Rotational Variability of Earth's Polar Regions: Implications for Detecting Snowball Planets

Abstract

We have obtained the first time-resolved, disk-integrated observations of Earth's poles with the Deep Impact spacecraft as part of the EPOXI mission of opportunity. These data mimic what we will see when we point next-generation space telescopes at nearby exoplanets. We use principal component analysis (PCA) and rotational light curve inversion to characterize color inhomogeneities and map their spatial distribution from these unusual vantage points, as a complement to the equatorial views presented by Cowan et al. in 2009. We also perform the same PCA on a suite of simulated rotational multi-band light curves from NASA's Virtual Planetary Laboratory three-dimensional spectral Earth model. This numerical experiment allows us to understand what sorts of surface features PCA can robustly identify. We find that the EPOXI polar observations have similar broadband colors as the equatorial Earth, but with 20%-30% greater apparent albedo. This is because the polar observations are most sensitive to mid-latitudes, which tend to be more cloudy than the equatorial latitudes emphasized by the original EPOXI Earth observations. The cloudiness of the mid-latitudes also manifests itself in the form of increased variability at short wavelengths in the polar observations and as a dominant gray eigencolor in the south polar observation. We construct a simple reflectance model for a snowball Earth. By construction, our model has a higher Bond albedo than the modern Earth; its surface albedo is so high that Rayleigh scattering does not noticeably affect its spectrum. The rotational color variations occur at short wavelengths due to the large contrast between glacier ice and bare land in those wavebands. Thus, we find that both the broadband colors and diurnal color variations of such a planet would be easily distinguishable from the modern-day Earth, regardless of viewing angle.

Full text

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arXiv:1102.4345v1 [astro-ph.EP] 21 Feb 2011
Draft version February 23, 2011
Preprint typeset using LATEX style emulateapj v. 8/13/10
ROTATIONAL VARIABILITY OF EARTH’S POLAR REGIONS: IMPLICATIONS FOR DETECTING
SNOWBALL PLANETS
Nicolas B. Cowan1,2, Tyler Robinson3,4, Timothy A. Livengood5, Drake Deming5,4, Eric Agol3, Michael F.
A’Hearn6, David Charbonneau7, Carey M. Lisse8, Victoria S. Meadows3,4, Sara Seager9,4, Aomawa L. Shields3,4,
Dennis D. Wellnitz6
Draft version February 23, 2011
ABSTRACT
We have obtained the first time-resolved, disc-integrated observations of Earth’s poles with the Deep
Impact spacecraft as part of the EPOXI Mission of Opportunity. These data mimic what we will see
when we point next-generation space telescopes at nearby exoplanets. We use principal component
analysis (PCA) and rotational lightcurve inversion to characterize color inhomogeneities and map
their spatial distribution from these unusual vantage points, as a complement to the equatorial views
presented in Cowan et al. (2009). We also perform the same PCA on a suite of simulated rotational
multi-band lightcurves from NASA’s Virtual Planetary Laboratory 3D spectral Earth model. This
numerical experiment allows us to understand what sorts of surface features PCA can robustly identify.
We find that the EPOXI polar observations have similar broadband colors as the equatorial Earth,
but with 20–30% greater apparent albedo. This is because the polar observations are most sensitive to
mid-latitudes, which tend to be more cloudy than the equatorial latitudes emphasized by the original
EPOXI Earth observations. The cloudiness of the mid-latitudes also manifests itself in the form of
increased variability at short wavelengths in the polar observations, and as a dominant gray eigencolor
in the south polar observation. We construct a simple reflectance model for a snowball Earth. By
construction, our model has a higher Bond albedo than the modern Earth; its surface albedo is so high
that Rayleigh scattering does not noticeably affect its spectrum. The rotational color variations occur
at short wavelengths due to the large contrast between glacier ice and bare land in those wavebands.
Thus we find that both the broadband colors and diurnal color variations of such a planet would be
easily distinguishable from the modern-day Earth, regardless of viewing angle.
Subject headings: Methods: observational, analytical, numerical; Techniques: photometric; Planets
and satellites: individual: Earth
1. INTRODUCTION
Observations of exoplanets will be limited to disc-
integrated measurements for the foreseeable future. This
is true whether a planet can be spatially resolved from
its host star (direct imaging) or not (as in current stud-
ies of short-period transiting planets).
long integration times yield invaluable information about
the spatially and temporally averaged composition and
temperature-pressure profile of the atmosphere.
Time-resolved photometry, on the other hand, tells us
about the weather, climate, and spatial inhomogeneities
of the planet. The time-variability of a planet occurs
Spectra with
1Northwestern University, 2131 Tech Drive, Evanston, IL
60208
Email: n-cowan@northwestern.edu
2CIERA Postdoctoral Fellow
3Astronomy Department & Astrobiology Program, University
of Washington, Box 351580, Seattle, WA 98195
4NASA Astrobiology Institute Member
5NASA Goddard Space Flight Center, Greenbelt, MD 20771
6Department of Astronomy, University of Maryland, College
Park MD 20742
7Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, Cambridge, MA 02138
8Johns Hopkins University Applied Physics Laboratory,
SD/SRE, MP3-E167, 11100 Johns Hopkins Road, Laurel, MD
20723
9Department of Earth, Atmospheric, and Planetary Sciences,
Dept of Physics, Massachusetts Institute of Technology, 77 Mas-
sachusetts Ave. 54-1626, MA 02139
on two timescales, rotational and orbital10, and yields
different information depending on whether it is observed
in reflected or thermal light.
Thermal phases inform us about the diurnal heating
patterns of the planet: the day-side temperature, the
night-side temperature, and the hottest local time on the
planet (Cowan & Agol 2008). Depending on whether it
has an atmosphere, such observations can constrain a
body’s rotation rate, as well as its average Bond albedo,
thermal inertia, emissivity, surface roughness and wind
velocities (Spencer 1990; Cowan & Agol 2011a).
Rotational variations in thermal emission are caused
by inhomogeneities in the planet’s albedo and thermal
inertia. This has been studied for minor solar system
bodies, where it can be used to break the degeneracy
between albedo markings and shape (eg, Lellouch et al.
2000).
Reflected phases are a measure of the disk-integrated
scattering phase function, telling us —for example—
about clouds and oceans on the planet, especially when
combined with polarimetry (Williams & Gaidos 2008;
Mallama 2009; Robinson et al. 2010; Zugger et al. 2010).
Rotational variations at reflected wavelengths can iden-
tify the rotation rate of an unresolved planet (Pall´ e et al.
2008).Once the rotation rate has been determined,
one can constrain the albedo markings on a world
10These are one and the same for synchronously rotating plan-
ets.

