Download Phase light curves for extrasolar Jupiter and Saturn

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Phase light curves for extrasolar Jupiter and Saturn
Ulyana A. Dyudina 1 , Penny D. Sackett, Daniel D. R. Bayliss2
Research School of Astronomy and Astrophysics, Mount Stromlo Observatory, Australian
National University, Cotter road., Weston, ACT, 2611, Australia
Anthony D. Del Genio
NASA/Goddard Institute for Space Studies, 2880 Broadway, New York, NY, 10025, USA
Carolyn C. Porco
CICLOPS/Space Science Institute, Boulder, CO, USA
Henry B. Throop and Luke ?. Dones
Southwest Research Institute,Boulder, CO, USA
and
Sara Seager
DTM, Carnegie Institute at Washington, DC , USA
ABSTRACT
1
NASA/Goddard Institute for Space Studies, 2880 Broadway, New York, NY, 10025, USA, Tony, how
do I make it clear that part of the research was done in GISS?
2
Victoria University of Wellington, New Zealand
–2–
We predict how a remote observer would see the brightness variations of giant
planets similar to those in our Solar System as they orbit their central stars. We
model the geometry of Jupiter, Saturn and Saturn’s rings for varying orbital
and viewing parameters. Broadband scattering properties for the planets and
rings are assumed to follow these observed by Pioneer and Voyager spacecraft,
namely, planets are forward scattering and rings are backward scattering. Images
of the planet with or without rings are simulated and used to calculate the diskaveraged luminosity varying along the orbit, that is, a light curve is generated.
We find that the different scattering properties of Jupiter and Saturn (without
rings) make a substantial difference in the shape of their light curves. Saturn-size
rings increase the apparent luminosity of the planet by a factor of 2-3 for a wide
range of geometries, an effect that could be confused with a larger planet size.
Rings produce asymmetric light curves that are distinct from the light curve of
the planet without rings, which could resolve this confusion. If radial velocity
data are available for the planet, the effect of the ring on the light curve can
be distinguished from effects due to orbital eccentricity. Non-ringed planets on
eccentric orbits produce light curves with maxima shifted relative to the position
of the maximum planet’s phase. Given radial velocity data, the amount of the
shift restricts the planet’s unknown orbital inclination and therefore its mass.
Combination of radial velocity data and a light curve for a non-ringed planet on
an eccentric orbit can also be used to constrain the surface scattering properties
of the planet and thus describe the clouds covering the planet. To summarize
our results for the detectability of exoplanets in reflected light, we present a
chart of light curve amplitudes of non-ringed planets for different eccentricities,
inclinations, and the viewing azimuthal angles of the observer.
Subject headings: get the keywords later
1.
Introduction
Modern telescopes and instrumentation are now approaching the precision at which
reflected light from extrasolar planets can be detected directly (Jenkins & Doyle 2003; Green
et al. 2003)more citations?. Since 1995, meanwhile, more than 100 extrasolar planets have
been detected indirectly by measuring the reflex motion of their parent star along the line
of site (radial velocity or Doppler method). Some of these planets have been confirmed by
measuring the parent star motion on the sky (astrometry) citation here or by measuring the
change in parent star brightness as the planet executes a partial eclipse of its host (transit
–3–
photometry) (Charbonneau et al. 2000). Two exoplanets were detected first by transit
photometry (Konacki et al. 2003) and another reference here? and then confirmed with
Doppler measurements. It is expected that in the next decade, as photometric techniques
are improved both from the ground and in space, direct detection of the reflected light from
extrasolar planets will not only prove useful in expanding the number of exoplanets known,
but also in detailing their characteristics.1
Reflected light from an exoplanet can be detected in two ways. First, with sufficient
spatial resolution, direct imaging can resolve the planet and the star in space, and the projection of its orbit traced as a function of time simultaneously with the measurement of the
planet’s phases in reflected light. To be detected, the planets must be at large enough orbital distances to be easily resolved from their parent, and yet close enough that the reflected
brightness, which decreases as inverse square of the orbital distance, is large enough to be
observed against the background in a reasonable amount of time. Even with coronagraphic
or nulling techniques to help block the light at the position of the star itself, success will
require optical point spread functions with very low level scattering wings to obtain sufficient
reduction of parent starlight at the position of the planet. Consequently, the first exoplanets
to be detected directly in this way are likely to be giants orbiting relatively nearby stars
at intermediate distances (1-5 AU), References: CELT Report 34, 2002, website reference; Lardiere, O., Salinari, P., Carbillet, M., Riccardi, A. & Esposito, 2003,
preprint (Codona & Angel 2004) where the contrast between planet and scattered starlight
is high enough and yet the total reflected light still measurable. Since direct imaging will
yield both the orbit and the luminosity of the planet simultaneously, it will be a robust detection method that will add greatly to our knowledge of planetary structure and evolution;
unfortunately, it is likely to be used for only a relatively small number of stars (10-1000) in
our Sun’s neigbourhood. In this paper, the applicability of our results to direct imaging is not
described in great detail, but our light curves can be used for planning future observations
aimed at directly detecting spatially-resolved planets in reflected light.
The second method of detecting reflected light is precise, integrated photometry, which
is the main subject of this paper. This technique searches for temporal variations in the
combined parent star and exoplanet (reflected) light curve as the planet changes phases during the orbit in the same way the Moon changes phase, producing a ”phase light curve.”
The vast majority of the light comes from the star itself, but if the periodic variations can
be extracted from the total light curve, the planetary phase as a function of time can be
deduced. Since the light from the parent star need not be blocked, given enough integration
1
Reviews and references on extrasolar planets and detection techniques can be found at
http://www.obspm.fr/encycl/encycl.html
–4–
time, planets much closer to their parent star can be detected than with spatially resolved
imaging. Futhermore, more distant planetary systems can, in principle, be studied in this
way since the angular size of the point spread function does not present a limitation. Current
instruments from the ground (Collier-Cameron references) and space (MOST references) are capable of detecting variations of the order 10−4 − 10−6 of the total star light
(Jenkins & Doyle 2003; Green et al. 2003), at the limit of what might be expected from
the class of ”hot Jupiter” planets, i. e., gas giants orbiting within 0.1AU of their solar-type
parent stars (Seager, etc references here?). With the advent of 30m-100m telescopes in
the next 10-15 years, one might expect these limits to improve by factors of 10 to 100 (need
references here).
The shape of the phase light curve depends on many parameters: the planet’s orbit, its
geometry relative to the observer, the scattering properties of the planet’s surface, and the
presence and geometry of rings. The amplitude of the light curve scales in a simple manner
with planet’s size and its orbital distance. When planning observations, it is important
to understand the variety of shapes and amplitudes of these phase light curves and the
conditions under which they will be generated: this is the purpose of this paper. In particular,
the effects of planet size, rings, and scattering properties, which convey information about
atmospheric structure and reflecting clouds, will be superimposed in the light curves and may
be indistinguishable from one other, causing confusion or degeneracies in interpretation. We
discuss these degeneracies and possible ways to eliminate them through detailed light curve
measurements or alternative observations.
