【物理应用】基于Matlab模拟高精度轨道传播器

1 内容介绍

Studying and modeling perturbations are key disciplines in astrodynamics. Although most of the solution techniques have been known for a long time, the literature only recently has begun to present (with increasing exactness) the precise methods needed to satisfactorily solve these problems. Leon Blitzer (1970) developed the Handbook of Orbital Perturbations and has graciously allowed me to use it as a basis for this chapter and Chap. 9. Blitzer was one of the early scientists to begin detailed analysis of the perturbative effects on near-Earth satellites, publishing several papers in the mid-1950s before satellites had become routine. I’ve also incorporated personal discussions with (listed alphabetically) Cefola, Fonte, Gabor, Junkins, and McClain to support this section. A complete list of their works would take many pages! Szebehely (1989:13–22)lists other major contributors to the field of perturbations and astrodynamics. Lambeck (1988:654–709) includes extensive references on perturbations. Consult these sources for more detail. Perturbations are deviations from a normal, idealized, or undisturbed motion. We tend to view the universe as highly regular and predictable. Yet accurate observational data often reveals unexplained irregularities of motion superimposed on the more regular or mean motions of the celestial bodies. Actual problems don’t conform exactly to the assumptions made for the ideal two-body and restricted three-body problems. Therefore, these idealized solutions only approximate the observed motion. The actual motion will vary from the theoretical two-body path due to perturbations caused by other bodies (such as the Sun and Moon) and additional forces not considered in Keplerian motion (such as a nonspherical central body and drag). Notice we use the term central body to show these perturbation techniques apply about any central body. For missions to other planets, the Earth may not be the central body. Indeed, the formulas work equally well for the Earth, Mars, Saturn, and so on⎯assuming, of course, that we have accurately determined the data constants and coefficients for each central body. Bate, Mueller, and White (1971:385-386) introduce perturbations as follows. We’re familiar with random perturbations in almost every area of life. Seldom does anything go exactly as planned; rather, it is perturbed by unpredictable circumstances. An excellent example is an aircraft encountering a wind gust. All directional controls and power are kept constant; yet, the path changes abruptly from the predicted course that was based on previous measurements. In this example, the wind gust is a random perturbing force. Fortunately, most of the perturbing forces considered in orbital mechanics are more predictable and analytically tractable than a wind gust. The usual perturbing forces we consider for the orbit problem are both random and predictable. They include the asphericity of the central body, atmospheric drag and lift, third body effects, solar-radiation pressure, thrust, magnetic fields, solid-Earth and ocean tides, Earth re-radiation (albedo), relativistic effects, and others. I’ll show how to analyze most of these later. Don’t get the idea that all deviations are small—they can be comparable to the primary attracting force (two-body gravitation). Whenever they’re this large, they cease to be perturbations because the fundamental nature of the problem is different. Examples are atmospheric drag during reentry and the third body in the restricted three-body problem. In fact, many interplanetary missions would miss their target entirely if the perturbing effect of other attracting bodies weren’t taken into account! Ignoring the effects of the central body’s oblateness on any satellite keeps us from accurately predicting its position over a long time. Without analyzing perturbations, we couldn’t explain or accurately predict the orbit of the Moon about the Earth. We’ll look at the three main approaches to examine the effects of perturbations: special perturbation techniques (using numerical methods—Sec. 8.5), general perturbation techniques (using analytical methods—Chap. 9), and semianalytical techniques (a combination of analytical and numerical techniques—Chap. 9). Special perturbation techniques numerically integrate the equations of motion including all necessary perturbing accelerations. Because numerical integration is involved, we can think of numerical formulations as producing a specific, or specialanswer that is valid only for the given data (initial conditions and force-model parameters). Although numerical methods can give very accurate results and often establish the “truth” in analyses, they suffer from their specificity, which keeps us from using them in a different problem. Thus, new data means new integration, which can add lengthy computing times. NASA began the first complex numerical integrations during the late 1960s and early 1970s. Personal computers now compute sufficiently fast enough to perform complex perturbation analyses using numerical techniques. However, numerical methods suffer from errors that build up with truncation and round-off due to fixed computer wordlength. These errors can cause numerical solutions to degrade as the propagation interval lengthens. Let’s look at an example to illustrate the strengths and weaknesses of a numerical method. Start with the equation of motion in Example 8-1, which is simple, first order, and defined with a given initial condition. Find the value of this system when time, t, equals 0.5s , assuming q = 1 rad/s. Obviously, this example permits a direct solution if we recognize the integral as (SIN(qt)/q).

2 部分代码

%--------------------------------------------------------------------------

%

% AccelSolrad: Computes the acceleration due to solar radiation pressure

%    assuming the spacecraft surface normal to the Sun direction

%

% Inputs:

%   r           Spacecraft position vector 

%   r_Earth     Earth position vector (Barycentric)

%   r_Moon Moon position vector (geocentric)

%   r_Sun       Sun position vector (geocentric)

%   r_SunSSB    Sun position vector (Barycentric)

%   Area        Cross-section 

%   mass        Spacecraft mass

%   Cr          Solar radiation pressure coefficient

%   P0          Solar radiation pressure at 1 AU 

%   AU          Length of one Astronomical Unit

%   shm         Shadow model (geometrical or cylindrical)

%

% Output:

%   a     Acceleration (a=d^2r/dt^2)

%

% Notes:

%   r, r_sun, Area, mass, P0 and AU must be given in consistent units,

%   e.g. m, m^2, kg and N/m^2. 

%

% Last modified:   2018/01/27   Meysam Mahooti

% 

%--------------------------------------------------------------------------

function a = AccelSolrad(r,r_Earth,r_Moon,r_Sun,r_SunSSB,Area,mass,Cr,P0,AU,shm)

%        Moon wrt Earth          pccor      rpc

%        Earth wrt Sun           ccor       rc

%        Moon wrt Sun            pscor      rps   

%        Satellite wrt Earth     sbcor      rsb  

%        Satellite wrt Sun       bcor       rb 

%        Satellite wrt Moon      sbpcor     rsbp

pccor = r_Moon;

ccor = r_Earth-r_SunSSB;

pscor = r_Moon-r_Sun;

sbcor = r;

bcor = r-r_Sun;

sbpcor = r-r_Moon;

if ( strcmp(shm,'cylindrical') )

    nu = Cylindrical(r,r_Sun);

else

    [nu,~] = Shadow(pccor,ccor,pscor,sbcor,bcor,sbpcor);

end

% Acceleration

a = nu*Cr*(Area/mass)*P0*(AU*AU)*bcor/(norm(bcor)^3);

3 运行结果

4 参考文献

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