第九章 莫比乌斯环型反引力场发动机方程组
Chapter 9 The sets of equations of the Möbius-loop-typed antigravitation engine
9.1 完全对称物体的惯性运动的正弦曲线形短程线方程
9.1 The equation of the sinusoidal geodesic of a well-balanced body in inertial motion
(1)
基于第8.3.2节,设一完全对称的物体B沿x轴作惯性运动.根据广义相对论,由于物体B有质量,它周围的空间是弯曲的.由于(8.3.2-1),物体B的路径形成正弦曲线:
y = A sin ( ωx + φo ) (9.1(1)-1)
其中 A 是振幅, ω 是圆频率, φ0 是初相.
Based on § 8.3.2, suppose that a well-balanced body called Body B is in inertial motion along the x-axis. According to the general theory of relativity, since Body B has mass, the space around it is curved. Because of (8.3.2-1), Body B's path forms a sinusoid:
y = A sin ( ωx + φo ) (9.1(1)-1)
where A is the amplitude, ω is the angular frequency, and φo is the initial phase.
(2)
关于振幅 A, 应该有下面的关系式:
A / c = Δt ,
式中 c 是真空中的光的速率, Δt 是观测者收到物体 B 的信息所需时间的最小不确定值.
For the amplitude of A, there should be the following relation:
A / c = Δt ,
where c is the speed of light, and Δt is the smallest range of uncertainty in the time that an observer takes when he or she receives information from Body B.
由量子力学里的不确定关系, E Δt = h / ( 4 ), 所以有 A / c = Δt = h / ( 4 E ), A = c h / ( 4 E ).由 E = m c2,有
A = h / ( 4 m c ). (9.1(2)-1)
The uncertainty relations in quantum mechanics give E Δt = h / ( 4 ), and hence there are the relations A / c = Δt = h / ( 4 E ), A = c h / ( 4 E ). From E = mc2, there is
A = h / ( 4 m c ). (9.1(2)-1)
(3)
命方程(9.1(1)-1)所表示的正弦曲线的波长为 λ, 则由德布罗意关系,有
λ = h / ( m vx ). (9.1(3)-1)
Let the wavelength of the sine curve described by Eq. (9.1(1)-1) be λ. From the de Broglie relations, there is
λ = h / ( m vx ). (9.1(3)-1)
因为 ω = 2 / T, T = λ / vx , 所以
ω = 2 vx / λ. (9.1(3)-2)
Since ω = 2 / T, and T = λ / vx , there is
ω = 2 vx / λ. (9.1(3)-2)
将(9.1(3)-1)代入(9.1(3)-2)得到 ω = 2 vx / ( h / m vx ), 即
ω = 2 m vx2 / h. (9.1(3)-3)
Substituting (9.1(3)-1) into (9.1(3)-2), one obtains ω = 2 vx / ( h / m vx ), i.e.
ω = 2 m vx2 / h. (9.1(3)-3)
(4)
将(9.1(2)-1)和(9.1(3)-3)代入(9.1(1)-1), 得
y = ( h / ( 4 m c ) ) s i n ( ( 2 m vx2 / h ) x + φo ). (9.1(4)-1)
(9.1(4)-1)可以称为完全对称物体的惯性运动的正弦曲线形短程线方程.
Substituting (9.1(2)-1) and (9.1(3)-3) into (9.1(1)-1), one obtains
y = ( h / ( 4 m c ) ) s i n ( ( 2 m vx2 / h ) x + φo ). (9.1(4)-1)
(9.1(4)-1) can be called the equation of the sinusoidal geodesic of a well-balanced body in inertial motion.
以上推导是基于参考文献[1]和文献[2]的第二章.
The above deduction is based on Reference [1] and Chapter 2 of Reference [2].
参考文献
References
[1] 张旭.《UFO：现象、理论、实验》.1990年8月.
发表于《天·地·人 科学文化纵横集》.吴之静主编.北京:科学普及出版社,1992 年9月. 第90~113页.ZHANG Xu. "UFO: Phenomena, theories and experiments". August, 1990. This article is published in Heaven, Earth and Man - Across Science And Culture. Chief compiler: WU Zhijing. Beijing: Popular Science Publishing House, September, 1992. Print. Pages 90~113. (In Chinese.)