The great interest in the potential of direct exoplanet detection through reflected phase
light curves is demonstrated by a number of works published in the last few years. Seager
et al. (2000) modeled the atmospheric and cloud composition and simulated the light curves
for close-in giant planets with various cloud coverage. more citations here Burrows,
Sasselov etc. Barnes & Fortney (2003) modeled the transit light curves for a planet with
rings. Arnold & Schneider (2004) simulated light curves for planets with rings of different
sizes, assuming plants with isotropically-scattering (Lambertian) surfaces and rings with
isotropically scattering particles, and provide a discussion of the possibility of ring presence
at different stages of planetary system evolution.
Our model is the first to use the observed anisotropic scattering of Jupiter, Saturn
and the rings. We find that anisotropic scattering yields light curves that are substantially
different from those assuming Lambertian planets and rings, and thus is more likely to give
an accurate description for extrasolar planets similar to Jupiter and Saturn. We do not
model the spectral light curve dependence; our model produces light curves for averaged
broadband visible light (∼0.4-0.7µm ).
–5–
In Section 2 we describe our model. In Section 3 we present our light curves for nonringed exo-Saturn and exo-Jupiter planets at variously inclined circular orbits, for a ringed
planet (with Saturn-like scattering properities) on circular orbits, and for both a non-ringed
exo-Saturn and an exo-Jupiter on eccentric orbits. In Section 4, we discuss the uncertainties
of our model and the detectability of the light curves by the modern and future instruments.
Conclusions are presented in Section 5.
–6–
2.
Model
Our model produces images (pixel maps 16 to 200 pixels across) of the planet and rings
at different geometries. The model traces the light rays from the star reaching each pixel of
the image. The light rays from the star illuminating the planet are assumed to be parallel,
consistent with both star and planet being negligible in size compared to the star-planet
distance. The rays coming towards the observer are assumed to be parallel, consistent with
a remote observer. We assume that the planet is a spheroid with Saturn’s oblateness of 10%
(note that Jupiter is 6% oblate). We account for the planet’s reflection, rings’ reflection,
rings’ transmission and the shadows. We do not account for the ring shine on the planet
and planet shine on the rings.
We use the following notation to match the common notation in Jupiter’s and Saturn’s
observational papers. For comparison of our results with a model by Arnold & Schneider
(2004) we also list their variable names:
Name
A, B
a, a1
DP
E
e
F
g1 , g2 , f
I
i
LP
L∗
M
RP
rpix
Θ
µ0 , µ
α
ω
σf −d
Alt. name*
γ
Lp , IR , IT
Lps , Lpr
i
rp
φ = Θ-90◦
µ0 , µ
α
Meaning
Coefficients of the Backstorm law
Semi-major axis of the planet’s orbit
Planet-star distance
Eccentric anomaly (polar angle parameterization)
Eccentricity
Intensity of Lambertian surface
(πF is the solar incident flux)
Parameters of Henyey-Greenstein function
Intensity (or brightness, or radiance) of the surface
units
n/d**
AU
km.
n/d**
n/d**
W/m2 /sr
Inclination of the orbit (0◦ :face-on, 90◦ :edge-on)
Luminosity of the planet at ∼ 0.4-0.7 µm
Luminosity of the star at ∼ 0.4-0.7 µm
Mean anomaly(time parameterization)
Radius of the planet
Pixel size
Orbital angle (±180◦ :min. phase,0◦ :max. phase)
Cosines of the incidence and emission angle
Phase angle
Argument of pericentre(90◦ :observed from
perticentre, -90◦ :from apocentre)
Full-disk albedo LP /(πRP2 F )??check!
degrees
W/sr
W/sr
n/d**
km.
km.
degrees
n/d**
degrees
degrees
*Alternative name used in Arnold & Schneider (2004).
n/d**
W/m2 /sr
n/d**
–7–
**Non-dimensional
2.1.
Reflecting properties of Jupiter, Saturn and rings.
Figure 1 demonstrates the scattering properties of the surface of Jupiter, Saturn, and
the rings used in our model. The brightness of the point on the surface I, normalized by
Fig. 1.— Examples of scattering phase functions for Lambertian surface, Saturn, Jupiter,
Saturn’s rings (as used in our model). The phase function for rings consisting of isotropically
scattering particles used by Arnold & Schneider (2004) is plotted for comparison.
the incident flux is plotted versus the phase angle α. F µ0 is the ‘ideal’ reflected brightness
of the white isotropically scattering (Lambertian) surface, where µ0 is the cosine of the
incidence angle, which is measured from a local vertical. πF is a solar flux at the planet’s
orbital distance. α = 0◦ means backward scattering, α = 180◦ means forward scattering.
–8–
Lambertian scattering with reflectivity 0.63 (matching Saturn’s full-disk albedo) is shown
as a horizontal solid line for comparison. The phase function for the rings consisting of
isotropically scattering particles used by Arnold & Schneider (2004) is shown for comparison
for the same ring opacity, albedo, and geometry as in example for our ring. Note the
logarithmic scale of the ordinate and the large amplitudes of the anisotropic functions. For
Jupiter we consider the scattering phase function depending only on α. For Saturn and rings
the scattering phase function depends on three angles (incident, emission and phase α, see
below), which can not be fully described by the one-parameter function of α shown in Fig.
1. Figure 1 only gives an example of reflection when the surface is illuminated by the Sun
at 2◦ above the horizon and the observer moves from the Sun’s location (α = 0◦ ) across the
zenith toward the point on the horizon opposite to the Sun (α = 180◦ ).
The strong forward scattering of the clouds covering Jupiter is represented by the twoterm Henyey-Greenstein function P (g1 , g2 , f, α).
I/F = µ0 · P (g1, g2 , f, α)
(1)
P (g1 , g2, f, α) = f PHG (g1 , α) + (1 − f )PHG (g2 , α)
(2)
The two terms are the two single-term Henyey-Greenstein functions representing forward
and backward scattering lobes.
PHG (g, α) = 2 ·
(1 − g 2 )
(1 + g 2 + 2g · cos(α))3/2
(3)
where α is the phase angle , f is the fraction of the forward versus backward scattering,
g1 is positive and shows the sharpness of the forward scattering lobe, g2 is negative and
shows the sharpness of the backscattering lobe. Note that we rewrote PHG compared to
the common notation where the scattering angle θ=180◦-α is used, which changes the sign
in front of 2g · cos(α). Unlike the common use of Henyey-Greenstein function for a singleparticle scattering, for which the function is normalized over the sphere of solid angles
around the particle, we only utilise this function as a convenient analytical expression to fit
the measured scattering off Jupiter’s surface. In this case the function is not normalized over
the hemisphere of solid angles above the surface. The Bond albedo (the ratio of reflected to
incident light for the whole planet) is imbedded into the scattering function. It is a rather
complicated integral over the planet’s surface and we do not calculate the value for Bond
albedo here. We use the Henyey-Greenstain parameters g1 =0.8, g2 =-0.38, f =0.9. We chose
these values to fit the data points from the Pioneer 10 images (Tomasko et al. 1978). Figure
2 show the fit. Pioneer images are taken with broadband blue (0.390–0.500 µm ) and red
(0.595–0.720 µm ) filters at different locations on Jupiter. The two types of locations are the
belts, usually seen as dark stripes on Jupiter (×-symbols on the plot), and zones, usually seen
–9–
Fig. 2.— Our fit of the Henyey-Greenstein function (solid line, g1 =0.8, g2 =-0.38, f =0.9) to
the Pioneer 10 data published by Tomasko et al. (1978), Tables IIa, IIb (labeled P10 on the
plot), and Pioneer 11 data published by Smith & Tomasko (1984) Tables IIa, IIb (labeled
P11 on the plot). ×-symbols represent the values for the belt (dark stripe) on Jupiter. Cross
symbols represent the values for the zone (bright stripe) on Jupiter. Black symbols show
the data for the red filter (0.595–0.720 µm ). Grey symbols show the data for the blue filter
(0.390–0.500 µm ).
