行星轨道运动的数值解与音乐解

Numerical Solution of Planets Orbital Motion and Its

Musical Correlation

课程：天体物理概观

Course：A General Survey of Astrophysics

姓名：缪佶朗

Name：Jilang MIAO

学号：PB09214027

Student No.：PB09214027

系别：核科学技术学院

Department：School of Nuclear Science & Technology

指导老师：向守平

Supervisor：Prof. Shouping XIANG

完成日期：2012/5/311

Date：May.31, 2012

1 Translated to English on Dec 24th, 2012

Abstract

Principles of planets motion were summarized by Kepler and are perfectly

explained by Newton’s Law of Universal Gravitation. In this essay, the principles

are derived and numerical solution and musical correlation are presented in the

following 3 aspects.

1. Proposal, selection and realization of iteration method.

Real-time simulation can be obtained from iteration methods based on trajectory

equation from Lagrange’s Equation and motion equation from Newton’s Law. Wit

analysis, test and comparison, the former was selected and simulation was

performed on Microsoft Visual Studio 2005 platform.

2. Plotting of relative motion trajectory

Assumed parameters were first used to validate the method. Then relative

motions of real planets were simulated. Trajectory of Mars and Venus in Earth sky

was presented.

3. Correlating motion data with music

Angular frequency was selected to compose music. And playing was realized

with beep function of the VS2005 tool. Besides, how to play the music with general

instruments were suggested.

Keywords：Planets orbit motion; Numerical solution; Relative motion trajectory;

Musical correlation

Chap 1. Fundamental Principles of Planets Motion

Conversion of angular momentum constrains a planet within a plane. And the

Lagrangian L = T − V =1

2m(r2 + r2θ2) +

GMm

r, therefore Lagrange Equation is

expressed as

{

d

dt

∂L

∂θ−∂L

∂θ=d

dtmr2θ − 0 = 0

d

dt

∂L

∂r−∂L

∂r=d

dtmr − (mrθ2 −

GMm

r2) = 0

(1.1)

Eq (1.1) implies 2nd

Kepler's law of planetary motions since θ does not appear

explicitly and r2θ is constant. Mark r2θ as A

(A=r2θ = 21

2rv = 2 × Area swept rate), Mark GM as u, and trajectory equation can

be derived from Eq.1.1 as follows.

d

dtmr − (mrθ2 −

GMm

r2) = 0 => r − rθ2 +

u

r2= 0 (1.2)

One can try solving 1

r(θ)

Since r2θ is marked as A, θ =A

r2

From, d

dt

1

r=−1

r2d

dtr, then, r = −r2

d

dt

1

r= −r2

dθ

dt

d

dθ

1

r= −r2θ

d

dθ

1

r= −A

d

dθ

1

r，

Therefore r =d

dtr =

dθ

dt

d

dθr = θ

d

dθ(−A

d

dθ

1

r) =

A

r2d

dθ(−A

d

dθ

1

r) =

−A2

r2d2

dθ2(1

r)

Substituting above θ, r into Eq(1.2) yields d2

dθ21

r+1

r=

u

A2

d2

dθ21

r+1

r=

u

A2 (1.3)

Eq(1.3) has the particular solution 1

r

=

u

A2， and the general solution for a

corresponding homogeneous equation is1

r= c1 cos(θ + φ). therefore, solution for

Eq(1.3) is expressed as 1

r= c1 cos(θ + φ) +

u

A2=

u

A2(ecos(θ + φ) + 1)。

And the trajectory equation is r =A2/u

ecos(θ+φ)+1 (1.4)

From elementary math, Eq(1.4) indicates a conic and parameters can be easily

determined.

Chap 2. Solution of motion equations

The most acceptable and easy-to-analyze form of motion equations should be

r=r(t), θ=θ(t). Theoretically, r=r(θ) can be determined from Eq(1.4) with reasonable

initial or boundary conditions and so can inverse solution θ=θ(r). Taking into account

2nd

Kepler’s Law, d

dtθ(r) =

A

r2. And t(θ) can be achieved by integrating dt =

r2

𝐴dθ(r).

However, t(θ) is too complicated to present physics image or compare with

observation. Therefore, numerical solution is given below.

1.1 Numerical solution based on trajectory equation

The most direct method results from trajectory equation,

and the following iteration approach can be obtained fromθ =A

r2

θ0 = 0, θ0 =A

r02 (2.1)

θi = θi−1 + θi−1∆t = θi−1 +A

r2i−1∆t (2.2)

ri = ri(θi), θi =A

ri2 (2.3)

where Eq(2.3) is the trajectory equation. This way is named as Kepler Method,

and the same initial condition as in Eq(2.1) is assumed for further discussion.