[2] 张旭.本网站第一部分:反引力场发动机. https://www.twistor-light-silhouettes.com
ZHANG Xu. Part 1 of this website: Antigravitation Engine Site. https://www.twistor-light-silhouettes.com
9.2 莫比乌斯环型反引力场发动机方程组
9.2 The sets of equations of the Möbius-loop-typed antigravitation engine
莫比乌斯环有挠率.下面来推导莫比乌斯环型反引力场发动机方程组.
The Möbius loop has torsion. Now let's derive the sets of equations of the Möbius-loop-typed antigravitation engine.
(1)
在区间 [
( / 2 ),
] ,即在区间
(
/ 2 ) ≤ x
In the interval
[(
/ 2 ), ],
i.e. in the interval (
/ 2) ≤ x
在区间 [ / 2 , ] ,
Δvy / vx = y / x .
In the interval [( / 2 ), ],
Δvy / vx = y / x .
由方程(9.1(3)-1)和(9.1(4)-1)可以知道,在区间 [ / 2 , ] ,有 x = h / ( 4 m vx ), y = h / ( 4 m c ), 因此有 y / x = vx / ( c ).
所以有
Δvy / vx = vx / ( c );
From (9.1(3)-1) and (9.1(4)-1), it can be known that
in the interval [( / 2 ), ], there are x = h / ( 4 m vx ), and y = h / ( 4 m c ), and hence there is y / x = vx / ( c ).Hence there is
Δvy / vx = vx / ( c );
即
Δvy = vx2 / ( c ). (9.2(1)-1)
i.e.
Δvy = vx2 / ( c ). (9.2(1)-1)
(2)
命自转体的轴为 y 轴.当自转体的等效质点转动相当于一个波长的距离时, vy 就增加一次, 类似于在略微下坡的平坦道路上只用右脚蹬自行车时,每蹬一次,车就增速一次.所以 有
a = Δvy / T , (9.2(2)-1)
式中 T 是周期.
Let the shaft of the rotating body be the y-axis. When the equivalent particle of mass of the rotating body rotates for a distance equal to a wavelength, vy increases once, just as the speed of the bicycle increases every time a boy pedals his bicycle, with only his right foot, on a slightly downhill flat road. Thus there is
a = Δvy / T, (9.2(2)-1)
where T is the period.
(3)
将(9.2(1)-1)代入上式,得到
a = ( vx2 / ( c ) ) / T. (9.2(3)-1)
Substituting (9.2(1)-1) into the above equation, one obtains
a = ( vx2 / ( c ) ) / T. (9.2(3)-1)
由关系式 T = λ / vx ,和德布罗意关系 λ = h / ( m v ),有
T = h / ( m vx2 ), (9.2(3)-2)
From the relation T = λ / vx , and the de Broglie relation λ = h / ( m v ), there is
T = h / ( m vx2 ). (9.2(3)-2)
将(9.2(3)-2)代入(9.2(3)-1),得到
a = ( vx2 / ( c ) ) / ( h / ( m vx2 ) ),
即
a = m vx4 / ( c h).
Substituting (9.2(3)-2) into (9.2(3)-1), one obtains
a = ( vx2 / ( c ) ) / ( h / ( m vx2 ) ),
i.e.
a = m vx4 / ( c h).
(4)
由于反引力加速度具有 "非全即无" 的性质,由上式得到莫比乌斯环型反引力场发动机方程组如下:
|
a反引力 = m (v线)4 / ( c h) |
(当 | m (v线)4 / ( c h) | > | ∑R | 时); |
(9.2(4)-1a) |
|
a反引力 = 0 |
(当 | m (v线)4 / ( c h) | ≤ | ∑R | 时); |
(9.2(4)-1b) |
式中a反引力是自转体金属部分产生的沿自转体的前方方向的反引力加速度, a反引力的方向是从使反引力场得以发生的扰动源指向自转体的方向(请参看文献[1]的第一章第1.2节), m是自转体中具有长程有序自由运动的那种粒子或粒子团的质量(这是因为反引力加速度是由自转体中具有长程有序自由运动的那种粒子产生的), v线 是自转体的等效质点(位于回转半径末端点)的切向速率,| ∑R |是不包含a反引力在内的沿反引力的前方方向的合加速度的绝对值.
