文章目录
图片上传之后不知为何帧率降低了许多。。。
日地月三体
所谓三体,就是三个物体在重力作用下的运动。由于三点共面,所以三个质点仅在重力作用下的运动轨迹也必然无法逃离平面。
三体运动所遵循的规律就是古老而经典的万有引力
F ⃗ = G m i m j r 2 e ⃗ r \vec F=\frac{Gm_im_j}{r^2}\vec e_r F =r 2 G m i m j e r
则对于m i m_i m i 而言,
m i d v ⃗ i d t = G m i m j r i j 3 r ⃗ i j m_i\frac{\text d\vec v_i}{\text dt}=\frac{Gm_im_j}{r_{ij}^3}\vec r_{ij}m i d t d v i =r ij 3 G m i m j r ij
且
d r ⃗ i d t = v ⃗ i \frac{\text d\vec r_i}{\text dt}=\vec v_i d t d r i =v i
将其写为差分形式
v ⃗ i = ∑ j ≠ i G m j r i j 3 r ⃗ i j d t r ⃗ i = v ⃗ i d t \begin{aligned} \vec v_i&=\sum_{j\not=i}\frac{Gm_j}{r_{ij}^3}\vec r_{ij}\text dt\ \vec r_i&= \vec v_i\text dt \end{aligned}v i r i =j =i ∑r ij 3 G m j r ij d t =v i d t
由于我们希望观察三体运动的复杂形式,而不关系其随对应的宇宙星体,所以不必考虑单位制,将其在二维平面坐标系中拆分,令v ⃗ = ( u , v ) \vec v=(u,v)v =(u ,v ),则
u i + = ∑ j ≠ i G m j ( x j − x i ) d t ( x i − x j ) 2 + ( y i − y j ) 2 3 v i + = ∑ j ≠ i G m j ( y j − y i ) d t ( x i − x j ) 2 + ( y i − y j ) 2 3 x i + = u ⃗ i d t y i + = v ⃗ i d t \begin{aligned} u_i&+=\sum_{j\not=i}\frac{Gm_j(x_j-x_i)\text dt}{\sqrt{(x_i-x_j)^2+(y_i-y_j)^2}^3}\ v_i&+=\sum_{j\not=i}\frac{Gm_j(y_j-y_i)\text dt}{\sqrt{(x_i-x_j)^2+(y_i-y_j)^2}^3}\ x_i&+= \vec u_i\text dt\ y_i&+= \vec v_i\text dt \end{aligned}u i v i x i y i +=j =i ∑(x i −x j )2 +(y i −y j )2 3 G m j (x j −x i )d t +=j =i ∑(x i −x j )2 +(y i −y j )2 3 G m j (y j −y i )d t +=u i d t +=v i d t
太阳、地球和月亮就是一个典型的三体系统,其中太阳质量为1.989 × 1 0 30 k g 1.989×10^{30}kg 1.989 ×1 0 30 k g,地球质量为5.965 × 1 0 24 k g 5.965×10^{24}kg 5.965 ×1 0 24 k g,月球质量为7.342 ✕ 1 0 22 k g 7.342✕10^{22}kg 7.342✕1 0 22 k g,万有引力常数为G = 6.67 × 1 0 − 11 N ⋅ m 2 / k g 2 G=6.67×10^{-11}N·m2/kg^2 G =6.67 ×1 0 −11 N ⋅m 2/k g 2。地月距离为3.8 × 1 0 8 m 3.8\times10^8m 3.8 ×1 0 8 m;日地距离为1.5 × 1 0 11 m 1.5\times10^{11}m 1.5 ×1 0 11 m;地球公转速度为28.8 k m / s 28.8km/s 28.8 km /s;月球公转速度为1 k m / s 1km/s 1 km /s,则各参数初始化为
m = [1.33e20,3.98e14,4.9e12]
x = np.array([0,1.5e11,1.5e11+3.8e8])
y = np.array([0,0,0])
u = np.array([0,0,0])
v = np.array([0,2.88e4,1.02e3])
由于地月之间的距离相对于日地距离太近,所以在画图的时候将其扩大100倍,得到图像
尽管存在误差,但最起码看到了地球围绕太阳转,月球围绕地球转。。。代码为
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import animation
m = [1.33e20,3.98e14,4.9e12]
x = np.array([0,1.5e11,1.5e11+3.8e8])
y = np.array([0.0,0,0])
