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标题: 太阳高度角的计算 | 任何时间 | 任何地点 [打印本页]
作者: xgm518 时间: 2024-9-27 16:30
标题: 太阳高度角的计算 | 任何时间 | 任何地点
太阳赤纬角https://zhuanlan.zhihu.com/p/630216177
故名思义，就是太阳直射在赤道上的角度，如下图所示：
当春分的时候，太阳直射赤道，赤纬角为0度太阳赤纬角的简化计算公式
<span class="MathJax_SVG" id="MathJax-Element-1-Frame" tabindex="0" data-mathml="(1)δ=23.45sin(2π⋅284+n365)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">(1)�=23.45sin(2�⋅284+�365)
式中 <span class="MathJax_SVG" id="MathJax-Element-2-Frame" tabindex="0" data-mathml="n" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">� 是计算日期在一年中的日子数，例如 <span class="MathJax_SVG" id="MathJax-Element-3-Frame" tabindex="0" data-mathml="n=1" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">�=1 就对应着1月1日
如果你需要更精确的赤纬角计算公式，可以使用由Spencer在1971年提出的公式[1]
<span class="MathJax_SVG" id="MathJax-Element-4-Frame" tabindex="0" data-mathml="(1.1)δ=0.006918−0.399912cos(Γ)+0.070257sin(Γ)−0.006758cos(2Γ)+0.000907sin(2Γ)−0.002697cos(3Γ)+0.00148sin(3Γ)(rad)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; text-align: left; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">(1.1)�=0.006918−0.399912cos(Γ)+0.070257sin(Γ)−0.006758cos(2Γ)+0.000907sin(2Γ)−0.002697cos(3Γ)+0.00148sin(3Γ)(���)式中 <span class="MathJax_SVG" id="MathJax-Element-6-Frame" tabindex="0" data-mathml="Γ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">Γ 被称作日角，由 <span class="MathJax_SVG" id="MathJax-Element-5-Frame" tabindex="0" data-mathml="n" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">� 来确定
<span class="MathJax_SVG" id="MathJax-Element-7-Frame" tabindex="0" data-mathml="(1.2)Γ=2π(n−1)365(rad)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; text-align: left; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">(1.2)Γ=2�(�−1)365(���)
正午太阳高度角对于纬度为 <span class="MathJax_SVG" id="MathJax-Element-9-Frame" tabindex="0" data-mathml="θ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">� 的地区，正午太阳高度角 <span class="MathJax_SVG" id="MathJax-Element-8-Frame" tabindex="0" data-mathml="hm" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">ℎ� 和太阳赤纬角的关系如下图所示
用公式表示：
<span class="MathJax_SVG" id="MathJax-Element-10-Frame" tabindex="0" data-mathml="(2)hm=90−(θ−δ)(o)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; text-align: left; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">(2)ℎ�=90−(�−�)(�)
太阳日照时长计算计算太阳日照时长就是计算昼弧与夜弧的占比，如下图
推导过程这里就略去了，直接放结果（感兴趣的同志可以参考Cooper[2]的计算） 对于纬度为 <span class="MathJax_SVG" id="MathJax-Element-11-Frame" tabindex="0" data-mathml="θ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">� 的地区，日照时长为：
<span class="MathJax_SVG" id="MathJax-Element-12-Frame" tabindex="0" data-mathml="(3)Ts=215arccos(−tan(δ)⋅tan(θ))" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; text-align: left; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">(3)��=215arccos(−tan(�)⋅tan(�))
使用这个结果可以用来计算日出和日落的时间，以中国为例：
我国使用的是北京时间（GMT+8），北京市的经度为 <span class="MathJax_SVG" id="MathJax-Element-13-Frame" tabindex="0" data-mathml="116.40o" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">116.40�
假定我所在的城市经度为 <span class="MathJax_SVG" id="MathJax-Element-14-Frame" tabindex="0" data-mathml="γ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">� ，我所在的城市太阳达到正午高度角的时间是：
<span class="MathJax_SVG" id="MathJax-Element-16-Frame" tabindex="0" data-mathml="(3.1)Tc=12:00+γ−120360⋅1440(min)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; text-align: left; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">(3.1)��=12:00+�−120360⋅1440(���)例如天津（经度： <span class="MathJax_SVG" id="MathJax-Element-15-Frame" tabindex="0" data-mathml="117.2o" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">117.2� ）的达到正午太阳高度角的时间是： <span class="MathJax_SVG" id="MathJax-Element-17-Frame" tabindex="0" data-mathml="12:04" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">12:04
那么根据日出到正午和日落到正午的时间相同，可以计算出日出时间和日落时间为：