Since the antigravitational acceleration has the nature of "whole or none", from the above equation, one can obtain the set of equations of the Möbius-loop-typed antigravitation engine as follows:
|
aantigravitational = m (vtangential)4 / ( c h) |
(if | m (vtangential)4 / ( c h) | > | ∑R | ); |
(9.2(4)-1a) |
|
aantigravitational = 0 |
(if | m (vtangential)4 / ( c h) | ≤ | ∑R | ); |
(9.2(4)-1b) |
where aantigravitational is the antigravitational acceleration, which is produced by the metal part of the rotating body, and whose direction is from the source of disturbance which helps to produce the antigravitation to the rotating body (See Reference [1], Section 1.2 of Chapter 1); m is the mass of the particle or the mass of the group of particles which is in the rotating body and which moves freely for a long distance and in good order (this is because in the rotating body the antigravitational acceleration is produced by the particles which move freely for a long distance and in good order); vtangential is the tangential speed of the equivalent particle of mass (at the end point of the radius of gyration) of the rotating body; | ∑R | is the absolute value of the resultant acceleration which is along the front direction of the antigravitational acceleration, excluding aantigravitational.
(5)
|
反引力场发动机运动的方向被视为前方方向,相反的方向被视为后方方向. (9.2(5)-1) |
|
The direction in which the antigravitation engine is moving is regarded as the forward direction, while the opposite direction is regarded as the backward direction. (9.2(5)-1) |
将方程
vx = 2 r / T
代入(9.2(4)-1a)和(9.2(4)-1b),就得到了在我的著作《UFO：现象、理论、实验 》[1]中建立起来 的反引力场发动机方程组:
|
aantigravitational = 16 3 m r4 / ( c h T4 ) |
(当 | 16 3 m r4 / ( c h T4 ) | > | Σa' | 时); |
(9.2(5)-2a) |
|
aantigravitational = 0 |
(当 | 16 3 m r4 / ( c h T4 ) | ≤ | Σa' | 时); |
(9.2(5)-2b) |
Substitution of the equation
vx = 2 r / T
into (9.2(4)-1a) and (9.2(4)-1b) yields the following set of equations of the antigravitation engine which was set up in my work "UFO: Phenomena, theories and experiments"[1].
|
aantigravitational = 16 3 m r4 / ( c h T4 ) |
(if | 16 3 m r4 / ( c h T4 ) | > | Σa' | ); |
(9.2(5)-2a) |
|
aantigravitational = 0 |
(if | 16 3 m r4 / ( c h T4 ) | ≤ | Σa' | ); |
(9.2(5)-2b) |
详情请看本网站
For detailed information, please see Chapters 1~3 of this website.
问题9.2(1) 太阳对于钱德勒摆动的激发和大气中准两年周期振荡
Problem 9.2(1) Solar excitation of the Chandler Wobble, and the quasi-biennial oscillation in the atmosphere
太阳表面的半径是R = 6.959 9×108 m,太阳赤道处的表面引力加速度是a引力 = 274 m/s2.求环绕着太阳赤道的表面环的自转周期.
The radius of the surface of the Sun is R = 6.959 9×108 m, and the surface gravitational acceleration at the equator of the Sun is agravity = 274 m/s2. Find the rotation period of the surface ring around the Sun's equator.
设在动态平衡中a反引力 ≈ a引力, r = R,并且m = me.于是有r = 6.959 9×108 m, a反引力 = 274 m/s2.
Assume that, in dynamic equilibrium, aantigravitational ≈ agravitational, r = R, and m = me. Then there are r = 6.959 9×108 m, and aantigravitational = 274 m/s2.
将这些数值代入(9.2(5)-2a),得到T = 3.736 03×107 s,即
T = 432.411 天. (9.2-1)
Substituting these values into (9.2(5)-2a), one gets T = 3.736 03×107 s, i.e.
T = 432.411 days. (9.2-1)
这与地球的钱德勒摆动周期是一致的,正如文献[1]所指出的那样.
This is consistent with the period of the Chandler Wobble, as was pointed out in Reference [1]
(15.3-3)里面的绝对值符号和(8.3-1)有助于解释为什么钱德勒章动的相位在2005年跳动了180度.
The symbol of absolute value in (15.3-3) and (8.3-1) help to explain why the phase of the Chandler Wobble jumped by 180 degrees in 2005.