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2Cowan et al.
(Russell 1906), indicating surface features like conti-
nents and oceans (Ford et al. 2001; Cowan et al. 2009;
Oakley & Cash 2009; Fujii et al. 2010). Finally, the spa-
tial distribution of landmasses can be inferred, and the
planet’s obliquity can be estimated if diurnal variations
are monitored at a variety of phases (Cowan et al. 2009;
Oakley & Cash 2009; Kawahara & Fujii 2010).
In this paper, we study rotational (a.k.a. diurnal)
variability of Earth’s poles at visible wavelengths. At
these wavelengths, the observed flux consists entirely
of reflected sunlight. Earthshine —the faint illumi-
nation of the dark side of the Moon due to reflected
light from Earth— was first explaied in the early 16th
century by Leonardo Da Vinci, and has been used
more recently to study the reflectance spectrum and
cloud cover variability (Goode et al. 2001; Qiu et al.
2003; Pall´ e et al. 2003, 2004; Monta˜ n´ es-Rodr´ ıguez et al.
2007), vegetation “red edge” signature (Woolf et al.
2002; Monta˜ n´ es-Rodriguez et al.
2005; Hamdani et al. 2006; Monta˜ n´ es-Rodr´ ıguez et al.
2006) and the effects of specular reflection (Woolf et al.
2002; Langford et al. 2009) for limited regions of our
planet. More recently, Pall´ e et al. (2009) measured the
disc-integrated transmission spectrum of Earth by ob-
serving the Moon during a lunar eclipse.
shots of Earth obtained with the Galileo spacecraft have
been used to study our planet from afar (Sagan et al.
1993; Geissler et al. 1995) and numerical models have
been developed to anticipate how diurnal variations in
disc-integrated light could be used to characterize Earth
(Ford et al. 2001; Tinetti et al. 2006a,b; Pall´ e et al.
2008; Williams & Gaidos 2008; Oakley & Cash 2009;
Fujii et al. 2010; Kawahara & Fujii 2010; Zugger et al.
2010; Robinson et al. 2010, Robinson et al. 2011).
In Cowan et al. (2009) we presented two epochs of
rotational variability for disk-integrated Earth seen
equator-on. We performed a principal component anal-
ysis (PCA) of the 7 optical wavebands to identify the
eigencolors of the variability. There were two compo-
nents that accounted for more than 98% of the color
variations seen. The two-dimensionality of the PCA in-
dicated that three major surface types were necessary
to explain the observed variability. The dominant eigen-
color was red, which we identified as being primarily sen-
sitive to cloud-free land. A rotational inversion of the
red eigenprojection yielded a rough map of the major
landforms of Earth: the Americas, Africa-Eurasia and
Oceania, separated by the major oceans: the Atlantic
and Pacific.
One concern with the diurnal light curve inversion
of Cowan et al. (2009) and Oakley & Cash (2009) is
the unknown obliquity of the planet: there is no good
Bayesian prior for the obliquities and rotation rates,
except that they will be slightly biased towards pro-
grade rotation (Schlichting & Sari 2007). Numerical ex-
periments have shown that a pole-on viewing geometry
might complicate retrieval of a planet’s rotational period
(Pall´ e et al. 2008), but once that periodicity is identi-
fied, it is possible to create albedo maps of the planet,
although without knowledge of the planet’s obliquity one
will not know what latitudes those maps correspond to
(Cowan et al. 2009).Idealized numerical experiments
show that —in principle— the obliquity can be extracted
if one observes diurnal variability at a variety of phases
2005;Seager et al.
Brief snap-
(Kawahara & Fujii 2010), but it is not yet clear how well
such a technique would work for a cloudy planet with
unknown surface types.
Finaly, Earth’s climate is sensitive to the latitudinal
configuration of continents and few percent changes in in-
solation (neither of which will be well constrained for ex-
oplanets) leading to bifurcations between temperate and
snowball climates (Voigt et al. 2010). Given an extra-
solar Earth analog, how can we use optical photometry
to distinguish between the two branches of this positive
feedback loop?
In this paper we report and analyze disk-integrated
observations of Earth’s polar regions obtained from the
Deep Impact spacecraft as part of the EPOXI mission of
opportunity. In § 2 we present the observations; in § 3
we discuss the color variability; in § 4 we make longitu-
dinal profiles of these colors; in § 5 we introduce a simple
model for snowball Earth and compare its diurnal color
variations to those of Earth’s polar regions; we summa-
rize our results in § 6 and state the implications of this
study for mission planning in § 7.
2. OBSERVATIONS
The EPOXI11mission reuses the still-functioning
Deep Impact spacecraft that successfully observed comet
9P/Tempel 1. EPOXI science targets include several
transiting exoplanets and Earth en route to a flyby of
comet 103P/Hartley 2.
The first round of EPOXI Earth observations were
taken from a vantage point very near the Earth’s equa-
torial plane (Cowan et al. 2009, Robinson et al. 2010a;
Livengood et al. 2011). In the current paper, we focus
on the later polar observations, summarized in Table 1.
As with the equatorial observations, they were obtained
with Earth near quadrature (phase angle α = π/2), a fa-
vorable phase for directly imaging exoplanets, since the
angular distance between the planet and its host star is
maximized12.
Deep Impact’s 30 cm diameter telescope coupled with
the High Resolution Imager (HRI, Hampton et al. 2005)
recorded images of Earth in seven 100 nm wide optical
wavebands spanning 300–1000 nm, summarized in Ta-
ble 2.
Although the EPOXI images of Earth offer spatial res-
olution of better than 100 km, we mimic the data that
will eventually be available for exoplanets by integrating
the flux over the entire disc of Earth and using only the
hourly EPOXI observations from each of the wavebands,
producing seven light curves for each of the two observing
campaigns, shown in Figures 1 and 2. The photometric
uncertainty in these data is exceedingly small: on the
order of 0.1% relative errors.
Details of the observations and reduction will be pre-
sented in Livengood et al.
aperture photometry, the measured disk-integrated flux
of Earth as seen from the Deep Impact spacecraft has
units of specific flux [W m−2µm−1]. We then apply the
following steps: 1) We multiply by the HRI filter band-
width of 0.1 µm to convert from specific flux to flux [W
(2011).After performing
11The University of Maryland leads the overall EPOXI mission,
including the flyby of comet Hartley 2. NASA Goddard leads the
exoplanet and Earth observations.
12Strictly speaking, this is only true for a planet on a circular
orbit.

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Rotational Variability of Earth’s Polar Regions3
TABLE 1
EPOXI Earth Observations
NameStart of ObservationsStarting
CML1
150◦W
150◦W
152◦W
59◦W
Sub-Observer
Latitude
1.7◦N
0.3◦N
61.7◦N
73.8◦S
Sub-Solar
Latitude
0.6◦S
22.7◦N
2.6◦N
4.3◦S
Dominant
Latitude2
0.5◦N
13◦N
34◦N
39◦S
Phase3
Illuminated Fraction
of Earth Disc3
77%
62%
53%
53%
Earth1: Equinox
Earth5: Solstice
Polar1: North
Polar2: South
2008 Mar 18, UTC 18:18:37
2008 Jun 4, UTC 16:59:08
2009 Mar 27, UTC 16:19:42
2009 Oct 4, UTC 09:37:11
57◦
77◦
87◦
86◦
1The CML is the Central Meridian Longitude, the longitude of the sub-observer point.
2The dominant latitude is that expected to contribute the most photons, assuming a uniform
Lambert sphere.
3The planetary phase, α, is the star–planet–observer angle and is related to the illuminated
fraction by f =1
2(1 + cosα).
TABLE 2
EPOXI Photometric Bandpasses
Waveband
350 nm
450 nm
550 nm
650 nm
750 nm
850 nm
950 nm
Cadence
1 hr
15 min
15 min
15 min
1 hr
15 min
1 hr
Exp. Time
73.4 msec
13.3 msec
8.5 msec
9.5 msec
13.5 msec
26.5 msec
61.5 msec
m−2]. The detailed bandpass shapes are not important,
provided that we use the same 0.1 µm top-hat band-
passes in computing the solar flux in step 3). 2) We
divide the flux observed from the spacecraft by (R⊕/r)2
to obtain the disc-averaged flux from Earth at the top of
the atmosphere, where R⊕is Earth’s radius and r is the
spacecraft range. 3) We use a Kurucz model13for the
solar specific flux at 1 AU, and convert it to flux in each
of the HRI wavebands using the 0.1 µm top-hat band-
passes. 4) Dividing the result of steps 2) and 3) by each
other, we obtain the top-of-the-atmosphere reflectance
of the planet at the observed phase. 5) We further di-
vide the reflectance by the scaled Lambert phase function
(f(α) =
3π(sinα + (π − α)cosα); Russell 1916), thus
obtaining the planet’s apparent albedo, A∗. The precise
definition of A∗is given in Equation 5; for now it is suf-
ficient to think of it as the average albedo of the planet,
weighted by the illumination and visibility of various re-
gions at that moment in time.
Unlike the observations presented in Cowan et al.
(2009), the viewing geometry is sufficiently different for
the two polar observations that we treat them separately.
In particular, a single exoplanet could be observed by
the same stationary observer with the two viewing ge-
ometries of Cowan et al. (2009) since the sub-observer
latitude was essentially equatorial at both epochs (see
Table 1). Although the sub-stellar latitude varies with
phase for non-zero obliquities, the sub-observer latitude
is constant, provided that one can neglect precession (for
more details on viewing geometry, see § 4). By contrast,
the two time series presented in the current paper have
sub-observer latitudes of 62◦N and 74◦S, respectively.
The time-averaged spectra for the EPOXI polar and
equatorial observations are shown in Figure 3, and the
2
13Ideally, one would obtain a spectrum of the planet’s host star
with the same instrument used for imaging the planet, but this was
not possible here for technical reasons.
Fig. 1.— North: Light curves obtained in the seven HRI-VIS
bandpasses by the EPOXI spacecraft when it passed above Earth’s
equatorial plane on March 27, 2009. The bottom-right panel shows
changes in the bolometric albedo of Earth. The sub-observer lon-
gitude at the start of the observations is 152◦W, the sub-observer
latitude is 61.7◦N throughout. The relative peak-to-trough vari-
ability ranges from 16% (450 nm) to 10% (750 nm).
Fig. 2.— South: Light curves obtained in the seven HRI-VIS
bandpasses by the EPOXI spacecraft when it passed below Earth’s
equatorial plane on October 4, 2009. The bottom-right panel shows
changes in the bolometric albedo of Earth. The sub-observer lon-
gitude at the start of the observations is 59◦W, the sub-observer
latitude is 73.8◦S throughout. The relative peak-to-trough vari-
ability ranges from 11% (350 nm) to 20% (650 nm).
apparent albedo values are listed in Table 3. The liter-
ature abounds with vaguely-defined “reflectance” mea-
surements: as a result of differing definitions of re-
flectance used by different EPOXI team members, the