u = np.array([0.0,0,0])
v = np.array([0,2.88e4,2.88e4+1.02e3])
fig = plt.figure(figsize=(12,12))
ax = fig.add_subplot(xlim=(-2e11,2e11),ylim=(-2e11,2e11))
ax.grid()
trace0, = ax.plot([],[],'-', lw=0.5)
trace1, = ax.plot([],[],'-', lw=0.5)
trace2, = ax.plot([],[],'-', lw=0.5)
pt0, = ax.plot([x[0]],[y[0]] ,marker='o')
pt1, = ax.plot([x[0]],[y[0]] ,marker='o')
pt2, = ax.plot([x[0]],[y[0]] ,marker='o')
k_text = ax.text(0.05,0.85,'',transform=ax.transAxes)
textTemplate = 't = %.3f days\n'
N = 1000
dt = 36000
ts = np.arange(0,N*dt,dt)/3600/24
xs,ys = [],[]
for _ in ts:
x_ij = (x-x.reshape(3,1))
y_ij = (y-y.reshape(3,1))
r_ij = np.sqrt(x_ij**2+y_ij**2)
for i in range(3):
for j in range(3):
if i!=j :
u[i] += (m[j]*x_ij[i,j]*dt/r_ij[i,j]**3)
v[i] += (m[j]*y_ij[i,j]*dt/r_ij[i,j]**3)
x += u*dt
y += v*dt
xs.append(x.tolist())
ys.append(y.tolist())
xs = np.array(xs)
ys = np.array(ys)
def animate(n):
trace0.set_data(xs[:n,0],ys[:n,0])
trace1.set_data(xs[:n,1],ys[:n,1])
tempX2S = xs[:n,1]+100*(xs[:n,2]-xs[:n,1])
tempY2S = ys[:n,1]+100*(ys[:n,2]-ys[:n,1])
trace2.set_data(tempX2S,tempY2S)
pt0.set_data([xs[n,0]],[ys[n,0]])
pt1.set_data([xs[n,1]],[ys[n,1]])
tempX = xs[n,1]+100*(xs[n,2]-xs[n,1])
tempY = ys[n,1]+100*(ys[n,2]-ys[n,1])
pt2.set_data([tempX],[tempY])
k_text.set_text(textTemplate % ts[n])
return trace0, trace1, trace2, pt0, pt1, pt2, k_text
ani = animation.FuncAnimation(fig, animate,
range(N), interval=10, blit=True)
plt.show()
ani.save("3.gif")
日地火
质量
M M M G M GM GM
与太阳距离公转速度地球
5.965 × 1 0 24 k g 5.965×10^{24}kg 5.965 ×1 0 24 k g 3.98 × 1 0 14 3.98×10^{14}3.98 ×1 0 14 1.5 × 1 0 11 m 1.5\times10^{11}m 1.5 ×1 0 11 m 28.8 k m / s 28.8km/s 28.8 km /s
火星
6.4171 ✕ 1 0 23 k g 6.4171✕10^{23}kg 6.4171✕1 0 23 k g 4.28 × 1 0 13 4.28×10^{13}4.28 ×1 0 13 1.52 A . U . = 2.28 × 1 0 11 1.52 A.U.=2.28\times10^{11}1.52 A .U .=2.28 ×1 0 11 24 k m / s 24km/s 24 km /s
m = [1.33e20,3.98e14,4.28e13]
x = np.array([0,1.5e11,2.28e11])
y = np.array([0.0,0,0])
u = np.array([0.0,0,0])
v = np.array([0,2.88e4,2.4e4])
def animate(n):
trace0.set_data(xs[:n,0],ys[:n,0])
trace1.set_data(xs[:n,1],ys[:n,1])
trace2.set_data(xs[:n,2],ys[:n,2])
pt0.set_data([xs[n,0]],[ys[n,0]])
pt1.set_data([xs[n,1]],[ys[n,1]])
pt2.set_data([xs[n,2]],[ys[n,2]])
