A power series inverse in time of the universal Kepler equation
Series inverse in powers of time and radius of convergence for universal form of Kepler equation, discussing recursion formulas for coefficients
Estes, R. H.↗
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Thermal analogies to Kepler's laws and their application to orbital heat transfer calculations
Integral expressions of radiant transfer in Keplerian orbits are developed in which quantities of heat are represented by areas swept over by thermal radii (presently introduced). The integrals are shown to be analogous in form to the solutions provided by Kepler's laws in orbital mechanics, when the radiation field is concentric with the gravitational field and possesses spherical symmetry. The more general case involving aspherical radiation fields and multiple heat sources is treated by graphical integration, using polar diagrams. Applications pertaining to the solar system and to the thermal balance of a conceptual Venus Radar Mapper Spacecraft are discussed.
Buna, T.↗
Autonomous orbital navigation using Kepler's equation
A simple method of determining the six elements of elliptic satellite orbits has been developed for use aboard manned and unmanned spacecraft orbiting the earth, moon, or any planet. The system requires the use of a horizon sensor or other device for determining the local vertical, a precision clock or timing device, and Apollo-type navigation equipment including an inertial measurement unit (IMU), a digital computer, and a coupling data unit. The three elements defining the in-plane motion are obtained from simultaneous measurements of central angle traversed around the planet and elapsed flight time using a linearization of Kepler's equation about a reference orbit. It is shown how Kalman filter theory may also be used to determine the in-plane orbital elements. The three elements defining the orbit orientation are obtained from position angles in celestial coordinates derived from the IMU with the spacecraft vertically oriented after alignment of the IMU to a known inertial coordinate frame.
Boltz, F. W.↗
Extension of Gauss' method for the solution of Kepler's equation
Gauss' method for solving Kepler's equation is extended to arbitrary epochs and orbital eccentricities. Although originally developed for near parabolic orbits in the vicinity of pericenter, a generalization of the method leads to a highly efficient algorithm which compares favorably to other methods in current use. A key virtue of the technique is that convergence is obtained by a method of successive substitutions with an initial approximation that is independent of the orbital parameters. The equations of the algorithm are universal, i.e., independent of the nature of the orbit whether elliptic, hyperbolic, parabolic or rectilinear.
Battin, R. H.↗
A general algorithm for the solution of Kepler's equation for elliptic orbits
An efficient algorithm is presented for the solution of Kepler's equation f(E)=E-M-e sin E=0, where e is the eccentricity, M the mean anomaly and E the eccentric anomaly. This algorithm is based on simple initial approximations that are cubics in M, and an iterative scheme that is a slight generalization of the Newton-Raphson method. Extensive testing of this algorithm has been performed on the UNIVAC 1108 computer. Solutions for 20,000 pairs of values of e and M show that for single precision, 42.0% of the cases require one iteration, 57.8% two and 0.2% three. For double precision one additional iteration is required.
Ng, E. W.↗
Milgrom's revision of cosmic dynamics: Amending Newton's laws or Keplers?
Milgrom's recent revision of Newtonian dynamics was introduced to eliminate the inference that large quantities of invisible mass exist in galaxies. Simple examples show that a Milgrom acceleration, in the form presented so far, imply other far-reaching changes in dynamics. The momentum of an isolated system is not conserved, and the usual theorem for center-of-mass motion of any system does not hold. Naive applications require extreme caution. The model fails to provide a complete description of particle dynamics and should be thought of as a revision of Kepler's laws rather than Newton's. The Milgrom acceleration also implies fundamental changes in cosmology. A quasi-Newtonian calculation adapted from Newtonian cosmology suggests that a Milgrom universe will recollapse even if the classical closure parameter theta is less than 1. The solution, however, fails to satisfy the cosmological principle. Reasons for the breakdown of this calculation are examined. A theory of gravitation needed before the behavior of a Milgrom universe can be predicted.
Felten, J. E.↗
Self-consistent models for the X-ray emission from supernova remnants - An application to Kepler's remnant
A tool for exploring the evolution of X-ray emission from young SNRs is presented which employs a novel approach to the problem of time-dependent ionization in a shock-heated plasma. The solution of this problem is coupled to a spherically symmetric hydrodynamic calculation for the evolution of a point explosion in a uniform medium. The method is applied to Kepler's SNR, and two narrowly constrained classes of models which can simultaneously fit the spectral and morphological features of the object are found. One of these is a Sedov model in which the emission arises from shocked ambient gas, and the other is a reverse-shock model in which the SN ejecta is the dominant source of radiation. The emission from one specific model in each class is compared with the radial surface brightness profile, the 0.2-4.5 keV broadband spectrum, and the 1-3 keV moderate-resolution spectrum of the remnant. Reasonable fits are obtained in both cases.