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4Cowan et al.
TABLE 3
EPOXI Earth Observations: Apparent Albedos
Name
Earth1: Equinox
Earth5: Solstice
Polar1: North
Polar2: South
350 nm
0.482 (0.016)
0.462 (0.021)
0.575 (0.028)
0.590 (0.022)
450 nm
0.352 (0.014)
0.346 (0.017)
0.464 (0.023)
0.475 (0.022)
550 nm
0.268 (0.010)
0.268 (0.011)
0.373 (0.015)
0.383 (0.023)
650 nm
0.256 (0.010)
0.257 (0.009)
0.368 (0.012)
0.372 (0.023)
750 nm
0.264 (0.014)
0.270 (0.014)
0.388 (0.012)
0.382 (0.022)
850 nm
0.290 (0.017)
0.298 (0.019)
0.425 (0.013)
0.414 (0.023)
950 nm
0.230 (0.015)
0.230 (0.017)
0.355 (0.012)
0.351 (0.017)
The number in parentheses represents the root-mean-squared (RMS) time-variability of the ap-
parent albedo in that waveband. For example, the Earth1 350 nm time series had a mean apparent
albedo of 0.482 and an RMS variability of 0.016, or 3%.
Fig. 3.— 24-hour average broadband spectra of Earth as seen
from the Deep Impact spacecraft as part of the EPOXI mission.
albedos reported in Cowan et al. (2009) were about 2/3
of the correct value. (Note that this uniform offset had
no impact on the color variations and analysis presented
in that paper.)
We present the corrected values of apparent albedo in
Table 3 and Figure 3. The major features of the time-
averaged broadband albedo spectrum of Earth are: 1) a
blue ramp shortward of 550 nm due to Rayleigh scatter-
ing, 2) a slight rise in albedo towards longer wavelengths
due to continents, and 3) a steep dip in albedo at 950 nm
due to water vapor absorption. Significantly, apart from
a 20-30% uniform offset, the polar and equatorial obser-
vations have indistinguishable albedo spectra.
3. DETERMINING PRINCIPAL COLORS
As in Cowan et al. (2009) we assume no prior knowl-
edge of the different surface types of the unresolved
planet. Our data consist of 25 broadband spectra of
Earth for each of two viewing geometries. For the equa-
torial observations (Cowan et al. 2009), we found sub-
stantial variability in all wavebands (though the near-IR
wavebands exhibited the most variability, leading to the
dominant red eigencolor). The polar observations also
show variability at all wavebands (Table 2), but as we
argue below, the intrinsic cause of this variability is not
necessarily the same surface types rotating in and out of
view.
The multiband, time-resolved observations of Earth
can be thought of as a locus of points occupying a 7-
dimensional parameter space (one for each waveband).
Principal component analysis (PCA) allows us to reduce
the dimensionality of these data by defining orthonormal
eigenvectors in color space (a.k.a. eigencolors). Quanti-
tatively, the observed spectrum of Earth at some time t
can be recovered using the equation:
A∗(t,λ) = ?A∗(t,λ)? +
7
?
i=1
Ci(t)Ai(λ),(1)
where ?A∗(t,λ)? is the time-averaged spectrum of Earth,
Ai(λ) are the seven orthonormal eigencolors, and Ci(t)
are the instantaneous projections of Earth’s colors on
the eigencolors. The terms in the sum are ranked by the
time-variance in Ci, from largest to smallest.
Insofar as the color variations are dominated by the
first few terms of the sum, the locus does not occupy the
full 7-dimensional color space, but a more restricted man-
ifold. The dimensionality of the manifold is one fewer
than the number of surface types rotating in and out of
view. E.g., a two-dimensional locus (a planar manifold)
requires three surface types; a three-dimensional locus
requires four surface types, etc. The general problem of
estimating the pure surface spectra based on the mor-
phology of such a locus of points is beyond the scope of
this paper. It is a form of spectral unmixing, and is an
area of active research in the remote sensing community
(e.g., Le Mou´ elic et al. 2009).
In practice, there are two different ways to perform
PCA, which may give quantitatively different results.
The analysis can be run using the covariance of the data,
Cov(X,Y ) = E[(X − E[X])(Y − E[Y ])], where E[X] is
the expected value of X; or it can be run using the cor-
relation of the data, Corr(X,Y ) = Cov(X,Y )/(σXσY),
where σX is the standard deviation of X. The correla-
tion matrix is a standardized version of the covariance
matrix; this is useful when the measured data do not all
have the same units, since division by the standard de-
viation renders them unitless. When the data are unit-
less to begin with, as is the case for our albedo mea-
surements, running covariance-PCA is preferable (e.g.,
Borgognone et al. 2001). In Cowan et al. (2009) we used
covariance-PCA, and we continue to do so here14.
3.1. Testing PCA on Simulated Data
Although PCA is a mathematically and numerically
robust technique for analyzing patterns in data, inter-
preting its results can be ambiguous. In particular, we
would like to verify to what extent there is a one-to-one
correspondence between the eigencolors output by the
PCA and real surfaces on Earth. To this end, we test the
PCA routine on a suite of simulated data produced by
14For this paper, we use the Interactive Data Language (IDL)
routine PCOMP (with the /COVARIANCE keyword set) to per-
form principal component analysis, while in Cowan et al. (2009)
we used the IDL routine SVDC, which performs singular value de-
composition.