1.2 Numerical solution based on Newton’s Law

Newton’s 2nd

Law yields F = −GMm

r3r = mr . If the coordinate is established with

origin located at the center and with x axis stretching from the center to the planet

and perpendicular to the velocity, Newton’s Law can be expressed as:

Fx = −GMm

r2= max

记 GM=u⇒ ax = −

u

r3x (2.4)

Fy = −GMm

r2= may

记 GM=u⇒ ay = −

u

r3y (2.5)

And the following iteration method can be directly obtained from relationship

between velocity and acceleration.

ax0 = −u

r02, vx0 = 0, x0 = r0; ay0

= 0, vy0= v0, y0 = 0; (2.6)

vxi = vxi−1 + axi−1∆t; vyi= vyi−1

+ ayi−1∆t (2.7)

xi = xi−1 + vxi−1∆t; yi = yi−1 + vyi−1∆t; (2.8)

ri2 = xi

2 + yi2; axi = −

u

ri3xi; ayi = −

u

ri3yi (2.9)

This way is named as Newton Method, and the same initial condition as in

Eq(2.6) is assumed for further discussion.

1.3 Comparison of the two methods

Newton’s Law results from Kepler’s, however, Eqs(2.6)~(2.9) possess no

advantage over (2.1)~(2.3) as high-level principles because they just solve a second

order differential equation by solving first order differential equations namely Newton

Method leave more computation amount for computers. Numerical solution

Eq(2.1)~(2.3) based on trajectory equations Eq(1.4) seems more concise, though

Eq(1.4) can be derived from Newton’s Law.

In order to select one iteration method for further simulation, real-time solution

is tried. Locations of the particle are marked continuously on the trajectory ellipse

plotted from Eq.(1.4) for comparison. Smoothness of solution from Kepler Method

and consistence with ideal ellipse of solution from Newton Method should be

confirmed. In fact, either of the method can reach high accuracy with large amount of

computation. This work concentrates on the dynamic solution. The comparison aims

to find the discrepancy between Eqs(2.1)~(2.3) and Eqs(2.6)~(2.9) if Δt is in the scale

of the time required to perform one iteration step.

(a) (b) (c)

Fig2.1 Comparison of the 2 methods

In Fig 2.1, the green point corresponds to particle moving according to Kepler

Method and the blue point to Newton Method. It can be observed that motion from

Newton Method does not converge. The time cost by first iteration step accounts for

insufficiently small in a period, the difference is amplified with further iteration and

the blue particle deviates from the ideal ellipse gradually. What’s worse, the deviation

causes less force and acceleration, longer period and it will fall behind the green

particle in phase.

(a) (b) (c)

Fig 2.2 Convergence and phase change of Newton Method

In fact, with the Newton Method, the 1st iteration assumes initial force (FIter)

which is along x-axis. Resultantly, velocity in y-axis direction induced by FReal will

not appear next step. And finally, motion toward the center is limited and particle

deviates from the ellipse (Fig 2.2 (a)). Alternatively, if the period is large enough,

namely iteration step comparatively is sufficiently short, the deviation will be well

restricted (Fig 2.2 (b)).

Newton Method operates at the cost of deviation. However, real-time simulation

can still be realized with it by applying period condition. Calculating and saving the

data for first cycle and later call these data to dynamically present planet motion is a

possible way. Period condition can be derived from Kepler’s Law but not Newton’s

though.

1.4 Plotting planets trajectories with Kepler Method

Kepler Method was applied to calculation of planets location, and dynamic

motion can be shown with multithread technology. Capture of the simultaneous

simulation of Mercury, Earth and Mars moving is given in Fig.2.3.

Fig 2.3 Capture of Mercury, Earth and Mars Moving

FReal

FIter

Chap 3. Relative motion and Ptolemy System

Apollonius, Hipparchus and Ptolemy all believed the theory of epicycle and

deferent. With the Kepler Method developed before, it can be shown that epicycle and

deferent demo is just another complicated description of Heliocentrism. To validate

ability of the method for relative motion description, ratio between periods of earth

and other planets is assumed to be 1:2, 2:3, 3:4 etc to give regular “epicycle and

deferent” curves. After the method is validated, relative motion trajectory can be

presented with credibility.

(a)period of Mars: 1 year (b) period of Mars: 2 years

(b) period of Mars: 0.5 year (d) period of Mars: 4 years

(e) period of Mars: a quarter year (f) period of Mars: 1.5 years

Fig 3.1 Relative motion of Mars with assumed periods

Fig 3.1 gives the assumed trajectory of Mars observed from Earth where orbits of

Mars and Earth are proportional to real parameters while period of Mars is assumed as

marked [1]

and rotation of Earth is ignored. In the figures, the red disc corresponds to

the Sun, the green curve to Earth’s orbit, the blue one to Mercury’s, and the black one

to motion of Mars relative to Earth (Earth located at center). Fig 3.1 validates Kepler

Method not just because the trajectories are regular and closed but also because the

point pair for relative location is selected from real-time simulation. In fact, the pairs

can be selected from many ways. For example the two planets are of the same phase

angle that is linearly increased with time. In trajectory given in Eq(1.4), area swept

rate is constant but ω =A

r2 varies with radius r. Therefore, how revolution angle θ

approximates the real one depends on the accuracy of iteration stability. Now with

Kepler Method validated in Fig. 3.1, “epicycle and deferent” curve of trajectory with

completely proportional data can be obtained.