大气中准两年周期振荡现象的平均周期大约为28~29个月.由(9.2-1),可以知道
2 T = 28.4 月 ≈ 2 年.
The mean period of the quasi-biennial oscillation in the atmosphere is 28 to 29 months. From (9.2-1), one can know that
2 T = 28.4 months ≈ 2 years.
因此钱德勒摆动和大气中准两年周期振荡现象可能都是由太阳的表面环激发的.
Hence both the Chandler Wobble and the quasi-biennial oscillation in the atmosphere may be excited by the Solar surface ring.
参考文献
[1] 张旭. 《UFO：现象、理论、实验》.1990年8月.
发表于《天·地·人 科学文化纵横集》.吴之静主编.北京:科学普及出版社,1992 年9月. 第112页.[1] ZHANG Xu. "UFO: Phenomena, theories and experiments". August, 1990. This article is published in Heaven, Earth and Man - Across Science And Culture. Chief compiler: WU Zhijing. Beijing: Popular Science Publishing House, September, 1992. Print. Page 112. (In Chinese.)
问题9.2(2) 太阳宇宙线加速机制
Problem 9.2(2) The acceleration mechanism of solar cosmic rays
太阳的赤道切向速率是v = 1996.94 m/s.在日冕中粒子碰撞的频率低.
假设太阳的自转导致带电粒子团以太阳的赤道切向速率v自转,并且假设在日冕里| mv4 / (ch) | > | ∑R |;求粒子在日冕里的加速度.
The Sun's equatorial speed is v = 1996.94 m/s. In the corona, frequencies of collisions between particles are low.
Assume that the rotation of the Sun makes groups of charged particles rotate at the Sun's equatorial speed of v, and assume that in coronal loops | mv4 / (ch) | > | ∑R |. Find the acceleration of particles in the corona.
由(9.2(4)-1a)可以知道,粒子的加速度与它的质量成正比.
It can be seen from (9.2(4)-1a) that the acceleration of a particle is proportional to its mass.
将v = 1968.05 m/s和m = me代入(9.2(4)-1a),得到电子的反引力 加速度是
a反引力= 2.3×107 m/s2.
Substituting v = 1968.05 m/s and m = me into (9.2(4)-1a), one gets the antigravitational acceleration of an electron:
aantigravitational = 2.3×107 m/s2 .
将v = 1996.94 m/s和m = mp代入(9.2(4)-1a),得到质子的反引力加速度是
a反引力=4.3×1010 m/s2.
Substituting v = 1996.94 m/s and m = mp into (9.2(4)-1a), one gets the antigravitational acceleration of a proton:
aantigravitational = 4.3×1010 m/s2.
因此在不到1秒钟,日冕环里的正离子就可以加速到近光速.
Hence in less than a second, positive ions within coronal loops can accelerate to nearly the speed of light
问题9.2(3) 太阳风的加速机制
Problem 9.2(3) The acceleration mechanism of the solar wind
在日冕中,尤其是在冕洞里,粒子碰撞的频率低.粒子流在中米粒里上升和下降的速度是v = 0.05 ~ 0.1 km/s[1].
In the solar corona, especially in coronal holes, frequencies of collisions between particles are low. The vertical speed of both downdrift and updrift of particles in the mesogranulation is v = 0.05 ~ 0.1 km/s[1].
假设在中米粒里粒子流的旋转通过磁场导致日冕中的粒子团以切向速率 v线 = 0.07 km/s自转,并且在日冕中| m (v线)4 / (ch) | > | ∑R |.求粒子在日冕中从速率 v0 = 0 m/s 加速到 vt = 700 km/s 时所运动的距离s.
Assume that the rotation of the flows of the particles in the mesogranulation makes, via the magnetic fields, particle balls in the corona rotate at the tangential speed of vtangential = 0.07 km/s, and assume that in the corona | m (vtangential)4 / (ch) | > | ∑R |. Find s, the distance which a particle in the corona travels as it accelerates from the speed v0 = 0 m/s to the speed vt = 700 km/s.
因为vt2 - v02 = 2 a反引力 s, 所以s = vt2 / (2a反引力). 由(9.2(4)-1a),距离 s 是
s = vt2 / (2 m (v线)4 / ( c h)) .