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Rotational Variability of Earth’s Polar Regions5
the Virtual Planetary Laboratory’s validated 3D spec-
tral Earth model (details can be found in Robinson et
al. 2011). The simulations used here were designed to
closely mimic the Earth1 EPOXI observations taken in
March 2008.
We run five different versions of the VPL 3D Earth
model: 1) Standard: this model is an excellent fit to
the EPOXI Earth1 observations; the remaining models
are identical, but in each case a single model element
has been “turned off”: 2) Cloud Free; 3) No Rayleigh
Scattering; 4) Black Oceans; 5) Black Land. We show
the results of this experiment in Appendix I; here we
simply state our conclusions:
1) PCA successfully determines the dimensionality of
the color variability and therefore the minimum number
of different surface types contributing to color variations.
In particular, n-dimensional variations require n+1 sur-
face types (N.B. we count clouds as a surface type).
2) Rayleigh scattering is important in determining the
time-averaged broadband colors of Earth, but does not
significantly affect its rotational color variability.
3) Cloud-free land surfaces, which are red, contribute
a red eigencolor to the diurnal variability. The presence
of relatively cloud-free land (deserts) near the equator
explains why the rotational map of the red eigencolor
(Figure 10 in Cowan et al. 2009) successfully identified
the major landforms and bodies of water on Earth.
4) Oceans are essentially a null surface, contributing
neither to the broadband colors of Earth, nor to the time-
variability of those colors, except insofar as the presence
of oceans corresponds to a shortage of land.
5) In the absence of land, the variability is gray, due
to large-scale inhomogeneities in cloud cover.
6) PCA necessarily outputs orthogonal eigencolors and
a good deal of Earth’s variability is due to clouds. There-
fore, if the first eigencolor is red, then the second eigen-
color may be blue even if there is no blue surface rotating
in and out of view; this is an improvement on the inter-
pretation of Cowan et al. (2009).
3.2. Results of PCA for Polar Observations
In Figures 4 & 5 we show the eigenvalue spectra for
time-variations in the 7 eigencolors identified by the PCA
of the EPOXI polar observations. The eigenvalue for a
given component is the projection of the data’s variance
onto that eigenvector; we plot here the square root of the
eigenvalues, which is a measure of the RMS variability of
the data projected onto an eigenvector. The variability
has been normalized in the figures such that the sum
of the variability for all seven components is unity. By
definition, the low-order principal components have the
largest variance.
For the North observation, there are two eigencolors
that dominate the color variations of Earth: the third
eigencolor contributes only ∼ 4% of the planet’s color
variability. As in Cowan et al. (2009), this means that
the colors of Earth populate a two-dimensional plane
rather than filling the entire seven-dimensional color-
space, and this requires at least three surface types. The
southern observation, on the other hand, is dominated by
a single eigencolor (the second eigencolor contributes to
variability at the < 10% level). This means that —for the
24 hours of observations— the colors of the planet pop-
ulated a one-dimensional line in the seven-dimensional
color volume, requiring only two surface types.
Fig. 4.— North Normalized variability in the 7 eigencolors of
Earth’s North polar regions, based on EPOXI observations taken
on March 27, 2009. The color variations of Earth during these
observations are well described as a combination of components 1
& 2.
Fig. 5.— South Normalized variability in the 7 eigencolors of
Earth’s South polar regions, based on EPOXI observations taken
on October 4, 2009. The color variations of Earth during these
observations are well described by component 1.
The eigencolors (the Ai(λ) from Equation 1) are
shown in Figures 6 & 7. The raw eigencolors are —by
definition— orthogonal and normalized (?7
1), and this is how we presented them in Cowan et al.
(2009). Here we have instead scaled the eigencolors by
their associated eigenvalues (?7
is the eigenvalue of the i’th component), so the dominant
components exhibit larger excursions from zero.
The eigenprojections (Ci(t) from Equation 1) are
shown in Figures 8 & 9. The standard deviation of an
eigenprojection corresponds to the variance or eigenvalue
of that component. By definition, the low-order eigen-
projections have the largest deviations from 0. Note that
in Cowan et al. (2009) we instead plotted the normalized
eigenprojections (?25
ier to compare the shapes of the eigenprojections but
masked their relative importance.
j=1A2
i(λj) =
j=1A2
i(λj) = vi, where vi
k=1Ci(tk)2= 1), which made it eas-

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6Cowan et al.
The North polar observations are dominated by two
eigencolors. At first glance, the two eigencolors are iden-
tical, only offset in the vertical direction, but they are
(by construction) orthogonal. The more important of
the two is blue, in that it is most non-zero at short wave-
lengths and nearly independent of what is going on at
long wavelengths; the second eigencolor is red: it is most
non-zero at long wavelengths and is largely insensitive to
variability in blue wavebands. Based on the findings pre-
sented in § 3.1, we may infer that clouds and continents
are rotating in and out of view as seen from this vantage
point. Furthermore, cloud-related variability appears to
be more important here than it was for the equatorial
observations, which had a dominant red eigenvector fol-
lowed by a blue, rather than vice-versa.
The South polar observations are dominated by a sin-
gle, gray eigencolor. Snow and clouds both have gray
optical albedo spectra, so either may be contributing to
the photometric variability. The absence of an impor-
tant red eigencolor is due to the relative dearth of con-
tinents in the southern hemisphere. The second eigen-
color is two orders of magnitude down in variance, or one
order of magnitude in variability. It indicates that red
and blue surfaces are trading places as the world turns
(A2(λ) is positive at short wavelengths, negative at long
wavelengths, and zero in between), but the forced or-
thogonality of the eigencolors makes this interpretation
ambiguous.
Fig. 6.— North Spectra for the eigencolors of northern Earth,
as determined by PCA. The two dominant eigencolors are the bold
solid and dotted lines.The eigenspectra have been normalized
by their eigenvalues, so the dominant components exhibit larger
excursions from zero.
4. ROTATIONAL MAPPING
In this section we address how to infer the longitudinal
color inhomogeneities of the unresolved planet based on
time-resolved photometry. Note that this is in principle
an independent question from that of identifying surface
types on the planet (§ 3). One could try to infer the sur-
face types on a planet without knowing or caring about
their spatial distribution; or one could simply produce
longitudinal color maps while remaining agnostic about
what these tell us about surfaces (where “surface” here
includes clouds). In practice, however, the two are inti-
mately tied: a planet only exhibits rotational variability
Fig. 7.— South Spectra for the eigencolors of southern Earth,
as determined by PCA. The two dominant eigencolors are the bold
solid and dotted lines.The eigenspectra have been normalized
by their eigenvalues, so the dominant components exhibit larger
excursions from zero.
Fig. 8.— North Contributions of northern Earth’s eigencolors,
as determined by PCA, relative to the average Earth spectrum.
The observations span a full rotation of the planet, starting and
ending with the spacecraft directly above 152◦W longitude, the
North Pacific Ocean.
Fig. 9.— South Contributions of southern Earth’s eigencolors,
as determined by PCA, relative to the average Earth spectrum.
The observations span a full rotation of the planet, starting and
ending with the spacecraft directly above 59◦W longitude, in the
South Atlantic Ocean.