Fig 3.2 Mars trajectory observed on Earth (for about 80 years)

If one observes Mars on Earth day and night for 79 years, the simulated

trajectory in the sky is given in Fig. 3.2. However, no record of Mars observed

trajectory has been found.

Besides, pentagram has always been related to human worship of Venus. The

relationship can be most credibly attributed to prehistoric astronomer observation of

Venus trajectory. They found that seen from Earth, Venus orbit repeats every five

years and the 5 junction points form an almost perfect pentagram [2]

. With validated

Kepler Method, simulated Venus trajectory is given in Fig. 3.3 and it is consistent

with observation quite well.

Fig 3.3 Venus trajectory observed on Earth (for about 40 years)

The right figure corresponds to human worship

In Fig 3.3, the red disc corresponds to the Sun, the green curve to Earth’s orbit,

the pink one to Venus’, and the black one to motion of Venus relative to Earth (Earth

located at center). It is different from Fig 3.2 that the trajectory of Venus is nearly

closed. As can be seen from Fig 3.3, the trajectory of relative motion becomes thicker

as time goes, which results from the quasi-closed curve. In fact, the radio between

periods of Earth and Venus is 365.26:224.7≈13:8, so evolution through eight times

Earth’s period or 13 times Venus’ period yields a closed curve. It is also consistent

with the observation that Venus orbit repeats every 8 years.

Chap 4. Musical correlation of planets motion

Numerical solution obtained above gives distance, velocity, phase and angular

frequency at any time. They can be applied to composing music.

Plotting “epicycle and deferent” curve in Chap 3 contributes to data evolving more

diversely with time. Frequencies and distances evolution of revolution and relative

motion to earth are given in Fig 4.1. Apparently the latter data are more suitable for

perfectly fluctuant and varied music.

4.1-(a) Earth’s revolution (for about 10 years)

4.1-(b) Venus’ relative motion to earth (for about 10 years)

Fig 4.1 Variation of potential music elements

Fig 4.1 gives how earth-sun distance and angular frequency of Earth’s revolution

evolve on its orbit (a) and how Venus-earth distance angular frequency of Venus’

relative motion on the observed trajectory (b) where rrelative = rEarth − rVenus, and

θRelative = θEarth − θVenus is conserved in iteration method given by Eq(2.2).

Data corresponding to that in Fig 4.1-(b) are selected for music. Before real-time

music playing, maximum and minimum of the data are located to transform the

frequencies to the range of general musical instruments. Then plotting observed

relative motion trajectory and playing “Song of Planets” can be realized

simultaneously with Beep() function and multithread technology.2

Music played from Beep() function has classic tone of electronic toys. The

problem can be avoided by playing violin or piano consistent with data obtained from

relative motion. To write music score for playing an instrument, music elements given

in Fig 4.1 (b) should be discretized to particular frequencies like 440Hz for A. And the

tune length can be selected according to tune diversity. Where the tune changes

rapidly, tune length is short for more during the same period and where the tune

changes smoothly, a long one can be played. However, discretization must lead to loss

of information. Experienced players can read music as given in Fig 4.2, which is

created by superposing curve from Fig 4.1-(b) to a staff and are able to play freely

with more elements conserved. Besides, distance data can be introduced to construct

chords.

Fig 4.2 Staff of “Song of Planets” for experienced players

So far, planets revolution and relative motion to Earth have been obtained with

numerical methods developed and validated. Motion of several planets and music

play can be simulated simultaneously. Instrument playing suggestions are offered.

Numerical methods: Kepler Method and Newton Method were proposed, compared

and validated carefully for finally music composing requirements. In fact, relative

motion trajectory can be obtained without necessarily real-time simulation. The pairs

for relative location can be selected from many situations where for example the two

planets are of the same phase angle that is linearly increased with time. However,

relative motion angular frequency from that way instead of real-time simulation

makes no sense. And that requires validating ability of Kepler method for relative

motion description.

2 “Song of Planets” is recorded and saved at http://home.ustc.edu.cn/~wsmjl/VenusEarth.mp3

Dynamic simulation video is saved at http://home.ustc.edu.cn/~wsmjl/VenusEarth.exe

References:

[1].Planets revolution data from http://jmstwxh.lamost.org/twcy11.htm

[2].Relationship between Venus and pentagram from

http://www.hudong.com/wiki/%E4%BA%94%E8%A7%92%E6%98%9F

Acknowledgement

The author would like to acknowledge Prof. Chen GAO at NSRL for his inviting

me to working at his Lab after his optics course. My initial computer skills and

interest for numerical methods were formed through the experiences in his group. The

author would like to than AP. Yongjie SUN for his high praise for my work of detector

simulation in Particle Detection essay. His encouragement drives me to go further.

Finally, the author would like to than Prof. Shouping Xiang for his hard work through

this term. Prof. Xiang attracts us with his abundant knowledge, exciting lecturing

skills and wonderful comments during classes. Learning the course successfully make

up for lack of astronomy in my secondary education.

May. 2012