ε <1. We attempt a statistical analysis of the albedo and redistribution eﬃciency for 24
transiting exoplanets that have at least one published secondary e clipse. For each planet, we show
how to calculate a sub-stellar equilibrium temperature, T0, and associated uncertainty. We then use
a simple model-independent technique to estimate a planet’s eﬀective temperature from planet/star
ﬂux ratios. We use thermal secondary eclipse measurements —tho se obtained at λ >0.8 micron—
to estimate day-side eﬀective temperatures, Td, and thermal phase variations —when available— to
estimatenight-sideeﬀectivetemperature. Westronglyruleoutth e“nullhypothesis”ofasingle ABand
εforall 24planets. If wealloweachplanet to havediﬀerent paramete rs,we ﬁnd that lowBond albedos
are favored ( AB<0.35 at 1σconﬁdence), which is an independent conﬁrmation of the low albedos
inferred from non-detection of reﬂected light. Our sample exhibits a wide variety of redistribution
eﬃciencies. When normalized by T0, the day-side eﬀective temperatures of the 24 planets describe
a uni-modal distribution. The two biggest outliers are GJ 436b (abno rmally hot) and HD 80606b
(abnormally cool), and these are the only eccentric planets in our sa mple. The dimensionless quantity
Td/T0exhibits no trend with the presence or absence of stratospheric in versions. There is also no
clear trend between Td/T0andT0. That said, the 6 planets with the greatest sub-stellar equilibrium
temperatures ( T >2400 K) have low ε, as opposed to the 18 cooler planets, which show a variety
of recirculation eﬃciencies. This hints that the very hottest trans iting giant planets are qualitatively
diﬀerent from the merely hot Jupiters. We propose an explanation o f this trend based on how a
planet’s radiative and advective times scale with temperature: both timescales are expected to be

means that lB≫l2

E/d. For magnetic ﬁeld ∼1T, and

lB∼25nm, using the estimate below Eq. (10) that lE∼

400nm we conclude that the width of the ﬂake should

exceedd >10µm. The magnetoplasmon modes (21) are

∼(lB/lE)2slowerthan electrons. Note that these modesare undamped since single-particle excitations cannot be

induced at frequencies below cyclotron frequency ωB.

Conclusions . Graphene p-njunctions are among the

most simple and promising applications of this material.

Single-electron properties of p-njunctions have been ex-

tensively studied. In the present paper we investigated

their collective excitations both with and without mag-

netic ﬁeld. We anticipate that plasmon modes will be

crucial for the optical response of graphene nanostruc-

tures and realistic samples containing electron-hole pud-

dles. High degree of experimental control should make

them of special interest to nanoplasmonics and electron-

ics. Among the most promising applications of plasmons

inp-njunctions we envisage a possibility of a “plasmon

transistor” [4]. In particular, by simply switching the

direction of electric ﬁeld from across the ﬂake to along

it (and back) the propagation of plasmons can be facil-

itated (or prevented). In addition, as follows from the

above Eqs. (1), (11), the plasmon velocity can be con-

trolled with simple change in the magnitude of electric

ﬁeld. This is in a sharp contrast to plasmons in metal-

lic nanostructures, whose spectra are typically ﬁxed once

devices are fabricated.

Acknowledgments. Useful discussions with M. Raikh

and O. Starykh are gratefully acknowledged. This

work was supported by DOE, Grant No. DE-FG02-

06ER46313. P.G.S. was supported by the SFB TR 12.

[*] Present address: School of Physics, University of Exete r,

EX4 4QL, U.K.

[2] M. Wilson, Phys. Today 59, No. 1, 21 (2006).

[3] A.K. Geim and K.S. Novoselov, Nature Mater. 6, 183

(2007).

[4] H.A. Atwater, Sci. Am. 296, 56 (2007).

[5] S.A. Maier, Plasmonics: Fundamentals and Applications

(Springer, New York, 2007).

[6] F. Stern, Phys. Rev. Lett. 18, 546 (1967).

[7] S. Gangadharaiah, A.M. Farid, and E.G. Mishchenko,

Phys. Rev. Lett. 100, 166802 (2008).