Page 7

Rotational Variability of Earth’s Polar Regions7
if it has a variegated surface and substantial spatial in-
homogeneities in the distribution of these surfaces.
4.1. Cloud Variability
As in Cowan et al. (2009), we wish to estimate disk-
integrated cloud variability, as this imposes a limit on
the accuracy of any rotational maps we create15. Af-
ter 24 hours of rotation the same hemisphere of Earth
should be facing the Deep Impact spacecraft, so the inte-
grated brightness of the planet’s surface should be nearly
identical, provided one has accounted for the difference
between the sidereal and solar day, as well as changes in
the geocentric distance of the spacecraft and in the phase
of the planet as seen from the spacecraft. Even after
correcting for all known geometric effects, the observed
fluxes at the start and end of our observing campaigns
differ by ∆A∗/?A∗? =3–6% and 0.4–1% for the North
and South polar observing campaigns, respectively. We
attribute this discrepancy to diurnal changes in cloud
cover.
Our 24-hour polar observation cloud variability of 4%
and 1%, respectively is comparable to our estimate of
cloud variability from previous EPOXI Earth observa-
tions (2% and 3%, Cowan et al. 2009) and somewhat
smaller than estimates from Earthshine observations.
For example, Goode et al. (2001) and Pall´ e et al. (2004)
found day-to-day cloud variations of roughly 5% and
10%, respectively. Although we are still very much in
the realm of small number statistics, it is conceivable that
Earthshine observations over-estimate diurnal changes in
cloud cover: under-estimating night-to-night calibration
errors would lead to over-estimating day-to-day cloud
variability.
Depending on the size of the telescope and cloud me-
teorology for a given planet, either photon counting or
cloud variability can dominate the error budget for rota-
tional inversion. For the purposes of rotational mapping,
we adopt effective Gaussian errors in the apparent albedo
of σA∗ = 0.01, comparable to the actual day-to-day cloud
variability on Earth.
4.2. Normalized Weight and the Dominant Latitude
Using the formalism of Cowan et al. (2009), the visibil-
ity and illumination of a region on the planet at time t are
denoted by V (θ,φ,t) and I(θ,φ,t), respectively, where θ
is the latitude and φ is the longitude on the planet’s sur-
face. V is symmetric about the line-of-sight, is unity at
the sub-observer point, drops as the cosine of the angle
from the observer and is null on the far side of the planet
from the observer; I is symmetric about the star–planet
line, is unity at the sub-stellar point, drops as the cosine
of the angle from the star and is null on the night-side of
the planet.
Following Fujii et al. (2010) and Kawahara & Fujii
(2010), we define the normalized weight,
W(θ,φ,t) =
V (θ,φ,t)I(θ,φ,t)
?V (θ,φ,t)I(θ,φ,t)dΩ, (2)
which quantifies which regions of the planet are con-
tributing the most to the observed light curve, under
15Note that diurnal cloud variability is not necessarily an ob-
stacle for PCA, since that analysis is not predicated on a periodic
signal.
the assumption of diffuse (Lambertian) reflection.
can be thought of as the smoothing kernel for the convo-
lution between an albedo map, A(θ,φ) and an observed
lightcurve, A∗(t).
To first order, the character of a lightcurve can be un-
derstood in terms of the shape and location of the weight
function. The normalization (denominator) of the weight
is a simple function of phase:
W
?
V (θ,φ,t)I(θ,φ,t)dΩ =2
3[sinα + (π − α)cosα]. (3)
As the planet rotates, W sweeps from East to West.
The width (in longitude) of the weight determines the
longitudinal resolution achievable by inverting diurnal
light curves: a broad W at full phase leads to a coarser
map than the slender W of a crescent phase. This of
course neglects the practical issues of inner working an-
gles and photon counting noise.
The peak of W lies half-way between the sub-stellar
and sub-observer points and corresponds to the location
of the glint spot. The latitude of the glint spot may
change throughout an orbit: the sub-observer latitude
is fixed in the absence of precession, but the sub-solar
latitude exhibits seasonal changes for non-zero obliquity.
The peak of the weight is the area of the planet that
contributes the most to the observed disc-integrated light
curve. E.g., a polar sub-observer latitude and an equa-
torial sub-stellar latitude would yield a weight with a
maximum at mid-latitudes. In detail, W is also tem-
pered by the usual sinθ dependence of dΩ (ie: there is
more area near the equator than near the poles). The
dominant latitude (the latitude where the most photons
would originate from in the case of a uniform Lambert
sphere) is therefore not simply the peak of W, but is
rather the average θ, weighted by W:
θdom(t) =
?
W(θ,φ,t)θdΩ =
?V IθdΩ
?V IdΩ.(4)
In Table 1 we list the dominant latitude for the four
EPOXI observations. Significantly, the dominant lati-
tude is temperate for the “polar” observations, despite
the exotic viewing geometry. This simple argument ex-
plains why the time-averaged colors of the polar EPOXI
Earth observations are so familiar: most of the pho-
tons will not originate from the snowy and icy regions
of Earth.
That being said, the mid-latitudes probed by the polar
observations are significantly more cloudy than the trop-
ics (yearly mean cloud cover in the tropics is 25–50%,
while at 45◦S cloud cover is 75–100%; see Figure 6a of
Pall´ e et al. 2008). As shown in Appendix I, clouds con-
tribute a uniform (gray) increase in albedo of 20–50%
between the cloud-free and standard VPL models, so a
latitudinal difference in cloud cover is a natural explana-
tion for the observed difference in albedo.
Furthermore, polar snow and ice necessarily contribute
more to the polar than the equatorial EPOXI observa-
tions. For example, the Earth1 and Polar1 EPOXI ob-
servations were both obtained at the same time of year
(March 2008 and 2009, respectively), so we may mean-
ingfully ask how the different viewing geometries affect
the contribution from snowy regions. If the global-mean
snowline for March lies at 55◦N, we find that 2% of the