– 10 –
as bright stripes on Jupiter (cross-symbols on the plot). The relative calibration of Pioneer 10
and Pioneer 11 (P10 and P11 labels above the points) is not as well constrained as the phase
dependence within each data set. Accepting the calibrations given in Tomasko et al. (1978)
for Pioneer 10 and Smith & Tomasko (1984) for Pioneer 11, our curve better represents
the observations in the red filter (black data points). The vertical spread of the points
of the same filter and location is mainly due to the differences in emission and incidence
angle between the points. This spread is not large, i. e., the reflectivity is not strongly
dependent on incidence and emission angles for the fixed α, except for 1/µ0 dependence
which is often quoted as Lambertian limb-darkening. However the phase angle dependence
is strongly non-Lambertian, which is important for the light curves. Besides the Pioneers’
data points, images of Jupiter at various angles were taken by Voyagers, Galileo and Cassini.
However we are not aware of other published data on the scattering phase functions for the
Jupiter’s surface. U.D. plans to work on obtaining the spectral phase functions from Cassini
nine-filter visible images in the immediate future. Data for α >150◦ do not exist because of
potential danger of pointing the spacecrafts’ cameras close to the Sun. Because of that the
forward-scattering peak cannot be well constrained with existing data. This does not create
large errors for the amplitudes of the light curves because at forward scattering the observed
crescent is small and the light curve has its minimum. We also tune g1 , g2 , and f values
to reproduce the full-disk albedos at the wavelengths of Pioneers’ red filter in the spectrum
observed by Karkoschka (1994) and Karkoschka (1998), as will be discussed in Section 3.1.
Scattering phase function and albedo of Saturn are represented by the Backstorm law:
I/F =
A µ · µ0 B
(
)
µ (µ + µ0 )
(4)
,where µ and µ0 are cosines of emission and incident angles measured relative to the local
vertical, and A and B are coefficients depending on the phase angle. The coefficients are
fitted by Dones et al. (1993) to Pioneer 11 observed phase functions (Tomasko & Doose
1984). We averaged the coefficients for zones, belts, blue and red filter published by Dones
et al. (1993), with equal weights. Table 1 shows the resulting coefficients.
Phase angle α
A
B
0◦
30◦ 60◦ 90◦ 120◦
1.16 1.09 1.01 0.95 1.03
1.30 1.29 1.30 1.30 1.28
150◦
2.04
1.37
180◦*
3.06*
1.46*
Table 1: Coefficients for Backstorm function averaged from Dones et al. (1993), table V.
*The coefficients at α=180◦ are not restricted by observations. We linearly extrapolated
these coefficients from the values at 120◦ and 150◦.
– 11 –
Pioneers did not observe Saturn at α > 150◦ . Corresponding coefficients in Dones
et al. (1993) are given for α up to 150◦ . To complete the phase curve we extrapolate the
coefficients at 180◦ linearly from their values at 120◦ and 150◦ . The phase curve given in
Fig. 1 corresponds to the case when the incidence angle is fixed to 2◦ (fixed µ0 ) and the
trajectory of the observer on the sky is in the plane including zenith and Sun’s location. This
cross-section of the fitted three-parameter phase function is not as smooth as the HenyeyGreenstein function for Jupiter. For example the dip at α = 90◦ is a result of approximation
by analytic function rather than a real observation.
The high-radial-resolution ring opacity and albedo is derived from Voyager 1 and 2
observations. The ring opacity for each pixel is taken at the point on the rings projected
to the pixel’s center. The ring reflection and diffuse transmission is provided as a tabulated
output of the Cassini ISS ring brightness model. The ring brightness model calculates
multiple scattering within the rings by Monte Carlo method. The input parameters are
the particle size and number density, which define reflected brightness at different angles
and locations on the rings. The output is fitted to Voyager ring images at the available
geometries. The ring structure restricted by the fitting is then used to predict brightness at
the angles not available from the observations. Carolyn and Henry, please check the
ring-scattering description above! The tabulated output depends on large number of
parameters: the three angles, the illuminated versus dark side of the rings, optical depth
and the albedo of the ring particles. Because of the data volume restrictions for such multidimensional table we use rather large steps in each parameter. These steps are clearly seen
in the ring phase function (dotted line) in Fig. 1. The phase function in the plot corresponds
to the same scattering geometry as the curves for Saturn and Jupiter. The optical depth at
the sample point on the rings (1.7 Saturn’s radii from the planet’s center) is 2.3, the albedo
of the ring particles is 0.56. The ring is observed from the illuminated side.
To produce the light curves of the planets we model images of the planet for a set of
locations along the orbit. For each image we integrate the total light coming from the planet
and the rings (if there are rings) to obtain the full-disk albedo σf −d .
2
pixels I/F · rpix
(5)
σf −d =
πRP2
,where rpix is the pixel size and RP is the planet’s radius. The full-disk albedo is the planet’s
luminocity LP normalized by the reflected flux from the Lambertian disk with the planet’s
radius at the planet’s orbital distance from the star, which is illuminated and observed from
the direction normal to the disk.
σf −d ≡ LP /(πRP2 F )
(6)
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2.2.
Model of Eccentric Orbit
In addition to modeling light curves from ringed planets traveling in circular orbits, we
also model the light curves for non-ringed planets moving in eccentric orbits. Although most
of the planets in our solar system orbit the sun with low eccentricities, the extrasolar planets
detected to date display a wide range of eccentricities 2 .
Eccentric orbits differ from circular orbits in two main respects. Firstly, the distance
between the star and the planet varies according to the planet’s orbital position. As the
planet moves from its pericentre to its apocentre it increases its separation from the star.
Secondly, the planet does not move with a constant angular speed around the star as it
would in a circular orbit. The planet’s angular speed varies throughout the orbit, being a
maximum at the pericentre and minimum at the apocentre.
To model the reflected light as a function of time we calculate the angular position of
the planet and the planet-star separation over a complete period using a solution to Kepler’s
equation (in notation of Murray & Dermott (2001)):
M = E − e sinE
(7)
where M is the mean anomaly (a parameterization of time), E is the eccentric anomaly (a
parameterization of the polar angle) and e is the eccentricity. Kepler’s equation is transcendental in E and therefore cannot be solved directly. However since we are only concerned
with one complete orbit to model a light curve, we are able to use the simple Newton-Ralpson
method to get a numerical solution to Kepler’s equation3 . Applying the Newton-Ralpson
method to Eq. 7 we get the iteration:
Ei+1 = Ei −
E − esinE − Mi
1 − ecosEi
(8)
Using this iteration we calculate the angular position and planet-star separation as functions
of time, and use this to model the amplitude of reflected light from the planet.
