Modelling thermal effects on eccentric asteroids: application to Didymos

The Hera mission, which will reach the Didymos system in December 2026, allows for a unique opportunity to study the thermal effects acting on a small body. Among these, the Yarkovsky effect is known to play a major role in the dynamics of asteroids in the Solar System. Thanks to the Inter-Satellite Link (ISL) between the main spacecraft and the two CubeSats, Milani and Juventas, which allow for a precise orbit determination process, scientists will be able to measure the Yarkovsky effect with unprecedented accuracy. This will require a precise model to be included in orbit determination software, such as Orbit14 from the University of Pisa [2]. In most applications, the Yarkovsky effect is computed assuming a circular orbit. However, the eccentricity of Didymos is estimated to be 0.38. Therefore, to have a precise model, one should compute the temperature of the body in the generic case of a non-circular orbit, resulting in a variation of the average temperature over time. Once the average temperature and its local surface deviations are known, it is possible to analytically compute the Yarkovsky effect in either its seasonal, diurnal or mixed variant. 

To compute the temperature of a body orbiting the Sun one needs to solve a heat diffusion problem. Usually, the problem is split into two parts, one concerning the average temperature 𝑇_av, while the other focuses on the surface deviations Δ𝑇 (see [1]).  Δ𝑇 can be solved for in terms of 𝑇_av. We developed an algorithm to compute the average temperature as a function of the mean anomaly of an asteroid on a non-circular orbit around the Sun, up to a desired accuracy. Here we present the application of this method to Didymos, including the estimate of the Yarkovsky diurnal, seasonal and mixed effects. We provide a comparison with the circular case and other estimates found in recent literature. 

We model Didymos as a sphere of uniform density ϱ and radius R, behaving as a Lambertian scatterer (i.e. each point on the surface re-emits the absorbed heat isotropically), with values of the absorption coefficient α and emissivity ε that are typical of S-class asteroids.  

Following [1], we adopt normalized variables. The most relevant quantities appearing in this formulation are the normalized radius and the thermal inertia. The former gives a measure of the size of the body when compared to the thermal length, while the latter is responsible for the delay in thermal re-emission that causes the Yarkovsky effect. 

The normalized average temperature 𝑇′_av is expressed as a power series of 𝜁=𝑒^𝑖𝑀 where 𝑀 is the mean anomaly of Didymos in its orbit around the Sun. The coefficients of the power series are functions of the eccentricity. They are expressed in a power series of the parameter 𝛽=𝑒 (1-√(1-𝑒²))^-1. The coefficients are computed iteratively up to a selected degree in 𝜁 and 𝛽. Since the eccentricity of the orbit of Didymos around the Sun is 0.38, one should consider the maximum order in the expansion to be large enough to account for small variations in the temperature occurring along the orbit.  

Once the normalized average temperature is estimated, it is used to compute the surface deviations Δ𝑇′, which in turn are used to obtain the Yarkovsky acceleration. The resulting acceleration is compared to the estimates present in the literature. 

The inclusion of the average temperature feature in the dynamical model is used in the computation of the three variants of the Yarkovsky acceleration, while its implementation in Orbit14 allows for the estimate of the parameters through an orbit determination process. This procedure is convenient to assess the relevance of the mixed terms corresponding to the coupling of the diurnal and seasonal variants. 

Finally, we discuss future work that is expected to stem from this research and its application in the context of the Orbit14 software and of the Hera radioscience experiment. 

Bibliography 

[1] Michel, P., Küppers, M., Bagatin, A. C., Carry, B., Charnoz, S., De Leon, J., ... & Carnelli, I. (2022). The ESA Hera mission: detailed characterization of the DART impact outcome and of the binary asteroid (65803) Didymos. The planetary science journal, 3(7), 160. 

[2] Lari, G., Schettino, G., Serra, D., & Tommei, G. (2022). Orbit determination methods for interplanetary missions: development and use of the Orbit14 software. Experimental Astronomy, 53(1), 159-208.  

[3] Vokrouhlick ́y, D. (1999). A complete linear model for the Yarkovsky thermal 
force on spherical asteroid fragments. A&A, 344:362–366.