Page 8

8Cowan et al.
weight is in snow-covered regions for the equatorial ob-
servation, while this fraction is 16% for the polar obser-
vation. In general, for global mean snow lines between
50–60◦N, the snow-covered regions contribute 7–9 times
more to the polar observation than to the equatorial ob-
servation. But such an argument is unlikely to work for
the Polar2 observation, which probes the relatively land-
free southern oceans, and the time-averaged Polar1 and
Polar2 broadband colors are indistinguishable. We there-
fore believe the increased weight of clouds to be the main
source of the 20–30% greater absolute value of A∗in the
polar observations compared to the equatorial observa-
tions.
Fig. 10.— Top: Land coverage map for modern Earth. The col-
ored lines indicate important latitudes for the EPOXI North (blue)
and South (red) polar observations. The dotted lines show the sub-
solar latitudes; the dashed lines show the sub-observer latitudes;
the solid lines show the dominant latitudes, which are expected to
contribute the most to the lightcurves. Bottom: The longitudinal
land coverage profiles for the EPOXI Polar observations.
In the top panel of Figure 10, we show a map of
land coverage on Earth and indicate the sub-solar, sub-
observer, and dominant latitudes for the North and
South polar observations. The bottom panel of the figure
shows the longitudinal land fraction profiles for the two
polar observations, obtained by integrating the 2-D map
by the weight function dictated by viewing geometry. It
indicates the location of the major landforms probed by
the polar observations and can be compared to the longi-
tudinal profiles of eigencolors presented in the following
section.
4.3. Lightcurve Inversion
The flux ratio primarily depends on the planet’s or-
bital phase, the observer–planet–star angle, and the ra-
tio (Rp/a)2. We define the apparent albedo, A∗, as the
ratio of the flux from the planet divided by the flux we
would expect at the same phase for a perfectly reflecting
Lambert sphere (see also Qiu et al. 2003). This amounts
to the average albedo of the planet, weighted by W:
A∗(t,λ) =
?
W(θ,φ,t)A(θ,φ,λ)dΩ =
?V IAdΩ
?V IdΩ.(5)
A uniform planet would have an apparent albedo that
is constant over a planetary rotation in the Prot≪ Porb
limit; a uniform Lambert sphere would further have a
constant apparent albedo during the entire orbit. For
non-transiting exoplanets, the planetary radius may be
unknown, in which case A∗can only be determined to
within a factor of R2
p, with a lower limit on the radius
obtained by setting A∗= 1.
Lightcurve inversion means inferring A(θ,φ,λ) from
A∗(t,λ). If observations only span a single rotation, or
if a planet has zero obliquity, one can only constrain the
longitudinal variations in albedo.
The visibility, V (θ,φ,t), and illumination, I(θ,φ,t),
can be expressed compactly in terms of the locations of
the sub-observer and sub-stellar points:
V (θ,φ,t) = max[sinθsinθobscos(φ − φobs)
+cosθcosθobs,0]
I(θ,φ,t) = max[sinθsinθstarcos(φ − φstar)
+cosθcosθstar,0],
(6)
where φobs(t) = φobs(0)−ωrott is the sub-observer longi-
tude, θobsis the constant sub-observer latitude, φstar(t)
and θstar(t) = arccos[cos(ξ0+ωorbt−ξobl)sinθobl] are the
sub-stellar longitude and latitude; ωrotand ωorbare the
rotational and orbital angular velocities of the planet, ξ0
is the initial orbital position of the planet, ξoblis the or-
bital location of northern summer solstice, and θoblis the
planet’s obliquity. It is non-trivial to compute φstar(t)
over a sizable fraction of an orbit, requiring a numerical
integration or use of the equation of time.
For the current application, however, the planet’s rota-
tion period is much shorter than its orbital period, so it
is sufficient to assume that θstaris constant and φstar(t)
advances linearly at one revolution per solar day (as op-
posed to the sidereal day used in computing φobs(t)). We
use Horizons16to compute the relative positions of the
Deep Impact spacecraft, Earth and the Sun at the start
of the various EPOXI campaigns.
Both of the EPOXI polar observations were obtained
with a viewing geometry very close to quadrature17. The
weight function therefore has a width of 90◦in longitude,
indicating that we would need a model with 8 longitudi-
nal slices of uniform albedo to achieve Nyquist sampling
of the rotational lightcurve, in the absence of specular
reflection (for more discussion on slice vs sinusoidal lon-
gitudinal profiles see Cowan & Agol 2008; Cowan et al.
2009). An 8-slice model with variable phase offset (prime
meridian) would have 9 model parameters; we instead use
sinusoidal maps with terms up to fourth order, which
also have 9 model parameters but which converge bet-
ter (Cowan & Agol 2008). The best-fit reduced χ2of
these models are somewhat lower than unity because the
σA∗ = 0.01 “error bars” that we use are much larger than
the point-to-point scatter in the lightcurves.
We estimate uncertainties in our longitudinal eigen-
color maps with a Monte Carlo test. Using our adopted
photometric error of σA∗ = 0.01, we generate 10,000 sta-
tistically equivalent instances of the observed lightcurves
assuming Gaussian, uncorrelated errors.
same PCA and lightcurve inversion on each of these mock
We run the
16http://ssd.jpl.nasa.gov/horizons.cgi
17For small obliquities, polar observations imply that the planet
is in a nearly face-on orbit, and therefore permanently at quadra-
ture. But there is no reason to assume low obliquity for terrestrial
planets, so polar observations need not be made at quadrature.

Page 9

Rotational Variability of Earth’s Polar Regions9
data sets and take the standard deviation in the resulting
maps to be the uncertainty in our fiducial maps.
Note that the rotational inversion may be performed
directly on the lightcurves shown in Figures 1 & 2, and
independently of the PCA described in § 3, yielding
albedo maps of Earth in various wavebands. Instead,
we combine PCA and lightcurve inversion as we did in
Cowan et al. (2009) and produce longitudinal maps of
the dominant eigencolors, shown in Figures 11 & 12.
Based on the numerical experiments of § 3.1, we ex-
pect both dominant eigencolors in Figure 11 to be track-
ing clouds and snow-covered land, while the red eigen-
color is also sensitive to cloud-free land. Since clouds
are more prevalent at these latitudes, the red eigencolor
does not faithfully locate the major landforms, as it did
in Cowan et al. (2009). The southern polar observation
(Figure 12) shows a broad maximum in the gray eigen-
color at a longitude of 90◦W, roughly corresponding to
the location of Patagonia and the Antarctic Peninsula.
Since snow-covered land is essentially indistinguishable
from clouds at these poor spectral resolutions, we must
remain agnostic about the source of this variability.
Fig. 11.— North: Longitudinal profiles of the two dominant
eigencolors of Earth based on the lightcurves in Figure 1, the eigen-
colors shown in Figure 6, and the known phase and rotational pe-
riod of Earth.
In general, the cloudy mid-latitudes keeps the eigen-
color maps from faithfully identifying major landforms
(eg, compare Figures 11 and 12 to Figure 10).
5. ALBEDO MODEL OF SNOWBALL EARTH
Since the polar regions of Earth are largely covered in
snow and ice, it is worth asking if one might confuse a
habitable planet like Earth with a snowball planet (ie:
one caught in the cold branch of a snow-albedo posi-
tive feedback loop. See, for example, Tajika 2008, and
references therein).In this section we describe a toy
model for the reflectance of such a snowball planet, and
compare the resulting photometry to the EPOXI polar
observations. Note that V´ azquez et al. (2006) presented
bolometric (white light) diurnal lightcurves for a model
snowball Earth, but these are not useful for the current
comparison.
5.1. Geography
Fig. 12.— South: Longitudinal profiles of the dominant eigen-
color of Earth based on the lightcurves in Figure 2, the eigencolors
shown in Figure 7, and the known phase and rotational period of
Earth.
The geography of our snowball Earth model is shown
in Figure 13. We use the same idealized paleogeography
for the Sturtian glaciation (∼750 Mya) as Pierrehumbert
(2005). Paleomagnetism only constrains the magnetic
latitude of continents, however, and we are at liberty to
choose any longitudinal distribution. The diurnal vari-
ability of the planet is determined solely by its longitudi-
nal geography, however. If the continents are spread out
uniformly in longitude, for example, the planet would
not exhibit any rotational variability in the absence of
heterogeneous cloud cover. We instead adopt the oppo-
site limit of a single mega-continent. This will tend to
exaggerate the amplitude of the diurnal variations in ap-
parent albedo, but the changes in color should be robust.
Assuming that sea-level was not grossly different
from today, and that only trace amounts of continent-
formation has occurred in the intervening 750 Myrs, con-
tinents should cover ∼ 25% of the planet, as today. At
first sight, assuming constant water levels during a global
glaciation seems inconsistent. However, it is only icecaps
(ice on land) that significantly change water levels, while
the geological evidence for snowball Earth episodes in-
stead require that the oceans be frozen at the equator.
In fact, this criterion effectively shuts down the planet’s
hydrological cycle, so the polar ice caps are not very dif-
ferent from the present day. In any case, the high lat-
itudes of a snowball Earth should be covered in snow,
regardless of whether the underlying regions are conti-
nent or ocean. Therefore, the precise fraction of land
vs ocean does not directly impact the observed diurnal
variability.
For a snowball planet, we assume that both oceans and
continents are covered in snow at latitudes greater than
30◦. Closer to the equator, we assume that continents are
bare, dry land due to the very low precipitation; while
oceans are covered in blue glacier ice that flows towards
the equator (Goodman & Pierrehumbert 2003).
5.2. Albedo Spectra
Climate models of Snowball Earth are concerned with
albedo only insofar as it modulates the energy budget
of the planet. That is to say, they care about the Bond
albedo, AB, the fraction of incident energy that is re-
flected back into space. We, on the other hand, are con-

