Reflected sunlight and emitted thermal radiation

One body can supply more than one kind of radiation

A sunlit body redirects some incident radiation toward its surroundings. This reflected or scattered light carries information about the illumination, the body’s optical properties and the directions of illumination and observation. It does not require the body to produce that light through its own internal energy source.

The same body emits thermal radiation associated with its temperature and material properties. Absorbed sunlight can help maintain that temperature, while internal energy and exchange with the surroundings can also matter. Reflection and thermal emission therefore describe different processes even when the Sun ultimately supplies much of the energy.

Wavelength is useful evidence, but does not by itself name the process. Reflected sunlight extends beyond visible wavelengths, and a sufficiently hot object can emit visible thermal radiation. A measured infrared band can contain reflected and emitted components together.

For an original accounting example, suppose a calibrated band signal is 10 arbitrary units after the chosen background subtraction. Contributions of 6 reflected plus 4 thermal units and 2 reflected plus 8 thermal units both sum to 10. These alternative sums do not constitute two complete physical surface models; they show why the total alone cannot determine its decomposition.

A brighter point is not automatically a larger body

Reflected flux depends on incident sunlight, the source’s cross-sectional scale, wavelength-dependent scattering, illumination geometry and distance to the observer. An unresolved point lacks a directly measured outline. Its total brightness must be interpreted through a model before it becomes a size estimate.

Geometric albedo compares a body’s brightness at zero phase angle with a specified ideal diffusely reflecting flat disk of the same cross-sectional area. It is a directional brightness comparison. Bond albedo instead describes the fraction of total incident stellar radiant power scattered back into space over all outgoing directions and wavelengths.

These two albedos answer different questions and cannot simply replace one another in an equation. A reflectance quoted at one wavelength and one pair of illumination and viewing angles is yet another specifically defined measurement. Always retain that definition with its numerical value.

As a synthetic comparison, hold the illumination, observer distance and normalized phase response fixed for two spherical-body models. Their reflected signal then scales with pR², where p is geometric albedo and R is radius. A model with p = 0.16 and R = 1 L₀ has the same product, 0.16 L₀², as p = 0.04 and R = 2 L₀. Equal brightness has allowed different sizes.

Use the appropriate area at each stage of energy accounting

Now choose a deliberately restricted energy-balance model. A sphere of radius R receives nearly parallel stellar radiation with irradiance F in watts per square metre at its location. Its distance from the star is fixed and much larger than the bodies’ radii. Intervening space is transparent.

The sphere intercepts that beam over its projected disk, with area πR². If its Bond albedo is A_B, absorbed stellar power is (1 − A_B)FπR². The symbol A_B here is a dimensionless reflected-energy fraction, between zero and one, rather than the geometric albedo used in the brightness example.

Assume uniform temperature, unit thermal emissivity, no internal heat and steady balance, with negligible incoming thermal radiation from the surroundings. Thermal power then leaves over the whole spherical area 4πR² and is 4πR²σT⁴. Temperature T is absolute temperature in kelvin; σ is the Stefan–Boltzmann constant.

Equating these powers cancels the radius and gives T_eq = [(1 − A_B)F/(4σ)]¹ᐟ⁴. The factor four comes from the ratio of emitting to intercepting area. It does not mean sunlight arrives equally at every surface point; uniform emission is an explicit redistribution assumption.

Calculate a model temperature before interpreting it

Set F = 1360 W m⁻² and A_B = 0.25 as original teaching inputs. Use the rounded constant σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴. The absorbed stellar power divided by the full sphere area is 0.75 × 1360/4 = 255 W m⁻².

Taking the fourth root of 255/σ gives T_eq ≈ 259 K. The irradiance value is a stipulated input, not a claim about a particular observed world or a measured surface temperature. The result inherits every assumption in the preceding model.

For the same isotropically radiating star, move the model to four times the distance while preserving its albedo and all other assumptions. Irradiance falls by a factor of sixteen to 85 W m⁻². Mean absorbed power per surface area is then 15.9375 W m⁻², and T_eq is about 129 K, half the earlier unrounded value.

The fourth-root dependence explains that scaling. A radius cancellation within this model does not establish that all equally illuminated real bodies have the same temperature. Temperature distributions, atmospheric radiation transfer, thermal emissivity, rotation and internal heat can change what the observations mean.

Combine bands and geometry through an explicit physical model

An all-wavelength emitted-power relation is different from a flux measured through one filter. Recovering total power or a temperature from finite-band data requires an emission spectrum, instrumental response and observing geometry. The earlier blackbody lesson supplies one restricted spectral model rather than a universal description of a surface.

Thermal observations can help constrain size and reflectivity when combined with reflected-light data. That improvement still depends on assumptions about shape, temperature distribution, emissivity and the illuminated region. Independent size constraints provide useful tests of the chosen model.

Historical work with Hayabusa2’s Optical Navigation Camera compared Ryugu observations at different geometries and with ground-based spectra. It illustrates the need for photometric corrections before interpreting a brightness difference as a material difference. No current mission capability or unique composition follows from that example.

A useful result states which band was measured, which physical contributions were modelled, and which assumptions connect the signal to a property. The next lesson examines the illumination and viewing angles that make even an unchanged body look different.

Worked examples

Try the reasoning

1. Does an infrared detection by itself establish purely thermal emission?

   Show the answer and reasoning

   No.

   Reflected and emitted components can both contribute; wavelength and a physical model must be considered together.

2. May geometric albedo replace Bond albedo without further justification in the total absorbed-power equation?

   Show the answer and reasoning

   No.

   One is a directional brightness comparison; the other is an integrated reflected-energy fraction.

3. Why does the uniform-sphere balance contain a factor four?

   Show the answer and reasoning

   The sphere emits over four times its projected intercepting area.

   The relevant areas are 4πR² and πR² under the stated geometry and redistribution assumptions.

4. Does a band total of 10 units determine the reflected and thermal contributions separately?

   Show the answer and reasoning

   No.

   Different nonnegative component sums can give the same total; additional measurements and modelling are required.

What to carry forward

• Reflection and thermal emission are processes that may contribute together in one band.
• Brightness-to-size inference requires an explicit reflectivity and geometry model.
• Equilibrium temperature is conditional energy accounting with defined areas, albedo and emission assumptions.