Orbit calculation is a method for roughly determining the orbits of celestial bodies.In orbit calculation, people do not need to make any initial estimate of the celestial body's orbit in advance, but determine the initial orbit of the celestial body based on a number of observation data and mechanical and geometric conditions, so as to track the celestial body in time, or as the initial value of orbit improvement.In order to calculate the six orbital elements (see the two body problem), there must be at least three optical observations, because each observation can only obtain two components of the celestial coordinates.
Orbit calculation uses a set of techniques to estimate the orbits of objects such as satellites, planets and spacecraft.Determining the orbit of a newly observed asteroid is a common use of these techniques, so an asteroid can track future observations and verify it before it is discovered.[1]
Chinese name
Track calculation
Foreign name
Orbit determination
Meaning
Determination of celestial orbits
Object
Satellites, planets, spacecraft
Technology involved
Mechanical and geometric conditions
Initial application
Comet research
Interpretation
A method for roughly determining the orbits of celestial bodies
The observation is to feed the original data into the orbit determination algorithm.Observations made by ground observers usually consist of azimuth, elevation, range and or range rate values of time markers.usetelescopeorradarDevice, because naked eye observation is not enough for accurate orbit determination.
After determining the orbit, mathematical propagation technology can be used to predict the future position of the orbital object.Over time, the actual path of an orbital object tends to deviate from the predicted path (especially if the object suffers from unpredictable disturbances such as atmospheric drag), and new orbit determination using new observations is used to re calibrate orbit knowledge.
For the United States and its partner countries, within the scope of optical and radar resources, the Joint Space Operations Center collects observations of all objects in Earth orbit.The observations are used to maintain the overall accuracy of the satellite catalog for new orbit determination calculations.Collision avoidance calculation can use this data to calculate the probability that one orbital object will collide with another.If the collision risk of the current orbit is unacceptable, the satellite operator can decide to adjust the orbit.(It is impossible to adjust the orbit when encountering extremely low probability situations; doing so will lead to the rapid depletion of propellant by the satellite.) When the number or quality of observations increases, the accuracy process of orbit determination also improves, and fewer "false alarms" have attracted the attention of satellite operators.includeRussiaandChinaOther countries, including, also have similar tracking equipment.[2]
History of the Development of Orbital Calculation Methods
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Orbit calculation is based on researchcometThe movement of began.Before Newton, the study of celestial motion basically had the nature of geometric description.Tycho first tried to calculate the orbit of the comet, but failed.The difficulty is that we can only observe the direction of the comet without knowing its distance from the earth. Due to the lack of guidance of mechanical laws, we cannot obtain the space orbit of celestial bodies from these directional data.stayNewtonLaws of motion andUniversal gravitationLaw discovery grub oKepler's lawWith the mechanical explanation, the strict mathematical expression of the elliptical motion was obtained, and finally the orbit of the comet could be determined using a few observations with short intervals.[3]
Laplace method
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Laplace methodThe first formal orbital calculation method was proposed by Newton.He calculated the orbits of celestial bodies by graphical method based on the data of three observations.Halley used this method to analyze 24 comets that appeared between 1337 and 1698. He found that the comets that appeared in 1531, 1607 and 1682 were the same comet, which was famousHalley's Comet。After that, Euler, Lambert andLagrangeThey have also done a lot of research on orbit calculation.Laplace published the first complete analysis method of orbit calculation in 1780.This method does not limit the number of observations. First, according to several observations, the apparent position of the celestial body on the celestial sphere at a certain time (such as right ascension and declination) and its primary and secondary derivatives are determined. Then, the space coordinates and velocities of the celestial body at that time are strictly and simply calculated from these six quantities, so as to determine the six elements of the conic orbit.In this way, Laplace transforms the orbit calculation into a problem of determining the initial value of a differential equation.From an analytical point of view, this is a good method, but the orbit calculation is a practical problem, and the accuracy of the results and the convenience of the calculation should be considered.Laplace method is not very convenient in practice.Because numerical differentiation will magnify the error, it requires very accurate observation data to obtain a reasonable derivative.Although many people have made some progress, they seldom use it in solving practical problems due to the complexity of calculation.