[8] O. Vafek, Phys. Rev. Lett. 97, 266406 (2006).

[9] E.H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418

(2007).

[10] A.H. Castro Neto et al.,Rev. Mod. Phys. 81, 109 (2009).

[11] J. Martin et al.,Nature Physics 4, 144 (2008).

[12] J.R. Williams, L. DiCarlo, and C.M. Marcus, Science

317, 638 (2007).

[13] Rigorous derivation of Eq. (2) is based on the “rela-

tivistic” stress energy-momentum tensor, see L.D. Lan-

dau and E. M. Lifshitz, Fluid Mechanics , Butterworth-

Heinemann, Oxford (1987), Ch. 15; M. Mueller, L. Fritz,

S. Sachdev, and J. Schmalian, arXiv:0810.3657.

[14] In the case of gate controlled junctions the image charg es

induced at the gates should be included into Eq. (4).

[15] T.A. Sedrakyan, E.G. Mishchenko, and M.E. Raikh,

Phys. Rev. B 74, 235423 (2006).

[16] P.G. SilvestrovandK.B. Efetov, Phys.Rev.B 77, 155436

(2008).5

[17] In addition even and odd solutions with n >0 have dif-

ferent singular behavior at |ξ| ≪1:δρ(even)∼/radicalbig

|ξ|,

δρ(odd)∼sign(ξ)//radicalbig

|ξ|. Atξ→ ±1 all solutions diverge

asδρ∼1//radicalbig

1−ξ2.

[18] L.M. Zhang and M.M. Fogler, Phys. Rev. Lett. 100,116804 (2008).

[19] I.L. Aleiner and L.I. Glazman, Phys. Rev. Lett. 72, 2935

(1994).

[20] V.A. Volkov and S. A. Mikhailov, JETP Lett. 42, 556

(1985).

arXiv:1001.0012v2 [astro-ph.EP] 20 Dec 2010Draft version May 20, 2018

Preprint typeset using L ATEX style emulateapj v. 8/13/10

THE STATISTICS OF ALBEDO AND HEAT RECIRCULATION ON HOT EXOPL ANETS

Nicolas B. Cowan1,2, Eric Agol2,

Draft version May 20, 2018

ABSTRACT

If both the day-side and night-side eﬀective temperatures of a pla net can be measured, it is possible

to estimate its Bond albedo, 0 < AB<1, as well as its day–night heat redistribution eﬃciency,

0< ε <1. We attempt a statistical analysis of the albedo and redistribution eﬃciency for 24

transiting exoplanets that have at least one published secondary e clipse. For each planet, we show

how to calculate a sub-stellar equilibrium temperature, T0, and associated uncertainty. We then use

a simple model-independent technique to estimate a planet’s eﬀective temperature from planet/star

ﬂux ratios. We use thermal secondary eclipse measurements —tho se obtained at λ >0.8 micron—

to estimate day-side eﬀective temperatures, Td, and thermal phase variations —when available— to

estimatenight-sideeﬀectivetemperature. Westronglyruleoutth e“nullhypothesis”ofasingle ABand

εforall 24planets. If wealloweachplanet to havediﬀerent paramete rs,we ﬁnd that lowBond albedos

are favored ( AB<0.35 at 1σconﬁdence), which is an independent conﬁrmation of the low albedos

inferred from non-detection of reﬂected light. Our sample exhibits a wide variety of redistribution

eﬃciencies. When normalized by T0, the day-side eﬀective temperatures of the 24 planets describe

a uni-modal distribution. The two biggest outliers are GJ 436b (abno rmally hot) and HD 80606b

(abnormally cool), and these are the only eccentric planets in our sa mple. The dimensionless quantity

Td/T0exhibits no trend with the presence or absence of stratospheric in versions. There is also no

clear trend between Td/T0andT0. That said, the 6 planets with the greatest sub-stellar equilibrium

temperatures ( T >2400 K) have low ε, as opposed to the 18 cooler planets, which show a variety

of recirculation eﬃciencies. This hints that the very hottest trans iting giant planets are qualitatively

diﬀerent from the merely hot Jupiters. We propose an explanation o f this trend based on how a

planet’s radiative and advective times scale with temperature: both timescales are expected to be