Feasibility and Accuracy of Thermophysical Estimation of Asteroid 162173 Ryugu (1999 JU3) from the Hayabusa2 Thermal Infrared Imager

Abstract

We present the results of a numerical study to prepare for the remote sensing of asteroid 162173 Ryugu (1999 JU3) using the Hayabusa2 thermal infrared imager (TIR). We simulated the thermal characteristics of the asteroid with a thermophysical model (TPM) using an ideal body with a smooth and spherical surface, and investigated its feasibility to determine the thermophysical properties of the asteroid under two possible spin vectors; $(\lambda_{\mathrm{ecl}}, \beta_{\mathrm{ecl}}) = (73^{\circ}, -62^{\circ})$ and $(331^{\circ}, 20^{\circ})$. Each of the simulated snapshots taken at various local times during the 1.5-year proximity phase was analyzed to estimate uncertainties of the diurnal thermal phase delay to infer the thermal inertia of Ryugu. The temperature in a pixel was simulated based on the specification of the imager and the observing geometry. Moreover, we carried out a regression analysis to estimate albedo and thermal emissivity from the time variation of surface temperature. We also investigated the feasibility of determining thermal phase delay in a first attempt using realistic rough surfaces. We found that precise determination of the thermal phase delay would be difficult in the $(331^{\circ}, 20^{\circ})$ spin type unless the surface was nearly smooth. In contrast, the thermal phase delay is likely to be observable even if the surface topography is moderately rough in the other spin type. From the smooth-surface model, we obtained a less than 20% error of thermal inertia on observation opportunities under the likely range of thermal inertia $\leq 1000~\mbox{J}\,\mbox{m}^{-2}\,\mbox{s}^{-1/2}\, \mbox{K}^{-1}$. The error of thermal inertia exceeded 50% under a realistic surface with roughness.

Appendix A: Mathematical Description of the Imaging Simulation

We now describe mathematical implementations for simulating TIR images. In this procedure, we calculated the components of the spin vector in the imaging coordinate system to determine the asteroid’s local equatorial longitude and latitude, which are combined with the reference point of the sub-solar point.

The base vectors of the imaging coordinate system $(x',y',z')$ are defined based on the heliocentric ecliptic coordinates of the asteroid $(x_{A},y_{A},z_{A})$ and the earth $(x_{E},y_{E},z_{E})$, which are given by (see Fig. 3)

$$\begin{aligned} \boldsymbol{e}'_{z} =& \left[ \textstyle\begin{array}{c} (x_{E} - x_{A})/| EA | \\ (y_{E} - y_{A})/| EA | \\ (z_{E} - z_{A})/| EA | \end{array}\displaystyle \right] , \end{aligned}$$

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$$\begin{aligned} \boldsymbol{e}'_{x} =& \left[ \textstyle\begin{array}{c} - (y_{E} - y_{A})/| EA' | \\ (x_{E} - x_{A})/| EA' | \\ 0 \end{array}\displaystyle \right], \end{aligned}$$

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$$\begin{aligned} \boldsymbol{e}'_{y} =& \boldsymbol{e}'_{z} \times \boldsymbol{e}'_{x}, \end{aligned}$$

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where

$$\begin{aligned} | EA | =& \sqrt{(x_{E} - x_{A})^{2} + (y_{E} - y_{A})^{2} + (z_{E} - z_{A})^{2}}, \end{aligned}$$

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$$\begin{aligned} | EA' | =& \sqrt{(y_{E} - y_{A})^{2} + (x_{E} - x_{A})^{2}}. \end{aligned}$$

(21)

The spin vector of the asteroid in the imaging coordinates is given by

$$ \boldsymbol{\mathit{spin}}' = (\boldsymbol{\mathit{spin}} \cdot \boldsymbol{e}'_{x})\boldsymbol{e}'_{x} + (\boldsymbol{\mathit{spin}} \cdot \boldsymbol{e}'_{y})\boldsymbol{e}'_{y} + (\boldsymbol{\mathit{spin}} \cdot \boldsymbol{e}'_{z})\boldsymbol{e}'_{z} , $$

(22)

where

$$ \boldsymbol{\mathit{spin}} = \left[ \textstyle\begin{array}{c} \cos \beta_{\mathrm{ecl}}\cos \lambda_{\mathrm{ecl}} \\ \cos \beta_{\mathrm{ecl}}\sin \lambda_{\mathrm{ecl}} \\ \sin \beta_{\mathrm{ecl}} \end{array}\displaystyle \right] . $$

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We obtain the components of spin vector in the imaging coordinate system:

$$\begin{aligned} \lambda '_{\mathrm{ecl}} =& \tan^{ - 1}\frac{\mathit{spin}'_{z}}{\mathit{spin}'_{x}} , \end{aligned}$$

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$$\begin{aligned} \beta '_{\mathrm{ecl}} =& \tan^{ - 1}\frac{\sqrt{(\mathit{spin}'_{x})^{2} + (\mathit{spin}'_{z})^{2}}}{\mathit{spin}'_{y}} . \end{aligned}$$