k_text.set_text(textTemplate % ts[n])
return trace0, trace1, trace2, pt0, pt1, pt2, k_text
得到
这个运动要比月球的运动简单得多——前提是开上帝视角,俯瞰太阳系。如果站在地球上观测火星的运动,那么这个运动可能相当带感
所以这都能找到规律,托勒密那帮人也真够有才的。
太阳系
由于太阳和其他星体之间的质量相差悬殊,所以太阳系内的多体运动,都将退化为二体问题,甚至如果把太阳当作不动点,那就成了单体问题了。
尽管如此,我们还是尽可能地模仿一下太阳系的运动情况
质量半长轴(AU)平均速度(km/s)水星0.0550.38747.89金星0.8150.72335.03地球1129.79火星0.1071.52424.13木星317.85.20313.06土星95.169.5379.64天王星14.5419.196.81海王星17.1430.075.43冥王星
除了水星偏心率为0.2,对黄道面倾斜为7°之外,其余行星的偏心率皆小于0.1,且对黄道面倾斜普遍小于4°。由于水星的轨道太小,偏不偏心其实都不太看得出来,所以就当它是正圆也无所谓了,最后得图
au,G,RE,ME = 1.48e11,6.67e-11,1.48e11,5.965e24
m = np.array([3.32e5,0.055,0.815,1,
0.107,317.8,95.16,14.54,17.14])*ME*6.67e-11
r = np.array([0,0.387,0.723,1,1.524,5.203,
9.537,19.19,30.7])*RE
theta = np.random.rand(9)*np.pi*2
x = r*np.cos(theta)
y = r*np.sin(theta)
v = np.array([0,47.89,35.03,29.79,
24.13,13.06,9.64,6.81,5.43])*1000
u = -v*np.sin(theta)
v = v*np.cos(theta)
name = "solar.gif"
fig = plt.figure(figsize=(10,10))
ax = fig.add_subplot(xlim=(-31*RE,31*RE),ylim=(-31*RE,31*RE))
ax.grid()
traces = [ax.plot([],[],'-', lw=0.5)[0] for _ in range(9)]
pts = [ax.plot([],[],marker='o')[0] for _ in range(9)]
k_text = ax.text(0.05,0.85,'',transform=ax.transAxes)
textTemplate = 't = %.3f days\n'
N = 500
dt = 3600*50
ts = np.arange(0,N*dt,dt)
xs,ys = [],[]
for _ in ts:
x_ij = (x-x.reshape(len(m),1))
y_ij = (y-y.reshape(len(m),1))
r_ij = np.sqrt(x_ij**2+y_ij**2)
for i in range(len(m)):
for j in range(len(m)):
if i!=j :
u[i] += (m[j]*x_ij[i,j]*dt/r_ij[i,j]**3)
v[i] += (m[j]*y_ij[i,j]*dt/r_ij[i,j]**3)
x += u*dt
y += v*dt
xs.append(x.tolist())
ys.append(y.tolist())
xs = np.array(xs)
ys = np.array(ys)
def animate(n):
for i in range(9):
traces[i].set_data(xs[:n,i],ys[:n,i])
pts[i].set_data(xs[n,i],ys[n,i])
k_text.set_text(textTemplate % (ts[n]/3600/24))
return traces+pts+[k_text]
ani = animation.FuncAnimation(fig, animate,
range(N), interval=10, blit=True)
plt.show()
ani.save(name)
由于外圈的行星轨道又长速度又慢,而内层的刚好相反,所以这个图很难兼顾,观感上也不太好看。
如果只画出木星之前的星体,顺便加上小行星带,可能会好一些。
通过这个图就能看出来,有一颗小行星被木星弹了过来,直冲冲地向地球赶来,幸好又被太阳弹了出去,可见小行星还是挺危险的,好在这只是个假想图。
Original: https://blog.csdn.net/m0_37816922/article/details/120699335
Author: 微小冷
Title: 用Python搓一个太阳系
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