Page 10

10Cowan et al.
Fig. 13.— The layout of surface types in our Snowball Earth
model.Regions within 60◦of the poles —both continents and
oceans— are covered in snow. The tropics are dry (bare land or
glacier ice) due to negative net precipitation.
cerned with the appearance of the planet as seen from
the outside, A∗(t,λ). It is beyond the scope of this pa-
per to make a detailed snowball earth spectral model.
We therefore assume diffuse (a.k.a. Lambert) reflection
and use a wavelength-dependent albedo, A(λ), that is
simply a function of location rather than a bidirectional
reflectance distribution function. A hard Snowball Earth
will not have appreciable expanses of liquid water to con-
tribute to glint. That being said, other elements, notably
clouds, are not strictly Lambertian (eg, Robinson et al.
2010).
For a cloud-free and airless model planet, the albedo
spectrum at each point on the planet is simply deter-
mined by the albedo spectrum of the surface type at that
location. Reflection from clouds and Rayleigh scattering
from air molecules complicates this picture, however. We
treat the albedo from these semi-transparent media as
follows:
A(λ) = 1−(1−ARa(λ))(1−Acl(λ))(1 −Asurf(λ)), (7)
where ARais the effective albedo due to Rayleigh scat-
tering, Aclis the albedo due to clouds, and Asurf is the
albedo of the surface type at that point on the planet.
This simple expression captures the essential behavior
of clouds and Rayleigh scattering: they always increase
the effective albedo of a region, but the effect is most
pronounced for a dark underlying surface.
Our model has 5 elements, each with a distinctive
albedo spectrum, shown in Figure 14. We use spectra
for dry land and snow from Robinson et al.
The snow albedo spectrum we use is for medium grained
snow, while the cold, dry climate of a snowball Earth
would create small-grained snow, as seen in Antarc-
tica (Hudson et al. 2006). There is no perceptible dif-
ference in the broadband albedo spectra of these two
kinds of snow at optical wavelengths, however. We use
the empirical albedo spectrum for blue glacier ice from
Warren et al. (2002). To mimic the thin clouds expected
on a frozen planet with reduced hydrological activity,
we take a generic cloud spectrum (Robinson 2009, priv.
comm.)and divide the albedo by 2.
clouds increases the Bond albedo of our model planet
but does not significantly change the color variability.)
(2011).
(Using thicker
Fig. 14.— The broadband optical albedo spectra for the com-
ponents of our snowball Earth model. The Bond albedos of the
surfaces are estimated using a Solar spectrum and integrating to
5 µm (snow: 0.8; glacier ice: 0.6; thin cloud: 0.3, dry land: 0.3,
Rayleigh scattering: < 0.1).
We distribute the clouds on the planet using a snapshot
of cloud maps from a snowball Earth general circula-
tion model (Abbot & Pierrehumbert 2010). Note that
this model was run using the same idealized geography
shown in Figure 12 and thus offers a good estimate of
the spatial —and in particular longitudinal— variations
in cloud cover. We estimate the disc-integrated effect
of Rayleigh scattering by comparing the Standard and
Rayleigh-Scattering-Free VPL models (described in Ap-
pendix I) and using Equation 7.
Fig. 15.— Time-averaged albedo spectrum of Snowball Earth.
The vertical bars show the RMS variability in each band. Rela-
tive peak-to-trough variability ranges from 42% (450 nm) to 6%
(950 nm).
Our snowball Earth model has a time-averaged Bond
albedo of 70%, which is self-consistent with the snow-
albedo feedback18:Wetherald & Manabe (1975) used
a 70% albedo to induce their “White Earth” solution
in a 2-D model.Pierrehumbert (2005) ran GCMs of
a hard snowball Earth and maintained the snowball
18Despite its name, a snowball planet’s albedo does not equal
that of snow (80%). The cold, dry atmosphere keeps the tropical
land bare (AB= 30%) and exposes glacier ice (AB= 60%) near
the equator.

Page 11

Rotational Variability of Earth’s Polar Regions11
state with albedos of 60–67%. Chandler & Sohl (2000),
on the other hand, ran snowball Earth global climate
models (GCMs) with bolometric albedos of 20–40%;
V´ azquez et al. (2006) have an average Bond albedo of
approximately 50% for the frozen Earth.
5.3. Time Variability
For directly-imaged exoplanets, the albedo cannot be
determined independently of the planet’s radius: pho-
tometry of reflected light will constrain the quantity
AR2
p. Therefore, while the general agreement in ABbe-
tween our toy model and self-consistent simulations is en-
couraging, one cannot in general use the absolute albedo
of an exoplanet as a diagnostic. Multiband observations
will tell us about the colors of the planet, however, and
with sufficiently high cadence observations it will be pos-
sible to measure the variations in apparent albedo due to
the planet’s rotation.
We compute the variations in apparent albedo for an
equatorial observer and a planet at quadrature.
time-averaged spectrum of our snowball Earth model
(Figure 15) is completely different from that of the mod-
ern Earth, regardless of viewing geometry (Figure 3).
The flat spectrum shortward of 650 nm is due to snow
and glacier ice, which are so reflective at these wave-
lengths as to make Rayleigh scattering imperceptible.
The drop in albedo at longer wavelengths is also driven
by snow and glacier ice.
For our snowball model, the shortest wavelengths ex-
hibit the most variability, as shown by the vertical bars
in Figure 15. This is because bare land and glacier ice
exhibit the largest contrast at short wavelengths, while
at longer wavelengths they both have near-IR albedos of
∼40%. This is in stark contrast to the case for the mod-
ern Earth, which exhibits variability at all wavebands.
We conclude that —given high-quality photometry— the
modern-day Earth could not be mistaken for a snowball
planet, regardless of the viewing geometry.
The
6. SUMMARY
We presented time-resolved, disc-integrated observa-
tions of Earth’s polar regions from the Deep Impact
spacecraft as part of the EPOXI Mission of Opportu-
nity. These complement the equatorial views presented
in Cowan et al. (2009). We found that both of the polar
observations have broadband colors similar to the equa-
torial Earth, but with uniformly higher albedos.
explained this in terms of the 2-D weight function for
disc-integrated observations of Earth, which was most
sensitive to the tropics for the equatorial observations,
and most sensitive to mid-latitudes for the polar obser-
vations.
We performed PCA on a suite of simulated rotational
multi-band lightcurves from NASA’s Virtual Planetary
Laboratory 3D Earth model. We found that PCA cor-
rectly indicates the number of different surfaces rotating
in and out of view. We found that while the red eigen-
color consistently tracks cloud-free land, a blue eigencolor
only tracks oceans when clouds are entirely absent from
the simulation. In the general (cloudy) case, a blue eigen-
color is simply tracking cloud inhomogeneities.
eigencolors, when they are present, track large cloud pat-
terns and/or snow-covered land.
We
Gray
We also performed PCA on the EPOXI polar obser-
vations. Comparing these eigencolors to known surface
types on Earth, we establish that the variability seen in
the North EPOXI Polar observation is due to clouds,
continents and oceans rotating in and out of view; the
lack of large cloud-free land (ie: deserts) at the latitudes
probed by these observations keep us from being able to
faithfully extract the positions of major northern land-
forms. The South polar observation, on the other hand,
was characterized by gray variability due to a large cloud
pattern in the south oceans.
Lastly, we constructed a simple reflectance model
for a snowball Earth, and found that both the time-
averagedbroadband colors and diurnal color variations of
a Snowball Earth (gray snow + near-IR roll-off; variabil-
ity at blue wavelengths) would be distinguishable from
the modern-day Earth (near-UV Rayleigh ramp + gray
clouds; variability at all wavelengths), regardless of view-
ing angle.
7. DISCUSSION
We listed the possible constraints one could get from
time-resolved photometry in Section 1, here we briefly
review the three forms of photometric characterization
used in this paper in the context of measurements from
planned space telescopes.
We have considered time-resolved, multi-band optical
photometry, which could be obtained with a space-based
high-contrast imaging mission. On its own, such a tele-
scope could discover nearby exoplanets, determine their
approximate orbits, and characterize their time-averaged
colors and time-variability of colors.
The time-averaged colors of a planet will be the most
accessible observable. We have shown in this paper that
the broadband colors would be sufficient to distinguish
between the modern day Earth and a snowball Earth. On
modern-day Earth, polar ice and snow don’t contribute
significantly to the time-averaged albedo of Earth —even
with polar viewing geometries— because of the glancing
angle of sunlight at those latitudes. The coldest regions
of a planet receive the least sunlight, and therefore con-
tribute correspondingly little to the disk-integrated prop-
erties of the planet.
The time-variability of the colors are harder to obtain,
requiring shorter integration times and therefore a larger
telescope, all other things being equal.
hand, variability measurements are more robust to con-
tamination from exo-zodiacal light. We have shown that
modern Earth —regardless of viewing angle— exhibits
photometric variability at all wavelengths (RMS variabil-
ity within a factor of 2 for all wavebands), while Snowball
Earth varies 7 times more at short wavelengths than at
long wavelengths. A more subtle analysis of the time
variability may even allow us to distinguish between the
equatorial and polar viewing geometries of Earth, be-
cause clouds play a larger role at mid-latitudes. From an
equatorial vantage point, the dominant eigencolor is red,
followed by blue; for the polar geometries, the ordering
of the red and blue eigencolors is flipped, or there is a
single dominant grey eigencolor.
If the planet’s radius can be estimated, then its albedo
can be put on an absolute scale and one can estimate its
Bond albedo. Transiting planets have very well char-
acterized radii, but nearby earth analogs will almost
On the other