The geometry of an ellipse means that as well as the inclination parameter, there is an
additional parameter in the observers azimuthal perspective of the system. This parameter
is the argument of pericentre ω, which is the angle (in the planet’s orbital plane) between
the ascending node line and pericentre (Murray & Dermott 2001). Viewing the system from
the line of the pericentre or the apocentre would give arguments of pericentre of 90◦ or 270◦
respectively.
2
Eccentricities of extrasolar planets are listed at http://exoplanets.org/almanacframe.html
3
Application of Newton-Ralpson method to Kepler’s equation is described in Murray & Dermott (2001)
– 13 –
3.
Results
We modeled a variety of geometries for the planetary system, i. e., different orbital
eccentricities for non-ringed planets, and different obliquities of the rings relative to the
ecliptic for the ringed planets on circular orbits. Also we modeled a variety of observer’s
locations for each of these cases, i. e., different orbital inclinations to the observer as well as
different azimuths of the observer relative to the orbit’s pericentre or relative to the rings.
Below we compare the light curves for these various geometries and discuss if these geometric
effects can be distinguished given the observed light curve.
3.1.
Jupiter versus Saturn with no rings
First we tested how different would the light curves be for the planets without rings
with the surface reflection of Jupiter and Saturn. We modeled the planets on a circular
orbit at various orbital inclinations. The low-inclination orbits (close to face-on) produce
smaller amplitudes of the light curves because the planet appears nearly half-phase most
of the time. The most prominent amplitudes and the most prominent difference between
Jupiter and Saturn occurs at the edge-on orbit where phases change from completely “new
moon” to completely “full moon”.
Figure 3 compares the edge-on light curves for the non-ringed Saturn, Jupiter, and a
Lambertian planet. The luminosity of the planet (ordinate) is normalized by the incident
flux to obtain the full-disk albedo σf −d as described in Eqs. 5 and 6.
The full-disk albedo can be converted into the planet’s luminosity LP as a fraction of
star’s luminosity L∗ for a planet of radius RP at the orbital distance DP .
LP /L∗ = (RP /DP )2 · σf −d
(9)
For example, for Saturn at 1 AU (RP /DP )2 ≈1.6×10−7 assuming RP =60.3×103 km. and 1
AU=150×106 km. The abscissa shows the azimuthal angle of the planet along the orbit in
the orbital plane, or the orbital angle Θ, starting at minimum phase (Θ=-180◦). Note that
only for the edge-on orbits the minimum phase along the orbit is a complete “new moon”.
At orbits with lower inclinations the minimum phase would be a crescent of the size between
the “new moon” and “half moon”. The corresponding maximum phase will be between the
“full moon” and “half moon” at Θ=0◦ . For the circular orbit, the plot can be converted into
time dependence by simply dividing Θ by 360◦ and then multiplying by the planet’s orbital
period.
We also display the data points for σf −d which we obtained from the groundbased
– 14 –
Fig. 3.— Comparison of light curves for Jupiter, Saturn , and Lambertian planet assuming
the planets are the same size and do not have rings. The data points are calculated from
the groundbased spectra published by Karkoschka (1994) and Karkoschka (1998) for Jupiter
(data points at positive α, dot-dashed line) and Saturn (data points at negative α, dashed
line). The black data points correspond to the red filter, the grey data points correspond to
the blue filter. The error bars display the range of σf −d within the spectral interval of each
filter (0.595–0.720 µm for red and 0.390–0.500 µm for blue) rather than standard deviation
within this spectral interval.
– 15 –
spectra by Karkoschka (1994) and Karkoschka (1998). For each filter we obtain the range of
σf −d present in the spectra at wavelenghts corresponding to each filter and plot this range as
an error bar. For the edge-on orbit Θ = α and the observational data points depending on α
can be directly on the Θ–dependence in Fig. 3 Because the light curve is symmetric relative
to θ=0◦ , positive and negative values of Θ are equivalent. We choose the positive side of the
curve to show the data points for Jupiter (also marked by dot-dashed line style for the error
bar) and negative side of the curve to show the data points for Saturn (also marked by the
dashed line style). Jupiter’s light curve is better matched by the red filter data (black error
bars), but is within the range of blue filter data (grey error bars). Saturn’s light curve is
more consistent with the blue filter data. We did not tune the Saturn’s surface scattering,
which is borrowed from ?. Study of the spectral dependence of the light curves is out of
scope of this paper and would be an important subject for future research. However from
Figs. 2 and 3 we must note that in shorter wavelengths Jupiter is less forward scattering
and both Jupiter and Saturn are darker in blue than in red at α ∼0◦ .
Light curve for Jupiter peaks much sharper at full phase (Θ=0◦ ) than Saturn’s or
Lambertian curve because of the sharp back scattering peak in the scattering phase function
of Jupiter’s surface ( α=0◦ in Fig. 1). Near zero phase (Θ = ±180◦ in Fig. 3) Jupiter is
much more luminous because of the large forward scattering of the surface (α=180◦ in Fig.
1).
Such difference in phase functions is commonly attributed to larger particle size for the
main cloud deck on Jupiter, e. g., Tomasko et al. (1978) suggest particle size larger than 0.6
µm to explain the forward scattering. Tony, do you know more relevant references.
3.2.
Ring effects
Presence of the rings has large effect on the light curves. The geometry of the ringed
planetary system needs two more parameters for complete description, i. e., even assuming fixed ring density structure (Saturn rings), the geometry becomes sensitive to the ring
obliquity (the angle between the ring and ecliptic planes), and to the azimuth of the observer relative to the rings. Together with a variable orbital inclination, this produce a
three-parameter space which we sample to explore possible light curves.
Figure 4 demonstrates the effect of the rings on the example of one geometry. The
cartoon on the right demonstrates the images of Saturn at several positions on the orbit as
it appears for the remote observer. The plot on the left shows the light curve for the ringed
planet (cross symbols) compared with the curve for the same geometry but with the rings
– 16 –
Fig. 4.— Ring effect for Saturn observed at 35◦ above the orbital plane. Saturn’s ring
obliquity is 27◦ . The observer’s azimuth on the ecliptic is separated from the intersection
of the ring and ecliptic planes by 25◦ . The plot on the left shows the light curves for the
planet without rings (solid curve) compared to the light curve for the ringed planet (cross
symbols). Values of Θ near the images on the cartoon correspond to the plot’s abscissa.
The brightness of the images on the cartoon is given in the units of I/F, as indicated on the
nonlinear greyscale bar.
– 17 –
taken off (solid curve). The ∼10% spread of the points in the light curve for the rings is
partially due to the large steps in the ring reflection table (see Fig.1) and should be treated
as a model uncertainty. For a large fraction of the orbit the rings increase the luminosity
of the planet (Θ <-20◦) because we see more reflecting surface. However, at Θ=0◦–100◦
the rings are not well illuminated but they shadow part of the planet. That produces a
lower luminosity than for the non-ringed planet. More importantly, the light curve becomes
asymmetric, which makes it distinguishable from any of the curves of the non-ringed planet
on a circular orbit. Obviously, planets on eccentric orbits may produce asymmetric curves
due to the planet-star distance variation. Such curve may be confused with the ringed planet
curve if no radial velocity data exist for this planet (see Section 3.3 for discussion).
