AE-C attitude determination and control prelaunch analysis and operations plan

A description of attitude control support being supplied by the Mission and Data Operations Directorate is presented. Included are descriptions of the computer programs being used to support the missions for attitude determination, prediction, and control. In addition, descriptions of the operating procedures which will be used to accomplish mission objectives are provided

Solar and Heliospheric Observatory (SOHO) Flight Dynamics Simulations Using MATLAB (R)

This paper describes a study to verify onboard attitude control laws in the coarse Sun-pointing (CSP) mode by simulation and to develop procedures for operational support for the Solar and Heliospheric Observatory (SOHO) mission. SOHO was launched on December 2, 1995, and the predictions of the simulation were verified with the flight data. This study used a commercial off the shelf product MATLAB(tm) to do the following: Develop procedures for computing the parasitic torques for orbital maneuvers; Simulate onboard attitude control of roll, pitch, and yaw during orbital maneuvers; Develop procedures for predicting firing time for both on- and off-modulated thrusters during orbital maneuvers; Investigate the use of feed forward or pre-bias torques to reduce the attitude handoff during orbit maneuvers - in particular, determine how to use the flight data to improve the feed forward torque estimates for use on future maneuvers. The study verified the stability of the attitude control during orbital maneuvers and the proposed use of feed forward torques to compensate for the attitude handoff. Comparison of the simulations with flight data showed: Parasitic torques provided a good estimate of the on- and off-modulation for attitude control; The feed forward torque compensation scheme worked well to reduce attitude handoff during the orbital maneuvers. The work has been extended to prototype calibration of thrusters from observed firing time and observed reaction wheel speed changes

Numerical Ricci-flat metrics on K3

We develop numerical algorithms for solving the Einstein equation on Calabi-Yau manifolds at arbitrary values of their complex structure and Kahler parameters. We show that Kahler geometry can be exploited for significant gains in computational efficiency. As a proof of principle, we apply our methods to a one-parameter family of K3 surfaces constructed as blow-ups of the T^4/Z_2 orbifold with many discrete symmetries. High-resolution metrics may be obtained on a time scale of days using a desktop computer. We compute various geometric and spectral quantities from our numerical metrics. Using similar resources we expect our methods to practically extend to Calabi-Yau three-folds with a high degree of discrete symmetry, although we expect the general three-fold to remain a challenge due to memory requirements.Comment: 38 pages, 10 figures; program code and animations of figures downloadable from http://schwinger.harvard.edu/~wiseman/K3/ ; v2 minor corrections, references adde

Topologically Massive Gravity and Ricci-Cotton Flow

We consider Topologically Massive Gravity (TMG), which is three dimensional general relativity with a cosmological constant and a gravitational Chern-Simons term. When the cosmological constant is negative the theory has two potential vacuum solutions: Anti-de Sitter space and Warped Anti-de Sitter space. The theory also contains a massive graviton state which renders these solutions unstable for certain values of the parameters and boundary conditions. We study the decay of these solutions due to the condensation of the massive graviton mode using Ricci-Cotton flow, which is the appropriate generalization of Ricci flow to TMG. When the Chern-Simons coupling is small the AdS solution flows to warped AdS by the condensation of the massive graviton mode. When the coupling is large the situation is reversed, and warped AdS flows to AdS. Minisuperspace models are constructed where these flows are studied explicitly

Ricci flow and black holes

Gradient flow in a potential energy (or Euclidean action) landscape provides a natural set of paths connecting different saddle points. We apply this method to General Relativity, where gradient flow is Ricci flow, and focus on the example of 4-dimensional Euclidean gravity with boundary S^1 x S^2, representing the canonical ensemble for gravity in a box. At high temperature the action has three saddle points: hot flat space and a large and small black hole. Adding a time direction, these also give static 5-dimensional Kaluza-Klein solutions, whose potential energy equals the 4-dimensional action. The small black hole has a Gross-Perry-Yaffe-type negative mode, and is therefore unstable under Ricci flow. We numerically simulate the two flows seeded by this mode, finding that they lead to the large black hole and to hot flat space respectively, in the latter case via a topology-changing singularity. In the context of string theory these flows are world-sheet renormalization group trajectories. We also use them to construct a novel free energy diagram for the canonical ensemble.Comment: 31 pages, 14 color figures. v2: Discussion of the metric on the space of metrics corrected and expanded, references adde

