Inverting Phase Functions to Map Exoplanets

By N. Cowan, E. Agol. 2008

156 citations recorded by the source corpus.

We describe how to generate a longitudinal brightness map for a tidally locked exoplanet from its phase function light curve. We operate under a number of simplifying assumptions, neglecting limb darkening/brightening, star spots, detector ramps, as well as time variability over a single planetary rotation. We develop the transformation from a planetary brightness map to a phase function light curve and simplify the expression for the case of an edge-on system. We introduce two models—composed of longitudinal slices of uniform brightness, and sinusoidally varying maps—which greatly simplify the transformation from map to light curve. We discuss numerical approaches to extracting a longitudinal map from a phase function light curve, explaining how to estimate the uncertainty in a computed map and how to choose an appropriate number of fit parameters. We demonstrate these techniques on a simulated map and discuss the uses and limitations of longitudinal maps. The sinusoidal model provides a better fit to the planet’s underlying brightness map, although the slice model is more appropriate for light curves which only span a fraction of the planet’s orbit. Regardless of which model is used, we find that there is a maximum of ~5 free parameters which can be meaningfully fit based on a full phase function light curve, due to the insensitivity of the latter to certain modes of the map. This is sufficient to determine the longitudes of primary equatorial hot spots and cold spots, as well as the presence of secondary maxima/minima.

Subfields: Astronomy and Astrophysics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/102aeca9ed3964618df24548ab28837c75ea4d9c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Astronomy and Astrophysics

1. Jon Fortney
2. N. Madhusudhan
3. A. Showman
4. D. Charbonneau
5. D. Deming
6. H. Knutson
7. M. Line
8. J. D'esert

In Statistical and Nonlinear Physics

1. N. Madhusudhan
2. Jon Fortney
3. H. Knutson
4. Sara Seager
5. J. Johnson
6. D. Charbonneau
7. M. Line
8. J. D'esert

Citations in the corpus

• 3.6 AND 4.5 μm PHASE CURVES AND EVIDENCE FOR NON-EQUILIBRIUM CHEMISTRY IN THE ATMOSPHERE OF EXTRASOLAR PLANET HD 189733b (2012, methodology): “As a check we also tried fitting the odd harmonics P/3 and P/6 to the data; as discussed in Cowan & Agol (2008), these harmonics cannot be produced by a static planetary brightness map, and their amplitudes therefore provide a useful test for whether or not we are over-fitting the data.”
• THERMAL PHASE VARIATIONS OF WASP-12b: DEFYING PREDICTIONS (2011, unknown): “If we instead interpret the phase curve as being caused entirely by longitudinal brightness variations on a spherical planet (the blue curves in Figure 8), it can be inverted into a longitudinal intensity map of the planet, following Cowan & Agol (2008).”
• starry: Analytic Occultation Light Curves (2018, background): “…utilize the time dependence of unresolved, disk-integrated light from planetary bodies: both rotational variability (Russell 1906; Lacis & Fix 1972; Cowan & Agol 2008; Oakley & Cash 2009) and occultations (Williams et al. 2006; Rauscher et al. 2007) yield the opportunity to constrain the presence…”
• ORBITAL PHASE VARIATIONS OF THE ECCENTRIC GIANT PLANET HAT-P-2b (2013, background): “We also find that the preferred functional form for the phase curve of planets on circular orbits described in Cowan & Agol (2008) provides a reasonable fit to our theoretical light curves if the orbital phase angle, ξ, is replaced by the true anomaly, f .”
• INFERENCE OF INHOMOGENEOUS CLOUDS IN AN EXOPLANET ATMOSPHERE (2013, background): “The key features of Kepler-7b’s phase-curve translate directly into constraints on maps (Cowan & Agol 2008) assuming a tidally-locked planet on a circular orbit.”
• Observations of Transiting Exoplanets with the James Webb Space Telescope (JWST) (2014, unknown): “Using rotational inversion techniques, the spatial resolution that can be obtained in longitude is intrinsically limited to about 5 cycles per planetary circumference (Cowan & Agol 2008).”
• SPITZER PHASE CURVE CONSTRAINTS FOR WASP-43b AT 3.6 AND 4.5 μm (2016, methodology): “Similar formulations have been used to model other exoplanet phase curves (e.g. Cowan & Agol 2008; Knutson et al. 2012; Lewis et al. 2013; Stevenson et al. 2014).”
• THE CLIMATE OF HD 189733b FROM FOURTEEN TRANSITS AND ECLIPSES MEASURED BY SPITZER (2010, background): “The lack of variation of the atmosphere indicates that the assumptions used in creating longitudinal maps of planets from phase functions are likely valid (Cowan and Agol 2008).”
• MULTIWAVELENGTH CONSTRAINTS ON THE DAY–NIGHT CIRCULATION PATTERNS OF HD 189733b (2008, methodology): “Fitting the Phase Curve We fit the observed phase variation with an “orangeslice” model of the planet consisting of Nslices longitudinal slices with a uniform intensity in both longitude and latitude (Cowan & Agol 2008).”
• THE STATISTICS OF ALBEDO AND HEAT RECIRCULATION ON HOT EXOPLANETS (2009, background): “Furthermore, the thermal phase observations should span a full planetary orbit: the light curve minimum is the most sensitive measure of ε, and should occur nearly half an orbit apart from the light curve maximum, despite skewed diurnal heating patterns (Cowan & Agol 2008, 2010).”
• Global Climate and Atmospheric Composition of the Ultra-hot Jupiter WASP-103b from HST and Spitzer Phase Curve Observations (2018, unknown): “We also fit the thermal phase variation with a sinusoid, which can be inverted to map the climate (Cowan & Agol 2008; Cowan & Fujii 2017).”