A new component of the tangential YORP caused by the roughness of the asteroid surface
Authors: Golubov Oleksiy , Lipatova Veronika 
Published: September 2022

&lt;p&gt;Abstract&lt;/p&gt;
&lt;p&gt;The tangential YORP effect (or TYORP) is a radiation pressure torque, which acts on small irregularities of the asteroid surface due to their non-uniform heating. This effect causes asteroids to change their rotation rates, and in many cases, it can be larger than other non-gravitational torques. Other works have already considered TYORP produced by smooth boulders of different shapes on an asteroid's surface. Here, we present the new results on the contribution to TYORP due to the small-scale roughness on the surface of boulders or regolith. We carry out numeric simulations of the 2D heat conduction problem on a sinusoidal surface, average the radiation pressure force in time and space, and thus numerically find the TYORP experienced by the asteroid surface. We also create an approximate analytic theory of heat conduction on a slightly non-flat sinusoidal surface and use it to compute TYORP. We study the small-scale roughness of asteroid Ryugu published in other works and use it to evaluate its TYORP. As a result, we find that the numeric and analytic estimates of the tangential YORP produced by a rough surface are in good qualitative agreement with each other. The contribution of different sinusoidal harmonics to the tangential YORP is additive to good accuracy. It allows us to derive an approximate analytic formula, which expresses TYORP in terms of the Fourier spectrum of the shape of the asteroid surface. The TYORP contribution of the small-scale roughness can be comparable to or even greater than TYORP produced by boulders.&lt;/p&gt;
&lt;p&gt;Introduction&lt;/p&gt;
&lt;p&gt;The tangential Yarkovsky&amp;#8211;O'Keefe&amp;#8211;Radzievskii&amp;#8211;Paddack effect (also tangential YORP, or TYORP) is caused by the recoil light pressure and drags the asteroid&amp;#8217;s surface in the tangential direction [1]. It has previously been simulated for different geometries of smooth boulders [2, 3, 4]. Here, we study the contribution to TYORP arising from the roughness of the asteroid surface on a centimeter-scale, including both non-smoothness of the rocks and the non-flatness of the regolith. We present the recent analytic results from [5] and as yet unpublished results of the numeric simulation of the effect.&lt;/p&gt;
&lt;p&gt;Analytic model&lt;/p&gt;
&lt;p&gt;We use the perturbation theory to solve the heat conduction equation for a surface whose shape is described by a sinusoidal wave of small amplitude. The resulting temperature is used to compute the light pressure recoil force acting on the surface and thus to evaluate TYORP.&amp;#160;&lt;/p&gt;
&lt;p&gt;The expression for TYORP is second-order in terms of the surface slopes, thus it gets vanishingly small if the surface is nearly flat. TYORP of surface roughness has other features previously observed for TYORP of boulders: it vanishes at very small and very big thermal parameters, it vanishes for very short and very long wavelengths of the sinusoidal perturbation of the surface, it is maximal if the thermal parameter is of the order of unity and the wavelength of the sinusoidal perturbation is of the order of the thermal wavelength.&lt;/p&gt;
&lt;p&gt;As a qualitative result of this approximate analytic theory, we also obtain that sinusoidal waves with non-commensurable wavelengths do not interfere with each other, and their TYORP is additive.&amp;#160;&lt;/p&gt;
&lt;p&gt;Numeric model&lt;/p&gt;
&lt;p&gt;We create a computer program that numerically solves the 2D heat conduction equation under the curved surface, and uses the resulting temperature field to compute TYORP. The results of the program qualitatively agree with the analytic model, but differ in exact numeric values due to simplifying assumptions incorporated into the analytic model. We fit the numeric results with an analytic expression and thus get an equation for TYORP as simple as our analytic theory, but more precise than it.&lt;/p&gt;
&lt;p&gt;Numeric simulations of TYORP for a surface composed of two sinusoidal harmonics show that different harmonics are almost independent, with the TYORP of a surface being almost equal to the TYORP of individual harmonics.&lt;/p&gt;
&lt;p&gt;Results&lt;/p&gt;
&lt;p&gt;To apply the theory, we take the high-resolution shape model of asteroid (162173) Ryugu from [6] and decompose its shape into the Fourier harmonics. The additivity of TYORP for different Fourier harmonics tested both analytically and numerically allows us to derive a simple mathematical expression for TYORP as an integral over the Fourier power spectrum of the asteroid surface roughness. Application of this expression to Ryugu results in a TYORP value that is greater than the YORP effect due to the large-scale asymmetry of Ryugu computed from its global shape model [7].&lt;/p&gt;
&lt;p&gt;Overall, TYORP produced by the asteroid surface roughness has the same order of magnitude as TYORP produced by boulders. This new component of the YORP effect needs to be computed to correctly predict the dynamics of asteroids, and here we show how this computation can be done.&lt;/p&gt;
&lt;p&gt;&lt;br /&gt;Acknowledgments&lt;/p&gt;
&lt;p&gt;This work was supported by the National Research Foundation of Ukraine, project N2020.02/0371 &amp;#8220;Metallic asteroids: search for parent bodies of iron meteorites, sources of extraterrestrial resources&amp;#8221;.&lt;/p&gt;
&lt;p&gt;References&lt;/p&gt;
&lt;p&gt;[1] Golubov O., Krugly Y. N., 2012, ApJL 752, L11&lt;br /&gt;[2] Golubov O., Scheeres D. J., Krugly Y. N., 2014, AJ 794, 22&lt;br /&gt;[3] Sevecek P., Broz M., Capek D., Durech J., 2015, MNRAS 450, 2104&lt;br /&gt;[4] Sevecek P., Golubov O., Scheeres D. J., Krugly Y. N., 2016, A&amp;A 592, A115&lt;br /&gt;[5] Golubov O., Lipatova V., A&amp;A, accepted&lt;br /&gt;[6] Otto K. A., Matz K. D., Schr&amp;#246;der S. E., et al., 2021, MNRAS 500, 3178&lt;br /&gt;[7] Kanamaru M., Sasaki S., Morota T., et al., 2021, Journal of Geophysical Research: Planets 126, e2021JE006863.&lt;/p&gt;