A numerical approach to finding general stationary vacuum black holes

The Harmonic Einstein equation is the vacuum Einstein equation supplemented by a gauge fixing term which we take to be that of DeTurck. For static black holes analytically continued to Riemannian manifolds without boundary at the horizon this equation has previously been shown to be elliptic, and Ricci flow and Newton's method provide good numerical algorithms to solve it. Here we extend these techniques to the arbitrary cohomogeneity stationary case which must be treated in Lorentzian signature. For stationary spacetimes with globally timelike Killing vector the Harmonic Einstein equation is elliptic. In the presence of horizons and ergo-regions it is less obviously so. Motivated by the Rigidity theorem we study a class of stationary black hole spacetimes, considered previously by Harmark, general enough to include the asymptotically flat case in higher dimensions. We argue the Harmonic Einstein equation consistently truncates to this class of spacetimes giving an elliptic problem. The Killing horizons and axes of rotational symmetry are boundaries for this problem and we determine boundary conditions there. As a simple example we numerically construct 4D rotating black holes in a cavity using Anderson's boundary conditions. We demonstrate both Newton's method and Ricci flow to find these Lorentzian solutions.Comment: 43 pages, 7 figure

Exact Relativistic Two-Body Motion in Lineal Gravity

We consider the N-body problem in (1+1) dimensional lineal gravity. For 2 point masses (N=2) we obtain an exact solution for the relativistic motion. In the equal mass case we obtain an explicit expression for their proper separation as a function of their mutual proper time. Our solution gives the exact Hamiltonian to infinite order in the gravitational coupling constant.Comment: latex, 11 pages, 2 figures, final version to appear in Phys. Rev. Let

Investigating Off-shell Stability of Anti-de Sitter Space in String Theory

We propose an investigation of stability of vacua in string theory by studying their stability with respect to a (suitable) world-sheet renormalization group (RG) flow. We prove geometric stability of (Euclidean) anti-de Sitter (AdS) space (i.e., $\mathbf{H}^n$) with respect to the simplest RG flow in closed string theory, the Ricci flow. AdS space is not a fixed point of Ricci flow. We therefore choose an appropriate flow for which it is a fixed point, prove a linear stability result for AdS space with respect to this flow, and then show this implies its geometric stability with respect to Ricci flow. The techniques used can be generalized to RG flows involving other fields. We also discuss tools from the mathematics of geometric flows that can be used to study stability of string vacua.Comment: 29 pages, references added in this version to appear in Classical and Quantum Gravit

Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic

We develop an iterative method for finding solutions to the hermitian Yang-Mills equation on stable holomorphic vector bundles, following ideas recently developed by Donaldson. As illustrations, we construct numerically the hermitian Einstein metrics on the tangent bundle and a rank three vector bundle on P^2. In addition, we find a hermitian Yang-Mills connection on a stable rank three vector bundle on the Fermat quintic.Comment: 25 pages, 2 figure

Influence of Annealing on the Interface Structure and Strain Relief in Si/Ge Heterostructures on (100) Si

Research work on the general problem of the nature and thermal stability of the Si/Ge semiconductor interface is reviewed. We report on our recent studies of the interface structure in [(Si)m(Ge)n]p superlattices and (Ge)n layers buried in Si as revealed by Raman scattering, extended X-ray absorption fine structure, and X-ray techniques. Strain relaxation and interdiffusion in the superlattices caused by annealing have been investigated, and it is found that considerable strain-enhanced intermixing together with partial relaxation of Ge-Ge bonds occurs even for very short anneal times at 700°C. Further annealing leads to diffusion at a much slower rate and to the eventual formation of an alloy layer. The Ge-Ge bond lengths in as-grown samples are that expected for a fully strained Ge layer. Similar studies of the (Ge)n layers reveal that two-dimensional pseudomorphic growth proceeds up to n = 5, probably mediated by a Si-Ge interface interdiffusion over one or two monolayers of approximately 20%. A n = 12 layer gave evidence of strain relaxation by the introduction of dislocations and clustering. Interdiffusion proceeds rapidly on annealing at 750°C