Aubers method and Gauss method
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Aubers methodUnlike the Gaussian method and Laplace, Obers and Gauss believe that if the space position of the celestial body at two different times can be determined according to the observation data, the corresponding orbit can also be determined.In other words, Obers and Gauss transform the orbit calculation into a boundary value determination problem.Therefore, the key problem is how to determine the position of celestial bodies in space according to three directional observations.This should not only consider the geometric characteristics of the orbit, but also apply the mechanical laws of celestial motion.The most basic of these conditions is that the celestial body must move on the plane passing through the sun.Since the viewing direction of the celestial body at three times is known from observation, once the orientation of the orbital plane is determined, the spatial position of the celestial body at three times will also be determined, except for some special cases.The correct orientation of the orbital plane is based on the fact that the three space positions determined can meet the mechanical laws of celestial motion, such as the area law.
Most comet orbits are close to parabola, so they are often treated as parabola when calculating orbits.The complete parabola orbit calculation method was proposed by Obers in 1797.He used Newton's hypothesis to obtain the relationship between the comet's geocentric distance; combined with the Euler equation that represents the radial and chord relationship of the celestial body at two times on the parabolic orbit, he calculated the comet's geocentric distance; thus, he calculated the parabolic orbit of the comet.Although Aubers method has many improvements, its basic principle has not changed, and it is still a commonly used method for calculating parabolic orbits.
On January 1, 1801, Piazzi discovered No. 1asteroid（Ceres）Soon Gauss figured out its elliptical orbit, and his method was published in 1809.Gauss uses the successive approximation method to first find out the ratio of the sector area enclosed by the celestial body's radial path to the triangle area, then use the mechanical conditions to find out the space position of the celestial body, and then find out the orbit from the space position.Gauss solved the problem of orbit calculation not only theoretically but also practically.It can be said that Gauss first solved the practical problem of determining the orbit with three observations.After Gauss, although some new methods have been proposed, the basic principle has not changed.
Orbit calculation of artificial satellite
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Artificial satellite orbitIn principle, the classical methods for calculating the orbits of asteroids can be used to calculate the orbits of artificial satellites.After considering the motion characteristics of artificial satellites, some new methods are proposed.Sputnik MovementFast, short period, and the timing error has a significant impact on the orbit calculation results.On the basis of Gauss method, Bartrakov improved the orbit calculation method with time recording error by adding observation data.The near Earth satellite orbits the earth more than ten times a day, and its period can be easily observed, so that the semi major diameter of the orbit is known. Accordingly, the orbit calculation method of the known semi major diameter is proposed.The artificial satellite is close to the earth, and the parallax phenomenon is obvious. It is easy to obtain the satellite geocentric distance by using two or more stations of synchronous observation, which can simplify the classical calculation method.In view of the large influence of satellite perturbation, the orbit calculation method considering perturbation has emerged.Although these methods are various, they are still nothing more than to obtain the radial diameter of two points or the radial diameter and velocity of one point from the observation data, thus obtaining the orbital elements.
Through laser ranging and Doppler velocity measurement of artificial satellite, multi station synchronous observation, or combined with optical observation and other methods, the satellite's radial and velocity can be directly obtained, and then the satellite's orbit can be obtained.Using high-speed electronic computers, complex iterative operations can be carried out.Therefore, it is more important to integrate various types of observation data for orbit improvement than to focus on the calculation of initial orbit.Modern technical conditions have enabled the satellite orbit after entering orbit to be close to the predetermined orbit.In this way, the predetermined orbit can be used as the initial orbit.[4]
Principles of Computer Computing Planet Orbits
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Taking the earth's orbit around the sun as an example, for the convenience of description, the following is expressed in vector form.
Because M is the mass of the sun, M is the mass of the earth, G is the constant of gravity, a is the acceleration of the earth, and is the unit vector.So we can calculate point by point from the initial value point of the earth in differential form according to the equal time interval dt (i.e. equal step size).If t=0, the initial value point of the earth isr0,v0And then, the earth reaches the first point from the initial point via dt time, and the recurrence formula is,
Since dt is artificially set and known, the approximate value of the earth at point 1v,randaIt can be calculated from the above formula. After the value of one point is calculated, the value of one point can be taken as the initial value, and the value of the next point can be calculated according to the step length dt. In this way, the nth point can be calculated.Since the smaller the dt value is, the higher the recursive precision is, we can control the calculation error accordingly.
