# 歲差

（重定向自分點歲差

## 歷史

### 近代

${\displaystyle p_{A}=5\,028.796\,195T+1.105\,434\,8T^{2}+O(T^{3})}$

${\displaystyle p=5\,028.796\,195+2.210\,869\,6T+O(T^{2})}$

## 成因

### 方程式

${\displaystyle {\overrightarrow {T}}={\frac {3Gm}{r^{3}}}(C-A)\sin \delta \cos \delta {\begin{pmatrix}\sin \alpha \\-\cos \alpha \\0\end{pmatrix}}}$

Gm = 攝動天體的標準引力參數
r = 攝動天體的地心距離
C = 圍繞地球自轉軸轉動的轉動慣量
A = 任何環繞地球赤道直徑的轉動慣量
C−A = 地球赤道隆起的轉動慣量(C>A)
δ = 攝動天體的赤緯 (赤道以南或北)
α = 攝動天體的赤經 (從春分點向東)

${\displaystyle T_{x}={\frac {3}{2}}{\frac {Gm}{a^{3}(1-e^{2})^{3/2}}}(C-A)\sin \epsilon \cos \epsilon }$

${\displaystyle a}$  = 地球(太陽)或月球的軌道半長軸
e =地球(太陽)或月球的軌道離心率

${\displaystyle {\frac {d\psi }{dt}}={\frac {T_{x}}{C\omega \sin \epsilon }}}$

${\displaystyle {\frac {d\psi _{S}}{dt}}={\frac {3}{2}}\left[{\frac {Gm}{a^{3}(1-e^{2})^{3/2}}}\right]_{S}\left[{\frac {(C-A)}{C}}{\frac {\cos \epsilon }{\omega }}\right]_{E}}$

${\displaystyle {\frac {d\psi _{L}}{dt}}={\frac {3}{2}}\left[{\frac {Gm(1-1.5\sin ^{2}i)}{a^{3}(1-e^{2})^{3/2}}}\right]_{L}\left[{\frac {(C-A)}{C}}{\frac {\cos \epsilon }{\omega }}\right]_{E}}$

${\displaystyle e''^{2}={\frac {\mathrm {a} ^{2}-\mathrm {c} ^{2}}{\mathrm {a} ^{2}+\mathrm {c} ^{2}}}}$

Gm=1.3271244×1020 m³/s² Gm=4.902799×1012 m³/s² (CA)/C=0.003273763
${\displaystyle a}$ =1.4959802×1011 m ${\displaystyle a}$ =3.833978×108 m ω=7.292115×10−5 rad/s
e=0.016708634 e=0.05554553 ${\displaystyle \epsilon \,\!}$ =23.43928°
i= 5.156690°

S/dt = 2.450183×10−12 /s
L/dt = 5.334529×10−12 /s

S/dt = 15.948788"/a   vs   15.948870"/a 取自威爾斯[24]
L/dt = 34.723638"/a   vs   34.457698"/a 取自威爾斯

## 數值

p = A + BT + CT² + … 只是下面公式的近似 p = A + Bsin (2πT/P)，此處P 是410個世紀的週期。

### 黄经总岁差与交角岁差

#### 岁差常数

${\displaystyle p=5\,029.0\,965''/{\text{c}}\pm 0.3''/{\text{c}}}$

${\displaystyle \epsilon _{0}=84\,381.412''\pm 0.005''}$

${\displaystyle p=5\,038.478\,75''/{\text{c}}\pm 0.000\,40''/{\text{c}}}$

${\displaystyle \epsilon _{0}=84\,381.405\,9''\pm 0.000\,3''}$

#### 数值计算

${\displaystyle p_{A}=5\,029.096\,6''t+1.111\,61t^{2}-0.000\,113t^{3}}$

${\displaystyle \epsilon _{A}=\epsilon _{0}-46.815\,0''t-0.000\,59t^{2}+0.001\,813t^{3}}$

## 改正

${\displaystyle \mathbf {r} (t_{0})=\left[\mathbf {P} (t_{i},t_{0})\right]\mathbf {r} (t_{i})}$

### 三次坐标旋转法

MHB2000模型使用了三个欧拉角来表示转换前后的坐标系的相对位置，并将岁差矩阵表达为三个旋转矩阵的乘积：

${\displaystyle \left[\mathbf {P} (t_{i},t_{0})\right]=\mathbf {R_{Z}} (\eta _{A})\mathbf {R_{Y}} (-\theta _{A})\mathbf {R_{Z}} (\zeta _{A})}$

1. 将瞬时天球坐标系的X轴从瞬时平春分点移开，绕瞬时平天球坐标系的Z轴（即沿瞬时赤道面）逆时针旋转${\displaystyle \zeta _{A}}$ 角，得到第一过渡坐标系；
2. 将第一过渡坐标系的X轴绕其Y轴（即沿子午圈）顺时针旋转${\displaystyle \theta _{A}}$ 角，得到第二过渡坐标系，此时第二过渡坐标系的Z轴和赤道面与协议天球坐标系的重合；
3. 将第二过渡坐标系的X轴绕协议天球坐标系的Z轴（即沿协议天球坐标系的赤道面）逆时针旋转${\displaystyle \eta _{A}}$ 角，使其与协议天球坐标系的春分点重合，此时整个坐标系与协议天球坐标系重合。

### 四次坐标旋转法

${\displaystyle \left[\mathbf {P} (t_{i},t_{0})\right]=\mathbf {R_{X}} (-\epsilon _{0})\mathbf {R_{Z}} (\psi _{A})\mathbf {R_{X}} (\omega _{A})\mathbf {R_{Z}} (-\chi _{A})}$

1. 将瞬时天球坐标系的X轴从瞬时平春分点移开，绕瞬时平天球坐标系的Z轴（即沿瞬时赤道面）顺时针旋转${\displaystyle \chi _{A}}$ 角，得到第一过渡坐标系；
2. 保持第一过渡坐标系的X轴不变，将其赤道面绕其X轴逆时针旋转${\displaystyle \omega _{A}}$ 角，得到第二过渡坐标系，此时第二过渡坐标系的赤道面的协议天球坐标系的黄道面重合；
3. 将第二过渡坐标系的X轴绕其Z轴（即沿协议天球坐标系的黄道面）逆时针旋转${\displaystyle \psi _{A}}$ 角，得到第三过渡坐标系，此时第三过渡坐标系的X轴与协议天球坐标系的春分点重合；
4. 保持第三过渡坐标系的X轴不变，将其赤道面绕其X轴顺时针旋转${\displaystyle \epsilon _{0}}$ 角，此时整个坐标系与协议天球坐标系重合。

• ${\displaystyle \chi _{A}=10.556\,403\,00''t-2.381\,429\,200''t^{2}-0.001\,211\,970\,0''t^{3}+0.000\,170\,663\,00''t^{4}-0.000\,000\,056\,000''t^{5}}$
• ${\displaystyle \omega _{A}=\epsilon _{0}-0.025\,754\,00''t+0.051\,262\,300''t^{2}-0.007\,725\,030\,0t^{3}-0.000\,000\,467\,00t^{4}+0.000\,000\,333\,700t^{5}}$
• ${\displaystyle \psi _{A}=5\,038.481\,507\,00t-1.079\,006\,900t^{2}-0.001\,140\,45t^{3}+0.000\,132\,851\,00t^{4}-0.000\,000\,095\,100t^{5}}$
• ${\displaystyle \epsilon _{0}=84\,381.406''}$

## 参考資料

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