日出：<span class="MathJax_SVG" id="MathJax-Element-18-Frame" tabindex="0" data-mathml="(3.2)Tup=Tc−Ts2" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; text-align: left; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">(3.2)���=��−��2日落：<span class="MathJax_SVG" id="MathJax-Element-19-Frame" tabindex="0" data-mathml="(3.3)Tdown=Tc+Ts2" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; text-align: left; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">(3.3)�����=��+��2例如天津在2023年5月20日的日出时间为：4：56；日落时间为19：12
（与真实值的误差仅有20秒）
太阳视运动轨迹计算在天球上，太阳的视运行轨迹如下图所示：
可以看到，对于不同的日期，太阳的运行轨迹发生了平移。根据我们刚刚计算的太阳正午高度角，可以推算出太阳轨迹与日期的函数关系。
Tip：接下来可能会涉及到一些繁琐的数学运算，不感兴趣的同志可以直接跳转到结果部分
日轨部分如红色弧线所示，日轨是在天球上的一个圆环，圆环的圆心 <span class="MathJax_SVG" id="MathJax-Element-20-Frame" tabindex="0" data-mathml="P" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">� 的坐标如下：
<span class="MathJax_SVG" id="MathJax-Element-22-Frame" tabindex="0" data-mathml="(3)P=(0,dcos(θ),−dsin(θ))" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; text-align: left; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">(3)�=(0,�cos(�),−�sin(�))假定太阳的运行角度为 <span class="MathJax_SVG" id="MathJax-Element-21-Frame" tabindex="0" data-mathml="φ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">� ，太阳相对于日轨中心的位矢为<span class="MathJax_SVG" id="MathJax-Element-23-Frame" tabindex="0" data-mathml="(4)Δr=Rsin(β)sin(φ)ex+Rsin(β)cos(φ)sin(θ)ey+Rsin(β)cos(φ)cos(θ)ez" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; text-align: left; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">(4)Δ�=�sin(�)sin(�)��+�sin(�)cos(�)sin(�)��+�sin(�)cos(�)cos(�)��二者相加，得到太阳的坐标为：<span class="MathJax_SVG" id="MathJax-Element-24-Frame" tabindex="0" data-mathml="(5){x=Rsin(β)sin(φ)y=Rsin(β)cos(φ)sin(θ)+dcos(θ)z=Rsin(β)cos(φ)cos(θ)−dsin(θ)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; text-align: left; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">(5){�=�sin(�)sin(�)�=�sin(�)cos(�)sin(�)+�cos(�)�=�sin(�)cos(�)cos(�)−�sin(�)根据图中几何关系， <span class="MathJax_SVG" id="MathJax-Element-25-Frame" tabindex="0" data-mathml="β=h+θ=90+δ" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">�=ℎ+�=90+� ， <span class="MathJax_SVG" id="MathJax-Element-26-Frame" tabindex="0" data-mathml="d=Rcos(β)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">�=�cos(�)
带入后即可得出对应纬度的太阳运动轨迹，编写如下Python程序做一个简单的可视化吧：
import numpy as npimport matplotlib.pyplot as pltfrom mpl_toolkits.mplot3d import Axes3D# 定义参数R = 1 # 半径beta = 70theta = 40beta = beta * np.pi / 180theta = theta * np.pi / 180d = R * np.cos(beta) # 计算d的值# 创建φ值phi = np.linspace(0, 2 * np.pi, 100)# 计算x, y, zx = R * np.sin(beta) * np.sin(phi)y = R * np.sin(beta) * np.cos(phi) * np.sin(theta) + d * np.cos(theta)z = R * np.sin(beta) * np.cos(phi) * np.cos(theta) - d * np.sin(theta)# 创建3D图形fig = plt.figure()ax = fig.add_subplot(111, projection='3d')# 绘制原点ax.scatter([0], [0], [0], color="g", label="Origin")# 绘制基础平面circle_x = R * np.cos(phi)circle_y = R * np.sin(phi)ax.plot(circle_x, circle_y, 0, color='lightblue')# 绘制轨迹，颜色根据z的值确定for i in range(len(z)): if z > 0: ax.plot(x[i:i + 2], y[i:i + 2], z[i:i + 2], color='red') else: ax.plot(x[i:i + 2], y[i:i + 2], z[i:i + 2], color='blue')# 显示图例red_patch = plt.plot([], [], marker="o", ms=10, ls="", mec=None, color='red', label="Day")[0]blue_patch = plt.plot([], [], marker="o", ms=10, ls="", mec=None, color='blue', label="Night")[0]plt.legend(handles=[red_patch, blue_patch], loc='upper right')# 显示图形plt.show()
天津，5月20日的太阳轨迹至此，我们就完成了太阳视运动轨迹的计算，有了这个轨迹我们可以很容易的算出太阳的两个方位角。太阳的方位角计算对于在北半球的我们，当太阳达到最大高度角的时候，太阳在我们的南方（也就是 <span class="MathJax_SVG" id="MathJax-Element-27-Frame" tabindex="0" data-mathml="y+" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">�+ 方向）
我们将太阳的两个方位角分别记为 <span class="MathJax_SVG" id="MathJax-Element-28-Frame" tabindex="0" data-mathml="(Θ,Φ)" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">(Θ,�) ，如下图所示：
容易有：
<span class="MathJax_SVG" id="MathJax-Element-29-Frame" tabindex="0" data-mathml="(6)sin(Θ)=x2+y2R,sin(Φ)=yx2+y2" role="presentation" style="display: inline-block; line-height: normal; font-size: 16px; text-align: left; word-spacing: normal; overflow-wrap: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;">(6)sin(Θ)=�2+�2�,sin(�)=��2+�2
北京市5月20日的太阳轨迹回到开头的问题，我们已经成功的参数化表达了太阳的运动轨迹，我们可以回答这个问题了：
北京，5月20日的太阳轨迹
使用Excel做了一个计算表格：
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