If the r value of the earth at t=T is to be calculated, the initial values r0, a0 and v0 when the calculation error is e and t=0 are known.We can divide the time interval from 0 to T into n dt, that is, let the calculation step dt=T/n, and then, according to the above, calculate from the initial value point when t=0 to the r value when t=n · dt=T according to the step dt.Then divide dt into two parts, that is, make the calculation step dt1=dt/2, and then calculate the new step value dt1 from the initial value point when t=0 to the r value when t=2 · n · dt1=T as r2. Compare the r value before and after the two parts, that is, see whether the condition r2 - r<e is met. If the condition is met, the value of r2 is the required r value. If the condition is not met, continue to score two parts,According to the new step value, calculate from the initial value point when t=0 to the time when t=T until the newly calculated r value meets the condition r2 - r<e, where r and r2 are the r values before and after the bisection respectively.Note that the vector comparison mentioned above includes its size comparison and its direction comparison.
The above is the vector expression. In actual calculation, the vectorv,r，aBy projecting on the x and y axes respectively, we can get vx, vy, rx, ry, ax, ay. Therefore, their initial values are v0x,v0y,r0x,r0y,a0x,a0y。The recurrence formula of the earth from the initial point to the next point through dt time is given below,
The conditions for controlling the error range are the r values in the x and y directions before and after the bisection.
The above is just the principle of computer calculation of earth orbit.In fact, every two minutes, the number of dt in the time range from 0 to T will double, and the workload of computer calculation will also double.Because the computing speed of the computer is limited, the number of bisections is also limited.In order to improve the calculation accuracy and reduce the calculation workload of the computer, there are some standardized methods (Note 1), which will not be described here.
It can be seen from the above that there are two main points for computer calculation of star orbits.One is to list the recurrence formula, and the other is to determine the conditions of error range.
See the next page for lunar orbit calculation.
Note 1: See the book "Computer Numerical Calculation Method and Program Design".This r book was compiled by Zhou Xu.ByChina Machine PressPublication.[5]
Lunar orbit calculation
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Since the motion of the earth directly affects the motion of the moon, let's first analyze the force on the earth, as shown in Figure 1-3.
In Figure 1-3, the o2x2y2z2 coordinate system is a moving coordinate system, and the origin is at the center of the earth.The coordinate system follows the earth for translation, and the three coordinate axes x2, y2, z2 are always parallel to the three coordinate axes x, y, z respectively.R1 is the position vector of the earth, r is the position vector of the moon, and r2 is the position vector of the moon relative to the earth.
F Lundi is the gravitational force of the moon on the earth, and F Taidi is the gravitational force of the sun on the earth.Let the included angles of r1 and x, y, z axes be α 1, β 1, γ 1 respectively, and the included angles of r and x, y, z axes be α, β, γ, r2 and x2, y2, z2 axes be α 2, β 2, γ 2 respectively, then the resultant force on the earth in the x, y, z direction is:
Therefore, the acceleration of the earth in the x, y, z directions:
The force on the moon is shown in Figure 1-4.The resultant force on the moon in the x, y, z direction is:
Where, F Taiyue is the gravity of the sun on the moon, and F Diyue is the gravity of the earth on the moon.Therefore, the acceleration of the moon is:
set upaThe initial value of is such that the earth and the moon start from their respective initial points at the same time, and after the dt time, the earth will reach its next point, so the following recursive formula can be obtained:
(See next page)
The six conditions for controlling the calculation error are:
Wherein, they are the earth coordinates calculated before and after the bisection.Once again, the above calculation of lunar orbit is just the computer calculation principle, and some standardized methods should be adopted for actual programming to improve the calculation accuracy and reduce the computer calculation workload.