Figure 5 demonstrates the variety of the light curves for Saturn-like planet at different
geometries. Different ring’s obliquities are shown on each plot as black curves. Black solid
curve shows the obliquity of 89◦ (rings nearly perpendicular to ecliptic). Black dotted curve
shows the obliquity of 45◦ . Black dashed curve shows the obliquity of 1◦ (rings are nearly
in the ecliptic plane). Grey curves show the same geometries for the solid, dotted and
dashed curves respectively but for the planet with no rings. The non-ringed curves are
different because Saturn’s oblateness (10◦ ) makes the planet’s cross-section appear smaller
when observing from equatorial (and ring’s) plane than from near the pole.
Different observer’s geometries are represented as different plots. Left column of the
plots show the orbit observed edge-on (orbital inclination i=90◦ ). Middle column corresponds
to the observer 45◦ above the ecliptic (i=45◦ ). Right column shows the orbit observed faceon (i=0◦ ). Plots in different rows demonstrate different azimuths of the observer relative to
the rings. Top row show the geometries when the azimuthal angle between the ring-ecliptic
intersection and the projection of the observer’s direction onto the ring plane is 90◦ . At
such observer’s azimuth the rings are seen at the maximum possible opening when other
geometric parameters (rings’ obliquity and orbital inclination) are fixed. Middle row shows
the observer’s azimuth of 45◦ . Bottom row shows the observer nearly at the ring plane
(azimuth of 0.1◦ ).
Several lessons can be learned from Fig. 5. First, in most cases rings increase the amplitude of the light curves by a factor of 2 to 3. Note that in the case of observations by
precise photometry all the constant flux would be subtracted together with the background
stellar light and only the variations on top of the light curve’s minimum will be seen. This
amplitude increase can interfere with the effects of the planet’s size and albedo. The ambiguity may be resolved spectrally because the rings spectrum should be rather flat, but the
planetary atmosphere is expected to be dark at a set of prominent gaseous absorption bands.
Second, curves for the ringed planet are asymmetric. There are two types of asymmetry.
– 18 –
Fig. 5.— Light curves for Saturn at different geometries. Different ring’s obliquities are
shown on each plot as black curves (solid for 89◦ , dotted for 45◦ , dashed for 1◦ ). Different
obliquities of the planet without rings are shown on each plot as grey curves (solid for 89◦ ,
dotted for 45◦ , dashed for 1◦ ). Each column of the plot shows different orbital inclinations
i. Each row of the plot shows different azimuth of the observer relative to the rings (labeled
Az) between the ring-ecliptic intersection and the observer’s azimuth in the ring plane
– 19 –
One, potentially easier detectable, is the offset of the curve’s maximum relative to Θ=0◦ ,
the maximum point for the non-ringed planet (marked by vertical lines in Fig. 5). The
offset of the maximum point itself occurs in rather small fraction of the plots. However most
of the curves would have their weighted maximum offset when approximated by a simple
symmetric curve, as it would probably be for the first light curve detections. If the radial
velocity data exist for the same planet, the exact timing of Θ=0◦ and the zero eccentricity of
the orbit would be known. Then the shifted maximum in the light curve would bring a direct
evidence of the rings. This offset is also noted in the ring simulation by Arnold & Schneider
(2004) (as a φ−shif t) and is a strong photometric signature of the rings. However we do not
exclude other processes capable of producing φ − shif t. Similar, although probably smaller,
φ − shif t may be produced by seasonal brightness variationon the planet. None of the giant
planets in the solar system have pronounced seasonal brightness variation, but extrasolar
planets may display seasons. Apparently the actual temporal variation of brightness in the
case of seasons may offset the brightness maximim. Another offset may be produced by the
global non-even brightness distribution over the planet. The example in the solar system is
the striped appearance of Jupiter. Jupiter’s brightness varies along the latitude by as much
as 50% in the broadband visible light (Smith & Tomasko 1984). In a realistic scenario when
the extrasolar planet’s poles are brighter than equator the planet appears globally brighter at
winter and summer and darker at mid-seasons. This may produce the φ − shif t as well. As
in the case of large light curve amplitude for the ringed planet discussed above spectroscopy
may help resolve the ambiguity between the rings and the processes on the planet’s surface.
Another type of asymmetry is the shape of the light curve. Resolving such details in
the light curve would require another order of magnitude in the instruments’ sensitivity.
However, the fine structure of the light curves may allow detection of the rings without
the radial velocity data. The main source of uncertainty of the ring detection once an
asymmetric light curve is seen is the possibility that the asymmetry can be created by the
planet’s eccentric orbit. The fine structure of the light curve for the planet on eccentric orbit
would be rather different than the structure for most of the curves for the ringed planet, as
will be discussed in Section 3.3.
Third, in the case of face-on observation of the orbit (right column in Fig 5) both the
radial velocity observation and the precise photometry for the non-ringed planet would give
no signal. The ringed planet would produce a double brightness maximum as the rings
are illuminated once from the observer’s side and once from the back side during the orbit
(dotted curve). We cannot see the rings at the ring’s obliquity 89◦ (solid curve) because the
observer is nearly in the ring plane and at the ring’s obliquity 1◦ (dashed curve) because the
star is always in the ring plane. These extreme geometries are not typical for the random
observations. The more typical intermediate geometry (obliquity 45◦ , dotted curve) resilts
– 20 –
in a double peak. This feature cannot be confused with a light curve for non-ringed planet
on a circular orbit. In rare geometries (equal height of the two peaks) and if only rough
shape of the maximums is resolved, the curve with two maximums may be confused with the
curve for half the orbital period. Typically the maximums differ in height, which indicates
that both maxima belong to a single orbit. Double maximums can be produced on eccentric
orbit, but have different fine structures of the light curve (see Section 3.3). The double peak
may also be created by the planet’s seasons or by uneven brightness distribution around the
planet. The example of possible double peak due to uneven brightness is the planet with
bright poles rotating “on the side” (rotation axis parallel to ecliptic) with the orbit observed
face-on.
– 21 –
3.3.
Eccentric orbits
Figure 6 shows the light curves for the planet HD 108147b assuming the radius and
atmospheric properties of Jupiter. Prescribing Jupiter’s scattering properties to such close-
Fig. 6.— Light curves for the planet HD 108147b with the radius and surface of Jupiter.
The argument of pericentre is ω = 319◦ . The corresponding time of the maximum phase
Θ=0◦ is shown by the vertical line ∼ 0.4 days after the pericentre (time zero). The orbital
inclinations change from face-on to edge-on orientation (i=1◦ to i=89◦ respectively). The
edge-on light curve for the planet with Jupiter’s radius and Saturn’s surface is shown as a
dashed line.
in planet is a very rough assumption. This planet has to be much hotter than Jupiter and
may not have clouds at all or may have dark silicate?, iron? clouds Sara, please correct
me. What would be the appropriate citation? . However if the clouds are pesent
on this planet, scattering of Jupiter may give a rough approximation on the phase function
– 22 –
and an order of magnitude estimate for the planet’s brightness. Planet’s luminosity LP
normalized by the star’s luminosity L∗ is plotted versus time. We plot the light curve over
one complete orbit, beginning at the pericentre (time zero). The orbital parameters are
those of the planet HD 108147b, discovered by precise radial velocity measurements (Pepe
et al. 2002). We model HD108147b since it has a high eccentricity (e=0.498) and a relatively
short semi-major axis (a=0.104 AU), and therefore the effects of the ellipticity on the light
curve are prominent. Since the inclination of the orbital plane is not determinable from
radial velocity measurements, we display various light curves at different inclinations between
i=90◦ (edge-on orientation) and i=0◦ (face-on inclination). The azimuthal location of the
pericentre relative to the observer can be derived from the radial velocity measurements.