Hughes, J. P.↗
Efficient inverse solution of Kepler's equation
A bicubic polynomial approximation to Kepler's equation for elliptic orbits is shown to provide accurate starting values for efficient numerical solution of this equation for eccentric anomaly. In the approximate equation, a cubic in mean anomaly is set equal to a cubic in eccentric anomaly. The coefficients in the two cubics are obtained as functions of eccentricity by specifying values of function and slope at the midpoint and the endpoint of the complete interval (0 to pi). The initial estimate of eccentric anomaly to use in an iteration formula is obtained by evaluating the cubic in mean anomaly and finding the single real root of the cubic in eccentric anomaly. Numerical results are presented which indicate that the estimate accuracy of this method is roughly an order of magnitude better than that of other recently-reported formulas.
Boltz, Frederick W.↗
The evolution of the radio emission from Kepler's Supernova remnant
High-resolution radio maps of Kepler's Supernova remnant (SNR) using all four arrays of the VLA have been obtained at wavelengths of 20 and 6 cm. They show the complete structure of the remnant; all features are resolved with sizes greater than about 2 arcsec, and the relative brightness of the smooth component near the center is about 1/4 the brightness of the rim. The results have been compared with earlier more limited data to measure changes in the remnant over a four-year time span. The SNR is expanding with a mean rate of R proportional to t exp 0.50 with considerable variations around the shell. Values range from R proportional to t exp 0.35 on the bright northern rim to R proportional to t exp 0.65 on the eastern part of the shell. The measurements are consistent with expansion into a variable circumstellar medium.
Dickel, John R.↗
Naturally occurring continued fractions in the variation of Kepler's equation
A family of functions involving integrals of universal functions is introduced. These functions have some interesting mathematical properties including the fact that they may be expressed as Gaussian continued fractions. A unique method of performing the integration is demonstrated which indicates why these functions may be important in the variation of Kepler's equation.
Shepperd, Stanley W.↗
Universal Lambert and Kepler algorithms for autonomous rendezvous
This paper describes Lambert and Kepler algorithms designed to be the core of an autonomous rendezvous guidance system for an onboard computer. Applications include robotic and piloted missions to the moon and planets. Flight software must be compact, fast, and totally reliable. Although high accuracy is not essential for flight, in double precision these algorithms are accurate to at least 14 places almost everywhere. Both are universal; they apply to elliptic, parabolic, hyperbolic, and even rectilinear trajectories. The algorithms are improvements to those published by Battin (1987).
Klumpp, Allan R.↗
Electron acceleration in Tycho's and Kepler's supernova remnants - Spectral evidence of Fermi shock acceleration
First model synchrotron spectra calculated with a self-consistent nonlinear shock model of first order Fermi acceleration are presented and compared with the observed radio spectra of Tycho's and Kepler's SNR. Excellent agreement is obtained, and the correct mean spectral indices of about -0.64 are easily reproduced. The model spectra are slightly concave, with hardening toward higher energies, and there is evidence for such an effect in the data, allowing the mean magnetic field strength to be estimated in each remnant. Improvements in both theory and observation could allow accurate values of magnetic fields to be inferred from sufficiently precise integrated synchrotron spectra.
Reynolds, Stephen P.↗
Kepler Equation solver
Kepler's Equation is solved over the entire range of elliptic motion by a fifth-order refinement of the solution of a cubic equation. This method is not iterative, and requires only four transcendental function evaluations: a square root, a cube root, and two trigonometric functions. The maximum relative error of the algorithm is less than one part in 10(exp 18), exceeding the capability of double-precision computer arithmetic. Roundoff errors in double-precision implementation of the algorithm are addressed, and procedures to avoid them are developed.
Markley, F. Landis↗
Planetary and Stellar Data Products Expected From The Kepler Mission
The Kepler Mission is a Discovery-class mission scheduled to be launched in the 2006-2007 time frame. It is a wide field of view photometer with a 95 m aperture designed to attain a photometric precision of 2 parts in 10^5 for the 12th magnitude stars. It will continually observe 100,000 main-sequence stars from 9th to 14th magnitude for a period of four years with a cadence of 4/hour. This database should be unique in its photometric precision, cadence, and duration of observations. Several hundred terrestrial-size planets will be detected if they are common around solar-like stars. Based on the current results of Doppler-velocity searches, over a thousand giant planets will also be found. A guest investigator program is planned that would provide the opportunity to observe thousands of other objects in the 105 square degree FOV. Such objects could include stars with micro-variability, other intrinsic variables, cataclysmic variables, eclipsing binaries (including x-ray binaries), and possibly AGN. A ground-based program to classify all 225,000 stars in the FOV and to do a detailed examination of a subset of the stars that show planetary companions is planned. Doppler-velocity observations will be made to find the presence of giant planets not seen in transit. The data will be rapidly released to the community for follow up observations and for changes to the guest investigator program.
