The upper atmosphere model draws heavily on the behavior of the earth's upper atmosphere which exhibits cyclic as well as irregular variations in density profile, temperature, pressure, and composition in unison with solar activities as deduced from the more recent land-based and satellite observations. The lift and drag model is designed specifically for inertially stabilized vehicles of the Mariner class, with possible extension to gravity gradient stabilized vehicles of the GEOS class. The model considers operation in the free molecular flow regimes with large Knudsen numbers. The vehicle is considered a composite structure with basic components having well-defined shapes, each with its own surface characteristics in terms of temperature, reflectivity, and accommodation of free stream molecules. The model takes into account both the calculation of precise aerodynamic force coefficients in terms of expansion of modified Bessel functions in speed ratios and angle of attack, and approximate force coefficients when the speed ratios approach infinity. Other considerations include specular and diffused reflectivity, shielding, and shadow effects

Khatib, A. R.

English

NASA Technical Reports Server

JPL Quarterly Technical Review Volume 1, Number 3 October 1971
Dynamic Upper Atmospheric Force Model on
Stabilized Vehicles for a High-Precision Trajectory
Computer Program
A. R. Khatib
Mission Analysis Division
This article summarizes the results of research carried out at JPL for the design
and implementation of dynamic upper atmosphere and l i f t and drag models into the
advanced Double-Precision Trajectory Program (DPTRAJ). The upper atmosphere
model draws heavily on the behavior of the Earth's upper atmosphere which exhibits
cyclic as well as irregular variations in density profile, temperature, pressure, and
composition in unison with solar activities as deduced from the more recent land-
based and satellite observations.
The lift and drag model is designed specifically for inertially stabilized vehicles of
the Mariner class, with possible extension to gravity gradient stabilized vehicles of
the GEOS class. The model considers operation in the free molecular flow regimes
with large Knudsen numbers. The vehicle is considered a composite structure with
basic components having well-defined shapes, each with its own surface characteris-
tics in terms of temperature, reflectivity, and accommodation of free stream
molecules. The model takes into account both the calculation of precise aerody-
namic force coefficients in terms of expansion of modified Bessel functions in speed
ratios and angle of attack, and approximate force coefficients when the speed ratios
approach infinity. Other considerations include specular and diffused reflectivity,
shielding, and shadow effects.
Introduction
Improvements in the tracking data and orbit determination accuracies at
JPL in conjunction with low-altitude orbiter missions having upper
atmospheric penetration have resulted in the need for accurate modeling of
the aerodynamic forces which influence the trajectories of these vehicles.
Accurate lift and drag force models require an accurate, dynamic atmo-
sphere model, as Earth satellite drag analysis have made it obvious that no
static model can approximate the atmospheric parameters in the general
case. Although modeling complexity is unconstrained in principle when
solving equations of motion with special perturbation techniques (i.e.,
numerical integration), practical limiting factors on the complexity are
125
https://ntrs.nasa.gov/search.jsp?R=19720003205 2020-03-23T13:58:14+00:00Z
determined by computer hardware performance (i.e., precision of the
numerical integration) and the accuracy of other trajectory force models. It
is also important to design efficient general models which may be used for a
wide spectrum of missions in this decade.
The atmospheric model selected draws heavily on the behavior of the
Earth's upper atmosphere due to the availability of data obtained in the past
decade. Although the amount of factual data on many of the important
parameters necessary for the determination of the general variation of
atmospheres of other planets is scarce, the mechanisms and inferences
implied in the generality of a model based on the behavior of the Earth's
upper atmosphere remain valid to the extent of assuming that atmospheric
behavior is grossly comparable when projecting dynamic effects to other
planets.
The lift and drag force model selected is one that is applicable in a free
molecular flow regime, where the atmospheric molecular mean-free-path is
larger than the characteristic dimension of the vehicle (i.e., large Knudsen
number). For practical considerations, the spacecraft is considered a
composite structure of basic well-defined shapes with analytic properties.
While ignoring multiple reflections, shielding factors are introduced to
approximate the nonlinear interaction between components. The analysis is
simplified by ignoring torques and inertia considerations, thus limiting the
model to inertially stabilized types of vehicles (e.g., Mariner class) with
possible extension to gravity gradient stabilized types (e.g., GEOS class).
Upper Atmosphere
Survey
Instruments and drag analysis methods of investigations revealed the
existence of both periodic and irregular variations in the basic parameters of
the Earth's upper atmosphere. These variations proved to be in unison with
solar activities.
Although the microscopic details of the process of radiation absorption
and reradiation are well understood, such processes are generally compli-
cated, especially when evaluating variable solar radiation effects on