For this planet the argument of pericentre ω = 319◦ , which means that we are are lucky
to be at the azimuth where the fullest planet’s phase Θ=0◦ is separated from pericentre by
only 41◦ . At such geometry the phase-induced and ellipticity-induced maxima on the light
curve amplify each other. As a result, the amplitude of the curve for the edge-on case is
about 3.5×10−5 , nearly five times larger than in the face-on case, where only the distance
variation matters. Because planets with edge-on orbital inclinations are better detectable by
radial velocity, the chances of seeing an edge-on planet with larger amplitude in follow-up
observations are better.
The light curve can be rescaled for another planet with the same orbital eccentricity but
with different semi-major axis a1 by multiplying the luminosity by (a/a1 )2 and multiplying
the time by the ratio of the orbital periods (a1 /a)3/2 .
Figure 7 shows an example of a model light curve of the planet on eccentric orbit. The
example light curve assumes the same orbital properties as the planet HD 108147b except
for the argument of pericentre ω = 60◦ , different from the observed ω = 319◦ . The example
demonstrates the key features of the light curves for eccentric orbits.
The light curve for the i=90◦ orientation shows a primary peak when the planet is
presenting a full phase to the observer. The amplitude of the reflected light at this peak
was approximately LP /L∗ =6.5 ×10−6 . The amplitude of the primary peak was considerably
reduced for lower inclinations, as the planet no longer produces a full phase at the maximum
phase position. Additionally, the position of this peak moves towards the pericentre as the
inclination is lowered.
The i=90◦ light curve also displays a secondary peak which is located close to the
pericentre. This peak is due to the planet’s reflected light being more intense at this point
due to it being close to the star at its pericentre. This secondary peak becomes the highest
peak at lower inclinations which is a result of the planet-star separation function becoming
dominate over the phase function at lower inclinations. In the extreme case of i=0◦ the phase
– 23 –
Fig. 7.— Light curves for the example eccentric orbit. The planet has the radius and surface
of Jupiter. The orbit is assumed to be the one of the planet HD 108147b. The argument of
pericentre is ω = 60◦ . The corresponding time of the maximum phase Θ=0◦ is shown by the
vertical line ∼ 3.3 days after the pericentre (time zero). The orbital inclinations change from
face-on to edge-on orientation (i=1◦ to i=89◦ respectively). The edge-on light curve for the
planet with Jupiter’s radius and Saturn’s surface scattering is shown as a dashed line.
– 24 –
function no longer affects the light curve (as the planet displays a constant half phase), and
therefore the light curve is purely a function of the planet-star separation.
The i=90◦ and i=75◦ light curves are complicated by the appearance of a sharp trough
where the amplitude of reflected light is reduced almost to zero. This trough is due to the
planet showing no phase at this point of its orbit. The trough is sharp due to the planet’s
high angular speed at this point in its orbit. The trough is not present in lower inclinations
as the minimum phase of the planet is no longer a near zero minimum.
The case of the double peak is rather rare. For most ω it does not appear at any orbital
inclinations. If the planet’s surface is not so forward-scattering as Jupiter’s (e. g., Saturn),
the second peak is even less likely to appear. When the second peak appears, it is necessarily
accompanied by the sharp trough immediately next to it. Such shape makes the doublepeaked light curve for the eccentric orbit distinguishable from the double-peaked light curve
for the ringed planet, when the two peaks are broad (see Section 3.2).
The single-peaked curves for the elliptic orbit and the ringed planet may be distinguishable even without the radial velocity data because the curve is smooth in the elliptic case.
Although both ringed planets and elliptic orbits may result in asymmetric shape of the curve,
the complicated shadowing of the ringed planet creates abrupt variations along the curve
(see Fig. 5).
The value of orbital inclinations may be restricted by a temporal shift in the lightcurve
maximum relative to Θ=0◦ in the case of eccentric orbit . As the inclination increase, the
maximum moves from the pericentre (time zero) towards the time of the maximum phase
Θ=0◦ (the vertical line in Figs. 6 and 7). This shift is larger in the case when the orbit is
seen such that pericentre is close to half-phase (ω ∼ 90◦ or ω ∼ 270◦ ) rather than pericentre
close to the ”new moon” (ω ∼ 180◦ ) or “full-moon” (ω ∼ 0◦ ) phases. Unlike the amplitude
of the curve, which is also an indicator of inclination, the shift cannot be alternatively
produced by larger planet’s size or albedo. However the shift can result from a different
strength of backscattering at the planet’s surface. The example of such ambiguity is the
edge-on Saturn’s light curve (dashed line in Fig. 7) for i=89◦ , which has a maximum where
the Jupiter’s curve of i ≈50◦ would have its maximum. The ambiguity may be resolved by
comparing the detailed shape of the curves.
The curve’s maximum shift may also serve to restrict the surface properties of the planet.
Although the shift can be confused with orbital inclination effect, a lower limit may be derived
for the strength of backward scattering peak for the planet’s surface. For each type of the
planet’s surface a maximum possible shift occurs in the face-on case (i=90◦ ). If larger shift
(from the pericentre to the curve’s maximum) is observed, it can only be explaned by sharper
– 25 –
backward scattering peak in the scattering phase function of the surface. This restriction , as
well as the orbital inclination restriction should be treated with caution because of possible
interference with rings (we did not model both rings and eccentricity effects simultaneously),
large-scale bright patches on the planet’s surface, or seasonal variation of the cloud coverage.
Figure 8 demonstrates the sensitivity of the curve’s maximum point shift to the orbital
inclination (ordinate) and the atmospheric scattering (three columns of the plots for the
three atmospheric types.) The right side of each plot (ω=-90◦) corresponds to observations
from the apocentre side of the orbit. At such observer’s position the planet is observed
at fullest possible phase (Θ=0◦ ) at pericentre, the eccentricity-induced and phase-induced
maxima on the light curve coincide, happen at pericentre and thus the shift between the
light curve maximum and pericentre is zero (as all the plots display) The left side of each
plot (ω=90◦) corresponds to observations from the pericentre side of the orbit. At such
observer’s position the planet is observed at smallest possible phase (Θ=180◦ ) at pericentre,
the eccentricity-induced maximum is shifted from the phase-induced maximum by half the
orbital period. The maximum of the combined light curve has a complicated dependence on
e, i and planet’s scattering. In a simple case for e=0 the eccentricity-induced maximum does
not exist, the curve’s maximum is always at the fullest phase and the time of the maximum
follows the observer’s position on the orbit ω, the shift reaching half phase at ω=90◦ (red at
the left on all e=0 plots).