Borucki, W. J.↗
The Kepler Mission: A Mission to Determine the Frequency of Inner Planets Neat the Habitable Zone of a Wide Range of Stars
The surprising discovery of giant planets in inner orbits around solar-like stars has brought into question our understanding of the development and evolution of planetary systems, including our solar system. To make further progress, it is critical to detect and obtain data on the frequency and characteristics of Earth-class planets. The Kepler Mission is designed to be a quick, low-cost approach to accomplish that objective. Transits by Earth-class planets produce a fractional change in stellar brightness of 5 x 10(exp -5) to 40 x 10(exp -5) lasting for 4 to 16 hours, From the period and depth of the transits, the orbit and size of the planets can be calculated. The proposed instrument is a one-meter aperture photometer with a 12 deg field-of-view (FOV). To obtain the required precision and to avoid interruptions caused by day-night and seasonal cycles, the photometer will be launched into a heliocentric orbit. It will continuously and simultaneously monitor the flux from 80,000 dwarf stars brighter than 14th magnitude in the Cygnus constellation. The mission tests the hypothesis that the formation of most stars produces Earth-class planets in inner orbits. Based on this assumption and the recent observations that 2% of the stars have giant planets in inner orbits, several types of results are expected from the mission: 1. From transits of Earth-class planets, about 480 planet detections and 60 cases where two or more planets are found in the same system. 2. From transits of giant planets, about 160 detections of inner-orbit planets and 24 detections of outer-orbit planets. 3. From the phase modulation of the reflected light from giant planets, about 1400 planet detections with periods less than a week, albedos for 160 of these giant planets, and densities for seven planets.
Borucki, W. J.↗
Kepler Mission
The first step in discovering, the extent of life in our galaxy is to determine the number of terrestrial planets in the habitable zone (HZ). The Kepler Mission is a 0.95 m aperture photometer scheduled to be launched in 2006. It is designed to continuously monitor the brightness of 100,000 solar-like stars to detect the transits of Earth-size and larger planets. The depth and repetition time of transits provide the size of the planet relative to the star and its orbital period. When combined with ground-based spectroscopy of these stars to fix the stellar parameters, the true planet radius and orbit scale, hence the relation to the HZ are determined. These spectra are also used to discover the relationships between the characteristics of planets and the stars they orbit. In particular, the association of planet size and occurrence frequency with stellar mass and metallicity will be investigated. Based on the results of the current Doppler - velocity discoveries, over a thousand giant planets will be found. Information on the albedos and densities of those giants showing transits will be obtained. At the end of the four year mission, hundreds of terrestrial planets should be discovered in and near the HZ of their stars if such planets are common. A null result would imply that terrestrial planets in the HZ occur in less than 1% of the stars and that life might be quite rare.
William J Borucki↗
Kepler Mission: A Discovery-Class Mission Designed to Determine the Frequency of Earth-Size and Larger Planets Around Solar-Like Stars
The first step in discovering the extent of life in our galaxy is to determine the number of terrestrial planets in the habitable zone (HZ). The Kepler Mission is designed around a 0.95 in aperture Schmidt-type telescope with an array of 42 CCDs designed to continuously monitor the brightness of 100,000 solar-like stars to detect the transits of Earth-size and larger planets. The photometer is scheduled to be launched into heliocentric orbit in 2007. Measurements of the depth and repetition time of transits provide the size of the planet relative to the star and its orbital period. When combined with ground-based spectroscopy of these stars to fix the stellar parameters, the true planet radius and orbit scale, hence the position relative to the HZ are determined. These spectra are also used to discover the relationships between the characteristics of planets and the stars they orbit. In particular, the association of planet size and occurrence frequency with stellar mass and metallicity will be investigated. At the end of the four year mission, hundreds of terrestrial planets should be discovered in and near the HZ of their stars if such planets are common. Extending the mission to six years doubles the expected number of Earth-size planets in the HZ. A null result would imply that terrestrial planets in the HZ occur in less than 1% of the stars and that life might be quite rare. Based on the results of the current Doppler-velocity discoveries, detection of a thousand giant planets is expected. Information on their albedos and densities of those giants showing transits will be obtained.
William Borucki↗
Kepler Mission: A Wide-FOV Photometer Designed to Determine the Frequency of Earth-Size and Larger Planets Around Solar-like stars
The first step in discovering the extent of life in our galaxy is to determine the number of terrestrial planets in the habitable zone (HZ). The Kepler Mission is designed around a 0.95 m aperture Schmidt-type telescope with an array of 42 CCDs designed to continuously monitor the brightness of 100,000 solar-like stars to detect the transits of Earth-size and larger planets. The photometer is scheduled to be launched into heliocentric orbit in 2007. Measurements of the depth and repetition time of transits provide the size of the planet relative to the star and its orbital period. When combined with ground-based spectroscopy of these stars to fix the stellar parameters, the true planet radius and orbit scale, hence the position relative to the HZ are determined. These spectra are also used to discover the relationships between the characteristics of planets and the stars they orbit. In particular, the association of planet size and occurrence frequency with stellar mass and metallicity will be investigated. At the end of the four year mission, hundreds of terrestrial planets should be discovered in and near the HZ of their stars if such planets are common. A null result would imply that terrestrial planets in the HZ occur in less than 1% of the stars and that life might be quite rare. Based on the results of the current doppler-velocity discoveries, detection of a thousand giant planets is expected. Information on their albedos and densities of those giants showing transits will be obtained.
Borucki, William↗