In the lunar orbit calculation, I have achieved that the calculation error for one day is e<0.001 m (that is, the calculation error in the x, y, z axis direction is e), that is, the calculation error for one year is e<365 × 0.001=0.365 m.To verifyUniversal gravitation formulaThe degree of difference between itself and the actual situation can take two sets of actual observations, one set of observations as the initial value for calculation, and the other set of observations for verification, that is, the accuracy of the orbit of the planet calculated by the universal gravitation formula.Let's use a group of actual observations (Note 2) as the initial value for calculation, and let the computer calculate the orbit of the moon.The initial value is:
Calculation results
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The time range of the above calculation is 2006.The maximum and minimum orbital radius of the moon refer to the average value.Observations are made byPurple mountain observatoryProvided by our staff.The above calculation takes the calculation time t=366 days, and the coordinate calculation error e<0.366 m.When calculating the cycle, the time calculation error is less than 0.05 seconds.Since the lunar data provided by the observatory is relative to the earth coordinate system, the relationship between the earth coordinate system and the moving coordinate system described in this article is that the yz plane of the earth coordinate system is rotated by a yellow and right angle with the x axis as the rotation axis, which is the moving coordinate system described in this article.In this paper, the intersection angle of yellow and red is 2326 '.
Note 2: The corresponding time of the observation value as the initial value for calculation is 7:47.5 on March 15, 2006 Beijing time.This time is exactly the time of eclipse of penumbra.For the convenience of calculation, this time is set as zero time in this paper.
The coordinates of the Earth and the Moon calculated from the time of the lunar eclipse in March 2006 to the time of the lunar eclipse in September 2006 are given below.The data are as follows.
Earth coordinates (in meters)
Viewpoint: quantum formula of celestial orbit
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hypothesis
Early 20th centuryBohrThe theory of space quantization (orbital quantization) proposed by et al. has caused a profound revolution in the physical world. Since then, people believe that there is an insurmountable gap between micro and macro.In fact, if we introduce the concept of time quantization, we will find that there is a profound and wonderful relationship between micro and macro.
It is hard to imagine how objects moving in the gravitational field with the same mathematical form can have different orbital properties. Let's make a general assumption: in the gravitational field
V1/r=－ P/r (P is the constant related to the system)
Under the action, the stability of the orbit of an object moving in the orbit has a small and sharp peak when its orbit meets the following formula, or more precisely when its orbit is near the following formula:
an=n2a0,(n=1,2,3……）， ⑴
Tn=n3T0,(n=1,2,3……）， ⑵
Where a0 and T0 are system related constants, and an and Tn are the semi major diameter and period of orbit n.
When V1/r is a coulomb field generated by a hydrogen like nucleus, the above formula is equivalent to Bohr's first and second assumptions, and can be deduced from each other, so it is unnecessary to verify it here.
When V1/r is the Newton field generated by the central celestial body, the author finds that a0 and T0 can be determined by the following formula:
a0 = k1M 1 ， ⑶
T0 = k2M 2 ， ⑷
Where M is the mass of the central celestial body, constant
It can be seen from Table 1 that the result of formula (1) is better than that of Bode's rule, but not as good as its variants of Bellagio's and Richardson's formulas, but they both specify coefficients rigidly, with complicated forms and unclear physical meaning, which is close to mathematical games.
What is also different from the Bode rule and its variants is that the n value taken in formula (1) is discontinuous.Is this a pity?Obviously, we should think of the orbits of comets and asteroids, which also meet the prerequisite for the establishment of formula (1).
In Table 2, there are several comets occupying one orbital number, which is often called the orbital band - does it correspond to the 'degeneracy of energy levels' of quantum mechanics?
2. Planetary satellite system
Table 3 and 4 show the verification of Ganymede system and Celestial Satellite system.
Readers may have found that the eccentricity of satellites in orbit is significantly greater than that of satellites occupying an orbit alone;However, the objects with large eccentricity in the solar system are also orbital objects.What a wonderful similarity!Obviously, there are internal links.
3. Galaxy cluster system
If the companion galaxy does orbit around the central galaxy, the results given in Table 5 are indeed exciting.The a0 value is obtained by fitting the observed value, and the M value is obtained by backward deduction of Formula ⑶.[5]