If the radial velocity data exist for the planet, the eccentricity is known, the argument
af pericentre ω is known, and the shift (map’s value) is observable. The orbital inclination
i (plots’ ordinate) is not known. The exception is the known inclination i ≈ 90◦ at transits
and i < (90 − ∆)◦ at non-transits, where ∆ is the small angle depending on the planetary
system’s geometry.
The restriction on i discussed above can be obtained by matching the measured time
shift valie along the vertical line corresponding to the known ω on the plot for appropriate
eccentricity. For example, for Jupiter’s case at e=0.4, ω ≈45◦ the observed shift of 0.2 of the
orbital period (green on the plot) would mean inclinations of 60–90◦ . The shift of 0.1–0.15
of orbital period (light blue) would mean orbital inclinations of 30–60◦. The shift smaller
then 0.1 of orbital period (purple to black) would mean orbital inclinations of 0–30◦. There
remains the ambiguity because of the unknown planet’s scattering properties. The three
examples of the scattering are shown in the three columns of the plot: strongly backward
scattering Jupiter, weaker backward scattering Saturn and the less realistic extreme case of
Lambertian surface. Backscattering stronger than Jupiter’s may exist on extrasolar planets,
which we did not model. This other extreme example would appear as another column of
– 26 –
Fig. 8.— Summary of the temporal shift of the lightcurve maximum from the pericentre
for eccentric orbits at different observational geometries for Jupiter (left column), Saturn
(middle column), and Lambertian planet.
– 27 –
plots on the left and would likely follow the trends in the plots as backscattering increases
from Lambertian on the right to Jupiter on the left.
The restriction on the planet’s scattering properties mentioned above can be seen comparing the plots for Jupiter, Saturn and Lambertian surface. In some geometries a strong
temporal shift may be observed at given e and ω. For example, if e=0.4 ω=60◦, and the shift
is 0.25 the orbital period (green), only the left plot corresponding to Jupiter displays green
anywhere along the ω=60◦ vertical line and thus can be consistent with the observation.
The difference between the maximum and minimum points in the amplitude of the light
curve, the contrast, gives a measure of the degree of variability the planet’s reflected light
will display. It is this variation that new generation space photometers may be able to
detect if they can achieve the required levels of precision. The highest degree of contrast
is shown by the i=90◦ inclination (see Fig. 7), while at lower inclinations the contrast is
reduced. However at very low inclinations, approaching i=0◦ , the contrast increases again
as the amplitude of reflected light at the pericentre increases. The contrast at i=0◦ is higher
than at mid inclinations. This means that measuring reflected light may provide a method
of detecting planets with face on orientation, so long as the planet’s orbit is highly eccentric.
This may therefore compliment other methods such as radial velocity measurements which
are unable to detect planets orbiting with face on orientations.
We plot the contrast over all possible orbital orientations in Figs. 9. These contour
maps display the degree of contrast for various inclinations and arguments of pericentre
at given eccentricities. These contour plots indicate that at a given orientation the more
highly elliptical orbits show much better contrast than circular orbits. However the degree
of contrast shown is strongly dependent both on inclination and argument of pericentre for
high eccentricity orbits. At high eccentricities, if these parameters are favorable (such as at
i=90◦ and ω=-90) then the contrast can increase by approximately five times from that of
less favorable orientations.
The backward scattering peak of Jupiter’s surface makes the planet much darker at low
inclinations than Saturn or Lambertian planet.
– 28 –
Fig. 9.— Summary of the observable light curve amplitudes for eccentric orbits at different
observational geometries for Jupiter (left column), Saturn (middle column), and Lambertian
planet with surface albedo 0.63 (right column).
– 29 –
4.
4.1.
Discussion
Uncertainties
Our model provides rather accurate description of Jupiter and Saturn. The largest
uncertainties are due to the luck of observations at high phase angles α >150◦ for the
Saturn’s, Jupiter’s and ring surfaces. The extrapolated phase functions may result in the
factor of a few error for the brightness at these angles. Luckily, the luminosity of the
planet at forward-scattering geometries is small because of the small area of the illuminated
crescent. Therefore such uncertainty does not influence large-amplitude features of the light
curve. The ring reflectivity may have the errors of the order 50% because many geometries ,
especially face-on illumination and face-on observation, are not restricted by the observation.
The face-on illumination never happens for Saturn’s rings because of its 27◦ obliquity. The
face-on observations are unavailable because all spacecraft visiting Saturn had trajectories
close to ecliptic. New observations by Cassini spacecraft arriving to Saturn in July 2004
will fill this gap in the data. Another source of the error is the coarse grid in the table
representing ring scattering (as discussed in Section 2). The grid-induced noise is usually
below 30% of the ring’s luminosity.
Appication of Jupiter’s and Saturn’s light curves to the close-in giant planets, which
bright light curves are easier observable, has a number of complications. The clouds on
these planets are expected to be composed of solid Fe, MgSi03 , Al2 O3 , and other condensates
stable at high temperatures (Seager et al. 2000; Burrows & Sharp 1999) instead of water
and ammonia ices (Weidenschilling & Lewis 1973). This clouds may be darker and the
corresponding light curve amplitudes be several times smaller. The scattering phase function
is mostly sensitive to the particle size but not the composition. Therefore forward scattering
typical for Jupiter and Saturn is expected at the cloud-covered planets even though the cloud
composition may differ. Remarkably, the direct spectral signature of ammonia or water ice
particles is found in small weather patches covering only a few percents of Jupiter’s area
(Simon-Miller et al. 2000; Baines et al. 2003). Most of the area is covered by clouds with
no signature of the ice on the particles’ surface. This may be due to the coating of the
condensate particles by photochemically produced materials. Wether the photochemistry on
extrasolar giant planets will result in similar coating is out of scope of this paper, but the
direct comparison of pure ,e. g., silicate clouds on extrasolar planets with pure ammonia
clouds on Jupiter and Saturn is probably oversimplified. The most important parameter for
the planet’s brightness is the presence or absence of the clouds, which depends on poorly
known even for Jupiter and Saturn vertical atmospheric circulation. Detection or nondetection of the light curve with expected amplitudes would be most indicative of such cloud
presence.
– 30 –
Carolyn and Henry, please check the few paragraphs below. The rings on the
close-in planets (which mere existence is arguable because of a tidal disruption) can only
consist of rock instead of 99% ice as Saturnian rings. Ice rings are stable against evaporation
outside 5-10 AU Henry and Luke, can you please give a reference. The rock ring
material would probably be several times darker, but will remain backward scattering for
the geometric reason (the rocks in the rings behave as moonlets and are bright at backwardscattering full-phase geometry). The ring density is hard to predict for the extrasolar planets.
It is possible that rock rings on on extrasolar planets are massive and optically thick. An
example in the Solar sistem is the Earth, which probably had massive rings while the Moon
was forming. The ring size may be very different from the Saturnian rings. This would have
a large effect on the ring luminosity. We do not model various ring sizes here and refer the
reader to the work on Arnold & Schneider (2004) who study the ring size effects. The optical
depth of the rings may be quite different from the Saturnian rings. It is important to note
that once the ring reaches optical depth of several the scattering becomes non sensitive to
the further increase of optical depths. For Saturn’s rings more than half of the area has an
intermediate optical depth of 0.5-3, which cannot be considered neither optically thin nor
optically thick. Because of such optical depths the dark side of Saturn’s ring is quite bright.