multicomponent atmospheres.
Several major considerations due to efforts during the past decade can
now be stated.
(1) Higher regions of the atmosphere are best represented by a vertical
isothermy in a state of diffusion equilibrium. The ratio of the
principal constituents is essentially fixed at the beginning of the
diffusion level. Heat conduction is the principal transport mode.
126
(2) The atmosphere is heated by all radiations of the solar spectrum.
Principal contributors in terms of the total kinetic energy of a
thermospheric column belong to the region 1 to 1700 A (ultraviolet
and soft X7ray). Thermal emission from Coronal condensation
clustering above Sun spots is an emitter of radiation (EUV). A single
wave length is usually used as an index of solar decimetric flux (F107).
The choice of this index is influenced by its high correlation relative
to solar activity and atmospheric temperatures.
(3) Although satisfactory explanations for meridional transport and
auroral zone heating due to magnetic storms are not available
(Reference 1), satellite drag analysis shows a linear correlation
between geomagnetic indices and the upper-atmospheric tempera-
tures with an observed time lag.
(4) The observed variations can be categorized as follows:
(a) Altitude and latitude position.
(b) Diurnal (planetary rotation).
(c) Monthly (27-day solar rotation).
(d) Semi-annual and seasonal (planetary orbit).
(e) Solar decimetric radiation (11-yr Sun-spot cycle).
(f) Magnetic storms (irregular) and auroral activity (high latitudes).
Major approaches used in constructing upper-atmosphere models include:
an empirical approach, derived from satellite drag analysis, generally in
agreement with physical laws, and a theoretical approach based on
conservation laws. The empirical one-dimensional model of Jacchia
(Reference 2) is a quasi-static, multiple-component model. The model is
based on the assumption of hydrostatic equilibrium, an empirical tempera-
ture profile, and a fixed set of boundary conditions for temperature and
number densities. Empirical temperature profiles reflecting atmospheric
temperature variations due to solar activities are used to compute number
densities. Knowledge of thermal diffusion ratios and diffusive equilibrium is
assumed. Contrary to observations, a nearly isopycnic layer with a constant
density at some level results in models based on the above approach as a
consequence of the use of a fixed set of boundary conditions (Reference 3).
Another approach taken by Harris and Priester (Reference 4) is in
computing the temperature from an energy conservation equation by
utilizing all known (and some virtual) heat sources to generate the
temperature profile, which is then used simultaneously with number
densities assumed in hydrostatic equilibrium. Other more recent models
include those by Thomas (Reference 5) and Friedman (Reference 6). The
model of Thomas is analytic and one dimensional, where the undimensional
conservation equations are transformed to an isobaric frame, thus enabling
solution by simple analytic Green functions. Friedman's model utilizes
simplified expressions for radiation heating in solution of the conservation
127
equations. This three-dimensional model allows boundary conditions to vary
until agreement with observed data is achieved. This last approach is more
physically realistic in that it accounts for horizontal heat conduction and
mass diffusion.
General Computer Model
Due to operational flexibility, the empirical model was chosen for the
high-precision trajectory program DPTRAJ (Reference 7). The user supplies
density and average molecular weight versus height for the full range of
exospheric temperatures. The table can be made up of patched set of one-
dimensional atmospheres corresponding to given top exospheric tempera-
tures. The range of temperatures extends from the absolute minimum night
time to the maximum observed and modeled temperatures. Empirical
equations relating variations of exospheric temperature to dynamic causes
are evaluated yielding the required temperature at the evaluation epoch.
The required parameters at the given height are thus obtained. The
empirical form of the temperature equation'(Reference 8) is
T = (Tmin + ATy/W, in kelvins
where
known temperature
/(0) = 1 + L co^6 ~ diurnal bulge effect, where the angle 6 is
measured from the bulge maximum to the
considered location, and L and M are
constants
= contribution of dynamic effects
(e +/sin 2irf>j) (FQ 7 sin
\gap - h(\ - exp
and a, b, c, d, e, f, g, h, and k are constants
128
AFJ07 = decimetric flux lessF]07, 10 22 W-m 2(cps)-1 per bandwidth
FJO? = average over three solar rotations
AF* = average over one solar rotation less the average over the
11-yr Sun-spot cycle
D , (D^) = days past observed min, (max) seasonal variations
a = the 3-h geomagnetic index, properly scaled for atmospheric
response time lag, 2 X10"5 gauss
The dynamic equation terms empirically account for the effects of the solar
rotation, solar cycle, semiannual, and magnetic storms, respectively.
Aerodynamic Forces
Survey
At higher altitude, when the ratio of the mean-free-path of free stream
molecules to a basic characteristic body dimension (Knudsen number) is
large, the rarefied gases do not behave like a continuum. Computation of
aerodynamic forces and moments are based on a concept from the kinetic
theory of gases known as the free molecular flow regime. The theory takes
into account uniform mass motion of gases superimposed upon the thermal
motion (Maxwellian distribution) of gas molecules. Ignoring collisions
between impinging and re-emitted molecules, the total force on a body is