Since vt2 - v02 = 2 aantigravitational s, there is s = vt2 / (2aantigravitational). From (9.2(4)-1a), the distance s is
s = vt2 / (2 m (vtangential)4 / ( c h)) .
不同质量的粒子产生的反引力 加速度不同.
对于电子,m = me,于是s = 7.0 × 106 km, 即10个太阳半径(太阳半径是6.959×105 km).
对于质子,m = mp,于是s = 3.8 × 103 km.作为比较,太阳光球层的厚度约为5×102 km,色球的厚度约为2.5×103 km.
Particles of different masses produce different antigravitational accelerations.
For an electron, m = me , and hence s = 7.0×106 km, i.e. 10 times the radius of the Sun (The radius of the Sun is 6.959 × 105 km).
For a proton, m = mp , and hence s = 3.8 × 103 km. As a comparison, the thickness of the solar photosphere is 5×102 km, and that of the chromosphere is 2.5×103 km.
与在冕洞外面相比,在冕洞里面,平均而言 | ∑R | 较小,因而(9.2(4)-1a)描述的规律起作用的连续时间比较长,粒子在反引力场中连续加速的时间比较长,结果形成了太阳风高速流.
Compared with outside the coronal holes, inside the coronal holes, on the average, | ∑R | is smaller, and hence the law described by (9.2(4)-1a) has a longer continuous length of time to work, and particles have a longer continuous length of time to accelerate in the antigravitational field, forming high velocity solar wind streams.
请参看太阳风"湍流漩涡"的图片[2].
Please see the picture of the "turbulent eddies" of the solar wind[2].
参考文献
Reference
[1] Mark Peter Rast, "Solar granulation: a surface phenomenon", Geophysical and Astrophysical convection, Edited by Peter A. Fox and Robert M. Kerr, Gordon and Breach Science Publishers, 2000. Print. ISBN-10:9056992589; ISBN-13:978-9056992583.
[1] Mark Peter Rast, "Solar granulation: a surface phenomenon", Geophysical and Astrophysical convection, Edited by Peter A. Fox and Robert M. Kerr, Gordon and Breach Science Publishers, 2000. Print. ISBN-10:9056992589; ISBN-13:978-9056992583.
中文
[2] http://www.cmen.cc/2014/yzts_0101/331164.html
English
[2] http://www.sciencedaily.com/releases/2013/01/130108145227.htm
9.3 自制一台莫比乌斯环型反引力场发动机
9.3 Making your own Möbius-loop-typed antigravitation engine
自制一台莫比乌斯环型反引力场发动机.
Make a Möbius-loop-typed antigravitation engine in the following way.
1) 用玩具电动机带动一个小轮(以下简称为自转体)作为自转装置,
1) A small wheel (hereafter called the rotating body), driven by a toy motor, serves as the rotation device.
2) 用自转装置中的电动机后面的电源装置作为扰动装置(即扰动Riemann-sphere-bubble流的装置),也就是定向装置.电源可以采用四节五号充电电池.
2) The power device behind the motor of the rotation device serves as the disturbance device, which disturbs the Riemann-sphere-bubble current, and hence it also serves as the direction-determining device. Rechargeable batteries of the power device supply the power.
3) 把导线连接好以便使得电动机轴上的金属小轮自转.
3) Connect the wires to make the metal wheel on the motor shaft rotate.
4) 用乐扣乐扣塑料保鲜盒作为封闭装置,将上述装置封闭起来.
4) A LocknLock food storage container serves as the sealing device, with which the devices mentioned above are sealed up.
5) 将上述装置水平地放在承载装置上.承载装置是一块泡沫塑料板.将上述装置放在圆形铝制澡盆里的水面上,成为一个“小船”,这小船就是一台 莫比乌斯环型反引力场发动机;这个小船可以称为反引力小船.
5) Put the above devices horizontally in the carrying device, which is a foam plastic board. Put the above-mentioned devices on the water in a round aluminum wash tub, and the devices are now a "boat". This boat is a Möbius-loop-typed antigravitation engine; and this boat can be called an antigravitational boat.
实验中可以看到反引力小船在一开始时往往旋转、停滞或在起始点附近随机运动,通常小船逐渐呈现出沿自转体的前方方向或后方方向的运动,运动的方式往往像是一蹿一蹿的,在运动的过程中可能会停滞,停滞之后运动的方向有可能会相反,有时在实验快结束时小船会突然增速.