If the rings of the extrasolar planet are denser than Saturn’s the curves will become more
asymmetric due to the difference between the dark and bright side of the rings. If the rings
are optically thin, some of the asymmetry (due to the shadows) will remain in the light
curve, but the asymmetry due to the different sides of the ring will decrease. Because most
of the Saturn’s rings is quite dense, the brightness of the illuminated side is close to the
“saturated” brightness for the infinite optical depth. The possible increased optical depth in
the extrasolar planet’s rings will not result in large increase in brightness at the light curve
maximum. Thus the effect of increased optical depth will not be nearly as prominent as the
effects of ring particle albedo or the ring size.
– 31 –
4.2.
Detectability of the light curves by modern instruments
Close-in giant planets are the easiest target for detection by precise photometry because
they are well illuminated by the star. The light curves for these planets may be different from
the light curves for Jupiter and Saturn because of different planet’s temperatures. However
our light curves provide a first-order approximation for the cloud-covered planet and thus
we discuss the light curves for Jupiter and Saturn as if they were extrasolar planets placed
at small orbital distances. The light curve amplitudes (Figs. 3 to 5) can be converted to the
measurable units of fractional luminosity LP /L∗ using Eq. 9. LP /L∗ depends on the orbital
distance as inverse square. The orbital distance values typical for the known extrasolar
planets are 0.03 to 3 AU, corresponding to conversion factors of 2.5×10−4 to 2.5×10−8 in
front of the order-of-unity light curve amplitudes for the planet with Jupiter’s radius of 71400
km. The current sensitivities are of order LP /L∗ =10−5–10−6 for MOST (Green et al. 2003)
and ??? for KEPLER (Jenkins & Doyle 2003) space photometers. Besides the instruments’
precision the natural fluctuations of the stellar luminosity may create a noise of the order
10−5 ???–?, depending on the ?age? and ?? of the star. The stellar noise decreases at longer
orbital periods and thus larger orbital distances numbers, the time (radius) dependence
formula Longer observations covering many orbital periods may help reducing the stellar
noise.
The direct imaging has a good potential to detect faint planets if they are distant from
the star. The ?OLA? ground-based telescope may achieve the precision of LP /L∗ =10−8 at
?AU and ? at ? AU.Penny, please add the proper numbers and citations. The
planet projected onto the sky is most separated from the star near the half-phase, i. e., at
Θ ∼ ±90◦ (Figs. 3 to 5). At these points the planet is not at its maximum brightness. For
the anisotropically scattering planet the amplitudes at half-phase are substantially smaller
than for Lambertian planet (see Fig. 3).
Rings may increase the luminosity by a factor of 2-3 and more if the rings are larger
than Saturn’s. This would increase the chance of the light curve detection. Larger planets
would also be brighter. There is an indication of a planet with the radius 1.3 times Jupiter’s
detected by transit put citation here. However such discovery is surprising because radii
substantially larger than Jupiter’s are not expected from the theory of the giant planets.
The theory predicts nearly constant size of ?? for the planets heavier than Jupiter. put a
reference. Searching for photometric signature of large rings at extrasolar planets would
be an easier task than searching for the planet itself.
– 32 –
5.
Conclusions
The following table summarises the detection criteria for the rings on extrasolar planets
by the light curves and differences of these light curves with the curves producedon by
non-ringed planet on eccentric orbit.
Distinction criterium
Light curve amplitude
Double peak
Rings
increased 2-3 times1
rarely;smooth peaks
Large-scale asymmetry
likely
Fine light curve structure
Curve’s max. shift
from Θ=0◦
(with radial velocity data)
abrupt changes
often
Eccentric orbit
increased,depends on e
rarely; sharp trough
near one peak
likely
smooth
never for curcular
orbit,
known3 for eccentric
Alternative
albedo, planet size
brightness asymmetry2 ,
seasons
brightness asymmetry2
seasons
brightness asymmetry2
seasons
backscattering3
1
The values are given for Saturn-sized (relative to the planet’sradius) rings.
Large-scale asymmetry of the brightness distribution around the planet, e. g., bright poles
may create double peak at face-on orbit.
3
The strengh of back scattering peak of the planet’s phase function changes the amount of
curve’s maximum shift relative to Θ=0◦ , which is known within the range defined by the
possible phase functions (see Fig. 8).
2
For eccentric orbits our comparison of Jupiter, Saturn and Lambertian planet gives the
following messages to the potential observers:
1)Anizotropically scattering planet is much fainter at halp-phase (Θ=± 90◦ ) than Lambertian planet.
2)Anizotropically scattering planet is also much fainter at low inclinations.
3)Eccentricity may increase the light curve amplitude by large amount (as summarized
for various geometries in Fig. 9)
4)Timing of the lightcurve maximum may restrict the orbital inclination and atmospheric backscattering (as summarized for various geometries in Fig. 8)
– 33 –
Acknowledgements
We thank R.A. West for useful references on Jupiter’s scattering. This research was
supported by Australian Planetary Science Institute and NASA Cassini project.
REFERENCES
Arnold, L. & Schneider, J. 2004, accepted to A&A
Baines, K. H., Carlson, R. W., Camp, L. W., & Team, G. 2003, Icarus, get the proper
citation
Barnes, J. W. & Fortney, J. J. 2003, Bull.–Am. Astron. Soc., 923
Burrows, A. & Sharp, C. M. 1999, ApJ, 512, 843
Charbonneau, D., Brown, T. M., Latham, D. W., & Mayor, M. 2000, ApJ, 259, 45
Codona, J. L. & Angel, R. 2004, ApJ, 604, L117
Dones, L., Cuzzi, J., & Showalter, M. 1993, Icarus, 105, 184
Green, D., Mattews, J., Seager, S., & Kuschnig, R. 2003, ApJ, 590
Jenkins, J. M. & Doyle, L. R. 2003, ApJ, 595, 429
Karkoschka, E. 1994, Icarus, 111, 174
Karkoschka, E. 1998, Icarus, 133, 134
Konacki, M., Torres, G., Jha, S., & Sasselov, D. 2003, Nature, 421, 507
Murray, C. D. & Dermott, S. F. 2001, Solar System Dynamics (Cambridge University Press)
Pepe, F., Mayor, M., Galland, F., Naef, D., Queloz, D., Santos, N., Udry, S., & Burnet, M.
2002, A & A, 388, 632
Seager, S., Whitney, B., & Sasselov, D. 2000, ApJ, 540, 504
Simon-Miller, A. A., Conrath, B. J., Gierasch, P. J., & Beebe, R. F. 2000, Icarus, 145, 454
Smith, P. H. & Tomasko, M. G. 1984, Icarus, 58, 35
Tomasko, M. G. & Doose, L. R. 1984, Icarus, 58, 1
– 34 –
Tomasko, M. G., West, R. A., & Castillo, N. D. 1978, Icarus, 33, 558
Weidenschilling, S. J. & Lewis, J. S. 1973, Icarus, 20, 465
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