How can a 1-pixel image of a rotating asteroid be used to measure its thermal inertia?

Question

Phys.org's Observatory in Chile takes highest-resolution measurements of asteroid surface temperatures ever obtained from earth discusses imaging of millimeter wave imaging of the surface of asteroid 16 Psyche by ALMA. Per Wikipedia, at over 200 km in diameter Psyche "is the most massive of the metal-rich M-type asteroids".

It links to the extensive paper in the open access The Planetary Science Journal; The Surface of (16) Psyche from Thermal Emission and Polarization Mapping.

The Phys.org summary discusses the thermal inertia of the surface. Roughly speaking that's how slowly an area on the surface heats up or cools down after local sunrise or sunset.

A straightforward way do measure thermal inertia of a roughly spherical body that's rotating is to track the thermal radiation of a given area as it transitions back and forth between sunlit and dark (day and night) and gauge the slowness of the temperature change.

Read about thermal inertia measurements of Mercury for example in:

• Why the thermal imaging of Mercury's surface requires a telescope on a jet flying through an eclipse?

and answers therein.

But then I read in Phys.org:

Typically, thermal observations from Earth—which measure the light emitted by an object itself rather than light from the sun reflected off of that object—are in infrared wavelengths and can produce only 1-pixel images of asteroids. That one pixel does, however, reveal a lot of information; for example, it can be used to study the asteroid's thermal inertia, or how fast it heats up in sunlight and cools down in darkness.

Question: How can a 1-pixel image of a rotating asteroid be used to measure its thermal inertia?

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Millimeter-wavelength emissions reveal the temperature of the asteroid Psyche as it rotates through space. Credit: California Institute of Technology

Source

Answer 1

On the basis of this review article by Delbo et al. (2015) about modeling asteroid properties, I think it might work something like this:

1. From optical measurements (where you're seeing reflected sunlight) and knowledge of the asteroid's orbit and its current orientation with respect to the Sun and Earth, you can work out its approximate size, shape, and rotation period, even from unresolved ("1-pixel") data. (Something like this is how we got the idea that 'Oumuamua is probably "cigar-shaped".) You do have to make assumptions about the albedo, though possibly that can be constrained by optical color measurements, and can presumably also be part of the subsequent thermal modeling.

2. You then try modeling the temperature variations and resulting thermal emissions of the asteroid as it rotates, given what you know about its shape and how it's illuminated by the Sun as it rotates (a "thermophysical model"). One of the parameters in the model is the thermal inertia, which can have a strong effect on the temperature variations, as shown in this figure:

Temperature at the equator of an asteroid spinning with a period of 6 hours (at a distance of 1.1 au from the Sun), for different values of thermal inertia (numbered curves, in units of J m$^{-2}$ s$^{-1/2}$ K$^{-1}$). [Fig.2 of Delbo et al.]

Varying the parameters of the model (including the thermal inertia) and comparing it to the data allows you to put some constraints on those parameters. Here's an example of a model prediction for mid-IR emission, along with the observations:

Predicted flux at 8 microns (dashed curve) for a thermophysical model of the asteroid Bennu over one rotation period, along with actual observed values (diamonds with error bars). [Fig.5 of Delbo et al., based on Emery et al. (2014); according to the abstract of that paper, the best-fitting model has a thermal inertial of $310 \pm 70$ J m$^{-2}$ s$^{-1/2}$ K$^{-1}$.]

(I would imagine that you could improve your modeling by observing the rotational variations in both reflected optical sunlight and emitted thermal radiation at different points in the asteroid's orbit, where both its distance from the Sun and its orientation relative to us will change. For example, an asteroid that makes a $90^{\circ}$ angle with respect to the Sun-Earth line will show us about half of its illuminated side and half of its night side, while an asteroid at opposition to the Sun will show the full illuminated side.)