At the beginning of the experiment, the antigravitational boat often rotates, stays at the same place, or moves about randomly near the starting place. Usually the boat gradually moves along the front or back direction of the rotating body in such a way that it looks like leaping once and once again. During the movement the boat might stop, and after the stop it might move in the opposite direction. Sometimes towards the end of the experiment the boat might suddenly move much faster.
|
鉴于(8.12-3), 黎曼球面泡量子导致微观和宏观的量子现象.由于宏观量子效应,当物体(例如小船)被黎曼球面泡控制的时候,它处于不确定时空之中,因此出现忽快忽慢和进、退、停滞等不确定现象. (9.3-1) |
| In view of (8.12-3), the Riemann-sphere-bubble quantum causes microscopic and macroscopic quantum phenomena. Because of the macroscopic quantum effect, when being controlled by the Riemann-sphere-bubble, an object (e.g. a boat) is in the uncertain spacetime, and hence it moves now fast, now slow, now forwards, now backwards, and sometimes it stops for a while. (9.3-1) |
| 反引力场发动机发出的反引力场是耗散结构.反引力场发动机处于量子运动中. (9.3-2) |
| The antigravitational field generated by an antigravitation engine is a dissipative structure. An antigravitation engine is in quantum motion. (9.3-2) |
自转的电流可以作为反引力场发动机的自转体.
A rotating electrical current can serve as the rotating body of an antigravitation engine.
另请参看本网站的第三章.
See also Chapter 3 of this website.
产生 | Σa' | 的物体可以成为势垒.地面、海面、云、山 、气流、水面上的灰尘等障碍物都有可能成为势垒.根据量子力学,势垒区域中的波函数具有节点,除非这个波函数处于基态.(见文献[1],第7章第7.3节.)
The object causing | Σa' | can be the potential barrier. Obstacles, such as the land, the sea, the cloud, the mountain, the air current, and dust on water, can all become the potential barrier. According to quantum mechanics, a wavefunction in the potential barrier region has nodes, unless it is of the ground state. (See Reference [1], Chapter 7, Section 7.3.)
9.4 加速螺线
9.4 The accelerating spiral
肯尼斯 · W. 福特指出,
|
有一个数学公式可以给出任意给定自旋角动量(以ħ为单位)具有多少可能的方向,即二倍角动量加1.[1] (9.4-1) |
Kenneth W. Ford pointed out that
|
the number of possible directions of any given spin angular momentum (with ħ as its unit) is calculated as twice the spin angular momentum plus one.[1] (9.4-1) |
假设A是一个天体,那么天体A的自转角动量很大.因此A的自转的 黎曼球面泡的角动量(以ħ为单位)极其大,根据(9.4-2), 黎曼球面泡的自转具有极其多的可能的方向,黎曼球面泡具有极其多的可能的自转赤道;或者换句话说,可以认为 黎曼球面泡的自转方向是可以连续变化的,根据量子力学中的态叠加原理,天体的黎曼球面泡沿各方向的自转同时并存.由(9.4-1),
Suppose A is a celestial body; then A's angular momentum of rotation is very large. Hence the angular momentum (with ħ as its unit) of A's rotational Riemann-sphere-bubble is extremely large. Therefore, according to (9.4-2), the possible directions of the rotation of the Riemann-sphere-bubble is enormous, and the Riemann-sphere-bubble has enormous possible equators, or in other words, the direction of the Riemann-sphere-bubble's rotation can be regarded as being able to vary continuously. According to the principle of superposition states in quantum mechanics, a celestial body's Riemann-sphere-bubble's rotational movements in every direction coexist at the same time. From (9.4-1), one can know that
|
宏观物体的黎曼球面泡沿各方向的自转同时并存,并根据(9.2(4)-1a)而产生沿各方向的反引力. (9.4-2) |
|
The spin of a macroscopic body's PN-event-particle exists in every direction at the same time, and, according to (9.2(4)-1a), generate antigravitation in every direction. (9.4-2) |
由(9.4-2),宏观自转体的 Riemann-sphere-bubble量子沿各方向的自转同时并存,并根据(9.2(4)-1a)而产生沿各方向的反引力.宏观Riemann-sphere-bubble涡旋沿各方向同时产生离心反引力.在宏观 PN 气球涡旋里沿径向作反引力加速运动的质点的轨道是一条螺线,这条螺线可以称为加速螺线.
