AC21 · Carry daylight into the night

Exoplanet Thermal Phase Curves

How does a wind change the brightness of an unresolved planet? Wind carries absorbed heat onto the nightside before the gas cools. As a planet orbits, different longitudes face us, changing its total brightness. The curve measures projected emission; separating wind speed, thermal inertia and cloud effects needs additional assumptions or observations.

Drag the wind and spin of a radiating planet. Follow transported heat, the same temperature field’s phase curve, and secondary eclipse.

Thermal planet / star · bolometric ppm $404$

Nightside net power / absorbed power $0.139$

Layer temperature contrast · kelvin $1.18\times10^{3}$

Physical time in this record · hours $0$

Send warm gas into the night

Wind relative to the body

2000

u_{\rm eq}\;[\mathrm{m\,s^{-1}}]

Body spin / orbital angular speed

1

\Omega_{\rm core}/n

Areal heat capacity

10000000

C\;[\mathrm{J\,m^{-2}K^{-1}}]

Bond albedo

0.2

A_B

Drag the blue wind arrow or amber spin ring. Parameter changes retain the layer; let time run to see it warm and cool.

Energy, coordinates and approximation limits

\omega_{\rm eff}=\Omega_{\rm core}+u_{\rm eq}/R_p-n

Positive flow is eastward in the substellar frame. Each latitude band advects and radiates independently. Wind and body spin are prescribed; momentum, clouds, chemistry and meridional transport are omitted. The clock runs one orbit in eighteen display seconds.

Changing heat capacity is an external configuration change. Its storage jump is saved separately. Presets create new steady records. Eclipse only hides emission from the observer; stellar illumination continues.

Relative energy residual since this record began: $0$

Follow the course and replay real MIRI data

Model assumptions

Synthetic single-layer thermal model with constant configured areal capacity, prescribed zonal angular flow, periodic longitude and independent latitude bands. The orbit is circular and aligned. Unit thermal emissivity and a separate bolometric Lambert reflector approximate distinct incoming and outgoing bands. No momentum, meridional transport, clouds, vertical transfer or chemistry is solved. False color shows the computed layer temperature. Presets start new steady records; other configuration changes retain temperature and record any capacity-related energy jump. An implicit conservative upwind step evolves the field. Eclipse affects observer visibility only. The real MIRI tutorial uses separately labeled reduced measurements and conditional uncertainties.

The globe is false-color layer temperature, not a resolved photograph. Cyan tracers show gas moving relative to the star, amber meridians follow the body, and the blue arrow shows wind relative to the body.

Freeze this field and observe a full orbit

Equatorial temperatures in this same field

Orange is thermal emission; green is the separately computed ideal Lambert reflection. Both are bolometric planet-to-star ratios. Phase zero is primary transit and secondary conjunction is halfway around; the vertical line marks the physical clock, while the frozen-field forecast is distinct from the evolving saved history.

Numerical resolution and what the night fraction means

The layer has 128 periodic longitude cells and 24 equal-area latitude bands. Its temperatures are piecewise constant; the observer integrates that field across a projected disk. Implicit upwind transport has numerical diffusion. At the default parameters, the nightside fraction changes by about 0.83 percent between 128 and 512 longitude cells. The night fraction subtracts the prescribed internal heat; during a transient it is not yet a steady redistribution efficiency.

Start with the essentials

Focus question: Can the same phase curve determine heat capacity and wind speed separately?
One-sentence intuition: The steady circular temperature field depends on the product of capacity and relative angular flow. Its light is then a projected hemisphere sum.

Core mathematical model

Relative flow

\omega_{\rm eff}=\Omega_{\rm core}+u_{\rm eq}/R_p-n

A synchronized body can have a moving atmosphere. Positive angular flow is eastward from the substellar point.

The radiating layer

C(\partial_tT+\omega_{\rm eff}\partial_\lambda T)=(1-A_B)F_\star\mu_s^++F_{\rm int}-\sigma T^4

The illuminating cosine is clipped at zero on the dark side. Every term is a power per area.

Projected emission

F_p=\frac{R_p^2}{D^2}\int_{\rm visible,unocculted}\frac{\sigma T^4}{\pi}\mu_o\,d\Omega

The observing cosine weights projected area. Secondary eclipse changes this visible region while the thermal equation continues.

Common difficulties

A maximum is a hemisphere sum

Typical misconceptionPeak timing directly locates the hottest parcel.

Better mental modelMany longitudes contribute together, weighted by projected area. Compare the temperature profile with the timing of the phase-curve peak.

Fast flow averages longitudes

Typical misconceptionRapid zonal flow makes every latitude equally hot.

Better mental modelEach latitude has its own mean stellar heating. This model has no exchange between latitude bands.

Run the experiment

01
Reverse the wind
Compare Eastward heat with Heat flows west. Predict the shift before changing presets.
What to observe: The temperature map mirrors, and the unobscured curve maximum changes sides of secondary conjunction.
02
Keep the steady product
Pause Eastward heat, double capacity and halve wind, then continue the clock.
What to observe: The retained field remains a steady solution because its capacity-flow product is unchanged. The capacity storage jump is saved as configuration energy.
03
Change the view
Switch to Observer view through eclipse, then inspect the same field in Flow view.
What to observe: The star hides planetary light on its path to the observer. Stellar illumination continues during secondary eclipse.

Exoplanet Thermal Phase Curves

A warm parcel crosses the evening edge

Core mathematical model

Relative flow

The radiating layer

Projected emission

Common difficulties

A maximum is a hemisphere sum

Fast flow averages longitudes

Run the experiment

Reverse the wind

Keep the steady product

Change the view

Related experiments and articles

Heat carried into an unseen night