## Thesis etd-05282012-112137 |

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Thesis type

Tesi di dottorato di ricerca

Author

CICALO', STEFANO

URN

etd-05282012-112137

Thesis title

On theoretical and experimental aspects of the rotation of celestial bodies

Academic discipline

MAT/07

Course of study

MATEMATICA

Supervisors

**tutor**Prof. Milani Comparetti, Andrea

Keywords

- asteroidi
- asteroids
- celestial mechanics
- interplanetary missions
- meccanica celeste
- missioni interplanetarie
- pianeti
- planets
- rotation
- rotazione

Graduation session start date

20/06/2012

Availability

Full

Summary

This Thesis will address two issues related to the rotation of celestial bodies: (1) The determination of the rotation state of Mercury and (2) The non-principal rotation secular evolution of an asteroid under the non gravitational YORP torque.

(1)Space missions can have as a goal the determination of the interior structure of a planet: this is the case for the ESA mission BepiColombo to be launched towards Mercury. Very precise range and range-rate tracking from Earth and on-board accelerometry will provide a huge amount of data, from which it will be possible to study the gravity field of Mercury and a large number of other parameters of interest (Mercury Orbiter Radio-science Experiment). Gravity can be used to constrain the interior structure, but cannot uniquely determine the interior mass distribution. A much stronger constraint on the interior can be given by also determining the rotation state of the planet. If the planet is asymmetric enough, the gravity field as measured by an orbiting probe tracked from the Earth contains signatures from the rotation of the planet. Are these enough to solve for the rotation state, to the required accuracy, from tracking data alone, without measurements of the surface? In order to give an analytical argument to the determination of the rotation state from gravimetry, a simplified analytical model is developed, and the symmetry breaking occurring when the shape of the planet deviates from spherical symmetry is characterized by explicit formulas giving order of magnitude. Moreover, a full cycle numerical simulation of the Radio-science experiment is performed, including the generation of simulated tracking and accelerometer data and the determination, by least squares fit, of the Mercurycentric initial conditions of the probe, of the Mercury gravity field and its rotation state, together with a large number of other parameters affecting the Mercurycentric dynamics. The conclusion is that there is no reason of principle prohibiting the determination of the rotation from gravimetry, the sensitivity of the measurements and the coverage are good enough to perform the experiment at the required level of accuracy. This will be important also in ensuring independent terms of comparison for the rotation experiment performed with high resolution camera. However, the design of the mission is currently under development and much care has to be taken in guaranteeing the scientific goals as long as the scenario will change.

(2) The secular effect of YORP torque, which is the consequent effect to infrared thermal emission from the surface of a body irradiated by the Sun, on the rotational dynamics of an asteroid in non-principal axis rotation is studied. The general rotational equations of motion are derived and approximated with an illumination function expanded up to second order. The main limitations of the analytical approach consists in treating the thermal conductivity and in modeling the illuminated-shaded portion of the body’s surface hit by the Sun. We used locally-convex bodies, i.e. we made the strong approximation that there is no self-shadowing and we assumed zero thermal conductivity. The resulting equations of motion can be averaged over the fast rotation angles to yield secular equations for the angular momentum, dynamic inertia and obliquity. We studied the properties of these secular equations and compared the results to previous research. The results and the conclusions are in agreement with the analytical results about principal axis rotators present in the literature, and they are only partially in agreement with the numerical results of Vokrouhlicky et al. (2007) about non-principal rotators, although they have in common all of the main features seen in that work.

Finally, an application to several real asteroid shapes is made, using discrete models with triangular facets. In particular we studied the predicted rotational dynamics of the asteroid Toutatis, which is known to be in a non-principal axis state. Evident features of chaotic behavior of the dynamical system are present, both due to the choice of the rotation initial conditions and to the choice of the particular shape, and so of the illumination model.

(1)Space missions can have as a goal the determination of the interior structure of a planet: this is the case for the ESA mission BepiColombo to be launched towards Mercury. Very precise range and range-rate tracking from Earth and on-board accelerometry will provide a huge amount of data, from which it will be possible to study the gravity field of Mercury and a large number of other parameters of interest (Mercury Orbiter Radio-science Experiment). Gravity can be used to constrain the interior structure, but cannot uniquely determine the interior mass distribution. A much stronger constraint on the interior can be given by also determining the rotation state of the planet. If the planet is asymmetric enough, the gravity field as measured by an orbiting probe tracked from the Earth contains signatures from the rotation of the planet. Are these enough to solve for the rotation state, to the required accuracy, from tracking data alone, without measurements of the surface? In order to give an analytical argument to the determination of the rotation state from gravimetry, a simplified analytical model is developed, and the symmetry breaking occurring when the shape of the planet deviates from spherical symmetry is characterized by explicit formulas giving order of magnitude. Moreover, a full cycle numerical simulation of the Radio-science experiment is performed, including the generation of simulated tracking and accelerometer data and the determination, by least squares fit, of the Mercurycentric initial conditions of the probe, of the Mercury gravity field and its rotation state, together with a large number of other parameters affecting the Mercurycentric dynamics. The conclusion is that there is no reason of principle prohibiting the determination of the rotation from gravimetry, the sensitivity of the measurements and the coverage are good enough to perform the experiment at the required level of accuracy. This will be important also in ensuring independent terms of comparison for the rotation experiment performed with high resolution camera. However, the design of the mission is currently under development and much care has to be taken in guaranteeing the scientific goals as long as the scenario will change.

(2) The secular effect of YORP torque, which is the consequent effect to infrared thermal emission from the surface of a body irradiated by the Sun, on the rotational dynamics of an asteroid in non-principal axis rotation is studied. The general rotational equations of motion are derived and approximated with an illumination function expanded up to second order. The main limitations of the analytical approach consists in treating the thermal conductivity and in modeling the illuminated-shaded portion of the body’s surface hit by the Sun. We used locally-convex bodies, i.e. we made the strong approximation that there is no self-shadowing and we assumed zero thermal conductivity. The resulting equations of motion can be averaged over the fast rotation angles to yield secular equations for the angular momentum, dynamic inertia and obliquity. We studied the properties of these secular equations and compared the results to previous research. The results and the conclusions are in agreement with the analytical results about principal axis rotators present in the literature, and they are only partially in agreement with the numerical results of Vokrouhlicky et al. (2007) about non-principal rotators, although they have in common all of the main features seen in that work.

Finally, an application to several real asteroid shapes is made, using discrete models with triangular facets. In particular we studied the predicted rotational dynamics of the asteroid Toutatis, which is known to be in a non-principal axis state. Evident features of chaotic behavior of the dynamical system are present, both due to the choice of the rotation initial conditions and to the choice of the particular shape, and so of the illumination model.

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