From (9.4-2), the spin, in every direction, of a macroscopic rotating body's Riemann-sphere-bubble quantum coexist at the same time, and, according to (9.2(4)-1a), generate antigravitation in every direction. A macroscopic Riemann-sphere-bubble vortex generates antigravitational centrifugal force in every direction at the same time. In a macroscopic Riemann-sphere-bubble vortex, the path of a particle in the antigravitational acceleration in the radial direction is a spiral, which can be called an accelerating spiral.
加速螺线是黎曼球面泡的短程线,它可以由极坐标方程来描述.
|
加速螺线 r = r0 + (1 / 2) a反引力 ( k θ )2 , θ = ( 1 / k ) t ; 其中 r 是极径, r0 是极径的初始值, a反引力 是反引力加速度, θ 是极角, k 是比例系数, t 是反引力导致的惯性系拖曳效应的连续持续时间. (9.4-3) |
1) The accelerating spiral is the geodesic of
the Riemann-sphere-bubble, which can be described by polar coordinate equations.|
The accelerating spiral r = r0 + (1 / 2) aantigravitational ( k θ )2 , θ = ( 1 / k ) t ; where r is the radial coordinates, r0 the initial values; aantigravitational is the antigravitational acceleration, θ the angular coordinate, and k the proportionality coefficient; t is continuous duration of the inertial-frame-dragging effect brought by antigravitation. (9.4-3) |
(9.4-3)可以称为加速螺线形短程线方程.加速螺线上的正弦曲线可以称为正弦加速螺线.
(9.4-3) can be called the equations of the geodesics in the shape of accelerating spirals. A sinusoid on an accelerating spiral can be called a sinusoidal-accelerating spiral.
可以用Mathematica程序对于(9.4-3)所描述的加速螺线作图.下面是一个例子.
Clear["Global`*"]; k = 1.1; a = 2; r0 = 0; r = r0 + 0.5*a*(k*θ)^2; Picture1 = PolarPlot[r, {θ, 0, 4*Pi}]; Print[Picture1]
The accelerating spiral described by (9.4-3) can be plotted in Mathematica. An example is given as follows.
Clear["Global`*"]; k = 1.1; a = 2; r0 = 0; r = r0 + 0.5*a*(k*θ)^2; Picture1 = PolarPlot[r, {θ, 0, 4*Pi}]; Print[Picture1]
由(9.4-3),
| 随着时间的流逝,加速螺线变得越来越像同心圆. (9.4-4) |
所以椭圆星系比螺旋星系老.
From (9.4-3),
| as time passes, the accelerating spiral becomes more and more like concentric circles. (9.4-4) |
Hence the elliptical galaxy is older than the spiral galaxy.
|
对于正弦加速螺线形涡旋而言,正弦曲线的振幅的二倍等于涡旋的厚度. (9.4-5) |
| In the case of a vortex in the shape of a sinusoidal-accelerating spiral, twice the amplitude of the sinusoid is equal to the thickness of the vortex. (9.4-5) |
[1] (美)肯尼斯·W.福特. 量子世界:写给所有人的量子物理.王菲,译.北京:外语教学与研究出版社,2008. ISBN 978-7-5600-7977-6 .第97页.
[1] Kenneth W. Ford. The Quantum world: Quantum Physics for Everyone. Harvard University Press, 2005. Print. ISBN 0-674-01832-X & ISBN 9780674018327. Chapter 5.
问题9.4(1) 银河系旋臂的倾角
Problem 9.4(1) The pitch of the spiral arm of the Milky Way Galaxy
设在极坐标系中, r 是极径, θ 是极角(见(9.4-3)), 速度的方向是角 β .
则根据力学, vθ = r · ( dθ / dt ), vr = dr / dt, β = arctan ( vθ / vr )[1]. 因此有
β = arctan ( r · ( dθ / dt ) / ( dr / dt ) ). (9.4(1)-1)
(参看力学>运动学>质点运动学>极坐标表示法.)
Suppose that in a polar coordinate system, r is the radial coordinate, and θ is the angular coordinate (see (9.4-3)), and that the direction of the velocity is an angle of β .
