If used on an object on the moon, how cold would an ultrawhite painted object get?

Question

Recently in the media https://www.google.com/amp/s/www.bbc.co.uk/news/science-environment-56749105.amp very white paint has been reported on that can reduce an object's temperature below ambient.

If used on an object on the Moon, assuming that it's located in a standard position (not on the poles, not in a deep pit) what temperature range would it experience?

Bonus points for addressing Mars as well!

Answer 1

TL;DR

...it's not clear with all the extra UV heating from the Sun and thermal heating from the Surface if you could just slap this paint on something on the Moon and it would get cold.

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The Purdue University press release The whitest paint is here – and it’s the coolest. Literally. and the BBC's 'Whitest ever' paint reflects 98% of sunlight are some sources of information.

The images below are from a low-resolution version of a figure in the paywalled paper Ultrawhite BaSO4 Paints and Films for Remarkable Daytime Subambient Radiative Cooling. I've included the full abstract below, but one key sentence is:

reaches an ultrahigh solar reflectance of 97.6%

and the other is

and a high sky window emissivity of 0.96

Presumably the "sky window" is a portion of the thermal IR spectrum for which Earth's sky is somewhat transparent; cf. a description of a "gedanken thermometer" in How cold is the Martian sky at night? Or the day for that matter?:

If you take a simple infrared "thermometer" which is actually a lot like a bolometer and point it at a clear sky, it says "cold!" because in the wavelength range that it's using (roughly 5 to 15 microns) the Earth's atmosphere is partially transparent and not really an effective blackbody radiator, so the "thermometer" is really somewhat exposed to the cold of space. Point it at a cloud, and it will register "not as cold". It's (roughly) why clear nights are colder than cloudy nights.

I've zoomed and sharpened a bit of the image below to show two of the paint's "secret sauces" that make it better at staying cool, which are also discussed at length in @Puffin's answer:

1. A suppressed absorptivity (=emissivity) at near IR wavelengths (1 to 3 microns), where the Sun still shines but we wouldn't see or appreciate it. A normal white paint would not need to be so reflective in near IR, but it's critical for staying cool.
2. Very high emissivity (=absorptivity) at thermal IR wavelengths, not only in Earth's "sky window" but throughout the thermal IR range, where it's at about 0.95.

From How much is really known about those liquid metal droplets orbiting the Earth? (e.g. sizes, composition, orbits...) Are any actually tracked?

The equilibrium temperature of a spherical cow in space can be derived from here (found here):

$$T^4 = \frac{\epsilon_{vis}}{\epsilon_{therm}} \frac{I_0}{4 \sigma}$$

where $I_0$ is the solar intensity at 1 AU of about 1361 W/m^2 and $\sigma$ is the Stefan-Boltsman constant 5.67E-08 W/m^2/K^4.

Plug in $\epsilon_{vis}$ of 1-0.976 = 0.024, and $\epsilon_{therm}$ of about 0.95, and you would get a space temperature of only about 197 K or -76 C.

However

Two things would interfere with this.

1. On the Moon you'd have radiative heating from the surface of the Moon below and that would require a heat shield, perhaps a multi-layer reflective coated polymer film stack like the one on the JWST. It will leak a bit, so you'll still have some heating.
2. Without Earth's atmosphere, you now have some UV light intensity that the paint on Earth doesn't have to worry about. It looks like below 400 nm the absorptivity skyrockets! (pun intended). This paint is black in the UV and will bask in the ultraviolet warmth of the Sun that we are mostly spared on Earth. It looks like a factor of 4 or 5 more (yellow vs red) in the plot in @Puffin's answer.

Conclusion

So it is not clear with all the extra UV heating from the Sun and thermal heating from the Surface if you could just slap this paint on something on the Moon and it would get cold. It will take a detailed thermal analysis and possibly a tweak to the UV reflectivity of the paint material itself.

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Abstract:

Radiative cooling is a passive cooling technology that offers great promises to reduce space cooling cost, combat the urban island effect, and alleviate the global warming. To achieve passive daytime radiative cooling, current state-of-the-art solutions often utilize complicated multilayer structures or a reflective metal layer, limiting their applications in many fields. Attempts have been made to achieve passive daytime radiative cooling with single-layer paints, but they often require a thick coating or show partial daytime cooling. In this work, we experimentally demonstrate remarkable full-daytime subambient cooling performance with both BaSO4 nanoparticle films and BaSO4 nanocomposite paints. BaSO4 has a high electron band gap for low solar absorptance and phonon resonance at 9 μm for high sky window emissivity. With an appropriate particle size and a broad particle size distribution, the BaSO4 nanoparticle film reaches an ultrahigh solar reflectance of 97.6% and a high sky window emissivity of 0.96. During field tests, the BaSO4 film stays more than 4.5 °C below ambient temperature or achieves an average cooling power of 117 W/m2. The BaSO4-acrylic paint is developed with a 60% volume concentration to enhance the reliability in outdoor applications, achieving a solar reflectance of 98.1% and a sky window emissivity of 0.95. Field tests indicate similar cooling performance to the BaSO4 films. Overall, our BaSO4-acrylic paint shows a standard figure of merit of 0.77, which is among the highest of radiative cooling solutions while providing great reliability, convenient paint form, ease of use, and compatibility with the commercial paint fabrication process.

Source ACS Applied Materials & Interfaces @ACS_AMI tweet

Answer 2

You need more information than in the OP today (28/04/2021), though you could look at it this way:

1. You didn't specify whether the "white paint" really just blocks visible light or also all solar illumination in the the nearby non-visible spectrum, mainly the very near infra-red, VNIR, to at least 2.5, maybe 4 microns, see the figure below. This will make a difference, refer to the rationale at item, 3, below.
2. For longer wavelengths still the radiation boundary with the local lunar features is controlled by the thermal infra-red emissivity (8 - 15 microns). You will need an assumption for this property also of the "ultrawhite" object, though paint is typically high emissivity, say 0.8.
3. with very low solar absorption the inbound heat flows, and thus the temperature, will be dominated by other means. Noting point 1 above, that VNIR may play a significant role, as definitely will radiation exchange at thermal infra-red wavelengths and also the conduction through whatever the object is supported on to the lunar surface. You actually have to get a handle on all potential heat input paths before you can decide whether to disregard any.
4. To have very low heat input and finite output would lead to low temperatures but the lower the heat input is will lead to progressively larger uncertainties over the equilibrium temperature for any given material properties uncertainties.