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The longitude of the prime meridian is defined so as to coincide with the sub-solar longitude in the asteroid-centric coordinate system (Fig. 13). Sub-solar longitude and latitude are defined as follows, including the orbital position of a planet:

$$\begin{aligned} \phi '_{\mathrm{SSP}} =& \tan^{ - 1}\frac{\mathit{ssp}'_{z}}{\mathit{ssp}'_{x}} , \end{aligned}$$

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$$\begin{aligned} \theta '_{\mathrm{SSP}} =& \tan^{ - 1}\frac{\sqrt{(\mathit{ssp}'_{x})^{2} + (\mathit{ssp}'_{z})^{2}}}{\mathit{ssp}'_{y}} , \end{aligned}$$

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where the sub-solar vector expressed in $(x',y',z')$ system is

$$ \boldsymbol{\mathit{ssp}}' = (\boldsymbol{\mathit{ssp}} \cdot \boldsymbol{e}'_{x})\boldsymbol{e}'_{x} + (\boldsymbol{\mathit{ssp}} \cdot \boldsymbol{e}'_{y})\boldsymbol{e}'_{y} + (\boldsymbol{\mathit{ssp}} \cdot \boldsymbol{e}'_{z})\boldsymbol{e}'_{z} $$

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and the normalized vector of sub-solar point expressed in $(x,y,z)$ system is

$$ \boldsymbol{\mathit{ssp}} = \left[ \textstyle\begin{array}{c} - x_{A}/r \\ - y_{A}/r \\ - z_{A}/r \end{array}\displaystyle \right] , $$

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where

$$ r = \sqrt{x_{A}^{2} + y_{A}^{2} + z_{A}^{2}} . $$

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Fig. 13

Definition of the prime meridian in the imaging coordinate system

The view seen from the direction of the spacecraft camera is made by projecting the 3D coordinates of asteroid in $(x',y',z')$ to a rendering plane $x'y'$. We redefine the $x'y'$ plane as a $XY$ plane for simple representation of symbols.

The 2D components are given by

$$\begin{aligned} \left[ \textstyle\begin{array}{c} X \\ Y \end{array}\displaystyle \right] =& \left[ \textstyle\begin{array}{c@{\quad}c@{\quad}c} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\displaystyle \right] \left[ \textstyle\begin{array}{c} x' \\ y' \\ z' \end{array}\displaystyle \right], \quad z' \ge 0 \end{aligned}$$

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$$\begin{aligned} \left[ \textstyle\begin{array}{c} x' \\ y' \\ z' \end{array}\displaystyle \right] =& \left[ \textstyle\begin{array}{c} R\sin \theta '\cos \phi ' \\ R\cos \theta ' \\ R\sin \theta '\sin \phi ' \end{array}\displaystyle \right] \equiv R \boldsymbol{\mathit{vec}}(\theta ',\phi ') \end{aligned}$$

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where $R$ is the radius of the target asteroid.

The asteroid-centric equatorial latitude $\varTheta$ and longitude $\varPhi$ at $(X, Y)$, where $X^{2} + Y^{2} < R^{2}$ is satisfied in the image coordinate, are calculated based on the definition of the projected plane in imaging coordinates and the radius of the target asteroid:

$$\begin{aligned} \varTheta =& \frac{\pi}{2} - \cos^{ - 1}(\boldsymbol{p} \cdot \boldsymbol{q}), \end{aligned}$$

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$$\begin{aligned} \varPhi =& \left\{ \textstyle\begin{array}{l@{\quad}l} \cos^{ - 1}\frac{(\boldsymbol{p} \times \boldsymbol{s}) \cdot (\boldsymbol{p} \times \boldsymbol{q})}{| \boldsymbol{p} \times \boldsymbol{s} | \| \boldsymbol{p} \times \boldsymbol{q} |} &\mbox{for}\ (\boldsymbol{p} \times \boldsymbol{s}) \cdot \boldsymbol{q} \ge 0 ,\\ - \cos^{ - 1}\frac{(\boldsymbol{p} \times \boldsymbol{s}) \cdot (\boldsymbol{p} \times \boldsymbol{q})}{| \boldsymbol{p} \times \boldsymbol{s} | | \boldsymbol{p} \times \boldsymbol{q} |} &\mbox{for}\ (\boldsymbol{p} \times \boldsymbol{s}) \cdot \boldsymbol{q} < 0, \end{array}\displaystyle \right. \end{aligned}$$

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where

$$\begin{aligned} \boldsymbol{p} =& \boldsymbol{\mathit{vec}}(\beta '_{\mathrm{ecl}},\lambda '_{\mathrm{ecl}}), \end{aligned}$$

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$$\begin{aligned} \boldsymbol{s} =& \boldsymbol{\mathit{vec}}(\theta '_{\mathrm{SSP}},\phi '_{\mathrm{SSP}}) \end{aligned}$$

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and

$$ \boldsymbol{q} = \frac{1}{R}\left[ \textstyle\begin{array}{c} X \\ Y \\ \sqrt{R^{2} - X^{2} - Y^{2}} \end{array}\displaystyle \right] . $$

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