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12Cowan et al.
certainly not be transiting. Instead, a radius estimate
will require an additional large space mission: either
an infrared high-contrast imaging telescope, or a space
based astrometry mission. In the first case, thermal and
reflected photometry can be combined to estimate the
planet’s radius. If thermal photometry is obtained at a
variety of phases, then the efficiency of heat transport to
the planet’s night-side may be estimated (Cowan & Agol
2011b), and the systematic uncertainty will be ∼ 2% due
to the unkown efficiency of latitudinal heat transport
(Cowan & Agol 2011a). The uncertainty in the radius
will therefore likely be dominated by the —known— un-
certainties in thermal and reflected photometry.
In the second case, the star’s astrometric wobble pro-
vides a mass measurement for the planet; by assuming
a planetary density, one can estimate the planet’s ra-
dius. The dominant source of uncertainty here is the
planet’s composition: given a mass, a planet’s radius may
vary by 50% (eg, Charbonneau et al. 2009; Batalha et al.
2011), leading to absolute albedo estimates only valid
to within a factor of 2. Transiting planet surveys will
likely reduce these systematic uncertainties by provid-
ing an empirical mass-radius relation for planets across
a wide range of masses. It is not clear to what extent
the Kepler mission (eg, Borucki et al. 2011) will help de-
fine the mass-radius relation: although the vast majority
of Kepler candidates are likely to be bona fide planets
(Morton & Johnson 2011), most will not have mass es-
timates. The smaller, and better characterized, radius
uncertainty would therefore most likely come from com-
bining optical and infrared photometry, rather than from
a mass measurement.
The Bond albedo would allow us to better distinguish
between the equatorial (AB ≈ 0.3) and polar (AB ≈
0.4) EPOXI observations, or between a snowball planet
(AB≈ 0.7) and a temperate one. In general, this quan-
tity would be very useful in determining a planet’s energy
budget and would go a long way towards constraining its
habitability.
This work was supported by the NASA Discovery Pro-
gram. We thank D.S. Abbot and R.T. Pierrehumbert for
providing us with cloud maps of snowball Earth. N.B.C.
acknowledges many useful discussions with S.G. Warren
about snowball Earth, and thanks W. Sullivan for en-
couraging him to complete his astrobiology research ro-
tation. E.A. is supported by a National Science Founda-
tion Career Grant.
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Rotational Variability of Earth’s Polar Regions13
APPENDIX I: PCA OF SIMULATED VPL DATA
For completeness, we begin by re-running the PCA on the actual Earth1 data obtained by the Deep Impact spacecraft.
This endeavor is not redundant, since there are many differences between our current analysis and that of Cowan et al.
(2009): 1) here we use a different solar spectrum in computing reflectance (the change is most noticeable in the 950 nm
waveband); 2) we are now using rigorously-defined apparent albedo; 3) we run the analysis individually on the Earth1,
rather than on both equatorial observations simultaneously; and 4) we do not apply the cloud-variability uncertainties
when running the PCA.
In Figure 16 we summarize the results of the PCA performed on the 2008 EPOXI Equinox data. These are the same
as the results presented in Cowan et al. (2009): the dominant eigencolor is red (most non-zero at long wavelengths),
while the second eigencolor is blue (most non-zero at short wavelengths). Note that the sign of the eigenspectra, and
hence its slope, is not important in describing its color. Although the two primary eigencolors shown in Figure 16 look
similar at first glance, they are in fact orthogonal, by definition.
We now run five different versions of the VPL 3D Earth model: 1) Standard: this model is an excellent fit to the
EPOXI Earth1 observations; the remaining models are identical, but in each case a single model element has been
“turned off”: 2) Cloud Free; 3) No Rayleigh Scattering; 4) Black Oceans; 5) Black Land.
1) The Standard model (Figure 17) produces eigencolors indistinguishable from those presented in Cowan et al.
(2009) or the control case above: a dominant red eigencolor followed closely by a blue eigencolor. The relative
importance of the eigencolors as a function of time, “eigenprojections”, also match very well. This should not be
surprising, given the excellent fit to the actual data (Robinson et al. 2011).
2) The Cloud Free model (Figure 18) has similar time-averaged colors to the Standard model, but is less reflective
at all wavelengths (∆A∗≈ −0.1). This is especially noticeable at long wavelengths, where Rayleigh scattering does
not operate. If the albedo were not on an absolute scale, as would be the case for a directly-imaged planet with no
reliable radius estimate, it would be difficult to distinguish this cloud-free planet from its cloudy counterpart. Unlike
the Standard case, however, the Cloud Free model shows very little variability at blue wavebands. As a result, the
Cloud Free model shows the same dominant red eigencolor as the Standard model, but the amplitude of excursions
for the blue eigencolor are much smaller than for the Standard model.
3) The No Rayleigh Scattering model (Figure 19) has red time-averaged colors, with a slight upturn in reflectance at
the bluest wavebands due to oceans. The eigencolors and eigenprojections are essentially the same as in the Standard
model.
4) The Black Oceans model (Figure 20) has time-averaged colors, eigencolors and eigenprojections indistinguishable
from those of the Standard model. This indicates that at gibbous phases oceans on Earth consist of a null surface
type, contributing neither to the time-averaged nor to the time-resolved disk-integrated colors. This does not preclude,
however, the importance of specular reflection at crescent phases.
5) The Black Land model (Figure 21) has similar time-averaged colors to the Standard model, but without the
upturn at near-IR wavelengths. There is a single dominant, gray eigencolor.

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14Cowan et al.
Fig. 16.— Earth1 EPOXI Observations Top Left: Time-averaged broadband spectrum. Top Right: Normalized variability spectrum
from PCA. Bottom Left: Eigencolors from PCA. The eigenspectra have been normalized by their eigenvalues, so the dominant components
exhibit larger excursions from zero. Bottom Right: Eigenprojections from PCA.

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Rotational Variability of Earth’s Polar Regions15
Fig. 17.— Standard VPL Simulation Top Left: Time-averaged broadband spectrum. Top Right: Normalized variability spectrum
from PCA. Bottom Left: Eigencolors from PCA. The eigenspectra have been normalized by their eigenvalues, so the dominant components
exhibit larger excursions from zero. Bottom Right: Eigenprojections from PCA.