Then according to mechanics, there are vθ = r · ( dθ / dt ), vr = dr / dt, and β = arctan ( vθ / vr )[1]. Hence there is
β = arctan ( r · ( dθ / dt ) / ( dr / dt ) ). (9.4(1)-1)
(See mechanics > kinematics > particle kinematics > in polar coordinates.)
由(9.4-3),当 r0 = 0 时,有
r = 0.5 a k2 θ2 .
因此 dr = 0.5 a k2 dθ2 ;所以有 dr / dθ = ( 0.5 a k2 dθ2 ) / dθ = ( 0.5 a k2 ) ( dθ2 / dθ ),即
dr / dθ = a k2 θ .
From (9.4-3), when r0 = 0, there is
r = 0.5 a k2 θ2 .
Hence dr = 0.5 a k2 dθ2 ; Thus there is dr / dθ = ( 0.5 a k2 dθ2 ) / dθ = ( 0.5 a k2 ) ( dθ2 / dθ ), i.e.
dr / dθ = a k2 θ .
由(9.4-3),有
dθ / dt = 1 / k .
因此有 dr / dt = ( dr / dθ ) ( dθ / dt ) = a k2 θ · ( 1 / k ),即
dr / dt = a k θ .
From (9.4-3), there is
dθ / dt = 1 / k .
Hence there is dr / dt = ( dr / dθ ) ( dθ / dt ) = a k2 θ · ( 1 / k ), i.e.
dr / dt = a k θ .
所以有
β = arctan ( 0.5 a k2 θ2 · ( 1 / k ) / ( a k θ ) ), 即
β = arctan ( 0.5 θ ). (9.4(1)-2)
Hence there is
β = arctan ( 0.5 a k2 θ2 · ( 1 / k ) / ( a k θ ) ), i.e.
β = arctan ( 0.5 θ ). (9.4(1)-2)
命 φ = ( / 2 ) - β , 则有
φ = arccot ( 0.5 θ ), (9.4(1)-3)
式中 θ 是加速螺线的极角, φ 被称为旋臂的倾角.
Let φ = ( / 2 ) - β ; then there is
φ = arccot ( 0.5 θ ), (9.4(1)-3)
where θ is the angular coordinate of the accelerating spiral, and φ is known as the pitch of the spiral arm.
由(9.4(1)-3)可以得到下表:
|
φ = arccot ( 0.5 θ ) |
||||||
|
θ |
1 |
1.5 |
2 |
2.5 |
3 |
趋于无穷大 |
|
φ |
32.5° |
23.0° |
17.7° |
14.3° |
12.0° |
趋于零 |
| (9.4(1)-4) | ||||||
From (9.4(1)-3), one can obtain the following table:
|
φ = arccot ( 0.5 θ ) |
||||||
|
θ |
1 |
1.5 |
2 |
2.5 |
3 |
approaches infinity |
|
φ |
32.5° |
23.0° |
17.7° |
14.3° |
12.0° |
approaches zero |
| (9.4(1)-4) | ||||||
银河系旋臂的图片显示出旋臂延伸绕银心约一圈半,即 θ ≈ 3 .假定 θ ≈ 3 ,于是由上面的表可以知道旋臂的倾角是 φ ≈ 12° .这与观测值是一致的.反引力使得旋臂不会越旋越紧.
The image of the Milky Way's spiral arms shows that the spiral arms extend around the Galactic Centre about one and a half times, that is, θ ≈ 3 . Assume that θ ≈ 3 ; then from the above table one can know that the pitch of the spiral arm is φ ≈ 12° . This is consistent with the observed value. Antigravitation prevents the spiral arms from becoming more and more tightly wound.
由(9.4(1)-4),当加速螺线的极角 θ 趋近于无穷大时,加速螺线趋近于圆周.因此
| 在反引力场中,物体的内部运动可以被视为圆运动. (9.4(1)-5) |
From (9.4(1)-4), when θ, the angular coordinate of the accelerating spiral, approaches infinity, the spiral approaches a circle. Hence
|
In the antigravitational field, the internal motion of a body can be considered as the circular motion. (9.4(1)-5) |
[1] 史可信.《力学(第二版)》. 北京：科学出版社.2008年1月.第20页.式(3-15)、(3-17).
[1] Shi Kexin. Mechanics (Second Edition). Beijing: Science Press. January, 2008. page 20. Equations (3-15) and (3-17). (In Chinese.)