made up of components from surface bombardment by impinging molecules
and components due to specular or diffuse re-emission of those molecules. In
the specular reflection the normal velocity component is reversed, while the
tangential component with shear effects on the surface remains unchanged.
In the diffuse reflection, all previous directional history is erased. Molecules
leaving the surface have speeds with Maxwellian distribution dependent on
re-emitted stream temperature, and direction controlled by the Knudsen
cosine law.
The recent methods developed by Heineman (Reference 9), Stalder
(Reference 10), and Blick (Reference 11) involve integration of components
of the momentum force, imparted to a differential plane area by impinging
and reflected molecules, in defined directions. The direction cosines chosen
are related to the component of force being computed, which in turn yield
the desired aerodynamic coefficients for the type of shape being considered.
129
The basic equation for the total incident momentum (front and rear sides)
per elemental area per unit time is
^ (c - fyexpC-^V- V)
7T3/2
o -I
- f Cj (c • £) exp (-02 V • V) dc^dc2 dA
where
I
mN
0
V
c
U =
desired direction of momentum force vector
p = number density
1/c = reciprocal of most probable molecular speed
thermal velocity
V + U = total velocity = sum of thermal and mass
velocities
Sc = (7/2) M = stream mass speed, and S is the speed
ratio, 7 is ratio of the specific heats, and M is the Mach
number
For example, application of the above equation yields the impinging front
side pressure on an elemental area of flat plate along N(see Figure 1).
Let
U = t/(sin0:N-cos0y)
A A
B = N
Therefore,
(sin2 9 + — }
\ 2S2/
[1 + erf (S sin 8)]
130
2, y
1, N
3, Z
ANGLE OF ATTACK a = 90 - 8
Figure 1. Coordinate system definition
Simplifying assumptions leading to approximate expressions were used by
Blick (Reference 11) to obtain aerodynamic coefficients for a larger set of
geometric shapes and bodies of revolution. Large speed ratios are assumed
(zero kinetic temperature) allowing elimination of error function and
exponential terms. As a result, the flow approaches that of an elastic or
inelastic Newtonian flow for specular or diffuse reflection, respectively.
General Computer Model
The vehicle is considered a composite structure of components with
known aerodynamic properties. The components include: flat plate, sphere,
cylinder, and segment of a sphere. For each component, the front and back
side surface temperatures, the effective areas, the accommodation coeffi-
cients, and the proportions of each type of reflection are assumed known.
The orientation of each component relative to slip stream is calculated and
the angle of attack is determined. Tables supplying shadow effects in terms
of reduction of effective area of each component, based on its orientation
relative to the slip stream, are supplied externally.
The total lift and drag forces are then calculated as a vector sum of forces
on each component. The fact that this model is designed for inertially
stabilized spacecraft simplifies the calculations. Forces are translated and
torque considerations are not needed except when the model is utilized in
conjunction with GEOS class vehicles. In the case of gravity gradient
stabilized GEOS, the restoring gravity torque is needed for the determina-
tion of the offset angle created by drag. The total aerodynamic force is then
expressed in the proper inertial frame and added to the other perturbative
forces in the equations of motion of the spacecraft for numerical integration
by DPTRAJ (Reference 7). More complete details of the model and analysis
are planned.
131
References
1. Thomas, G., and Ching, B., A Study of Atmospheric Heating Applied
to the Phenomenon of Geomagnetic Heating, Aerospace Report TR
0158, 1968.
2. Jacchia, L. G., Static Diffusion Model of the Upper Atmosphere with
Empirical Temperature Profiles, SAO Special Report 170. Smithso-
nian Astrophysical Observatory, Cambridge, Mass., 1964.
3. Johnson, W. J., A Study of the Upper Atmospheric Models, NASA
TM X-53782. National Aeronautics and Space Administration, Wash-
ington, 1968.
4. Harris, I., and Priester, W., "The Upper Atmosphere in the Range
from 80 to 800 km," Proposal for CIRA (Cospar International
Reference Atmosphere), 1964.
5. Thomas, G. E., Analytic Solutions of the Heat Conduction Equation
in the Thermosphere, Air Force Report SAMSO-TR-67-118, 1963.
6. Friedman, M. P., A Three-Dimensional Model of the Upper Atmo-
sphere, SAO Special Report 250. Smithsonian Astrophysical Observa-
tory, Cambridge, Mass., 1967.
7. Khatib, A. R., "Double-Precision Trajectory Program: DPTRAJ," in
The Deep Space Network, Space Programs Summary 37-49, Vol. II,
pp. 52-55. Jet Propulsion Laboratory, Pasadena, Calif., Jan. 31, 1968.
8. Khatib, A. R., and Brenkle, J. P., Consideration of Upper Atmospheric
and Aerodynamic Force Models for DPTRAJ, Technical Memoran-
dum 392-25, 1970 (JPL internal document).
9. Heineman, M., "Theory of Drag in Highly Rarified Gases," Communi-
cation on Pure and Applied Math, Vol. 1, No. 3, pp. 259-273, 1948.
10. Stalder, J., et al., Theoretical Aerodynamic Characteristics of Bodies
in a Free-Molecular-Flow Field, NACA TN 2423. National Advisory
Committee for Aeronautics, Washington, 1951.
11. Blick, E. F., Forces on Bodies of Revolution in Free Molecular Flow
by the Newtonian-Diffuse Method, McDonnell Aircraft Corporation
Report 6670,1969.
132


Dynamic upper atmospheric force model on stabilized vehicles for a high-precision trajectory computer program

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server