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通过测量到达角起伏可以计算得出大气相干长度,为了消除仪器观测中的抖动影响,差分像运动法被提出。像运动的实质是到达角起伏,也即波前倾斜项的起伏,将望远镜上两个子孔上的到达角做差分,可以将望远镜的抖动去掉。显然,差分到达角的方差为两孔径上的到达角方差之和(纯湍流引起的到达角起伏,已经扣除仪器抖动)减去两孔径上到达角的相关函数,物理意义十分明确。Sarazin & Roddier在研究DIMM原理时给出的计算公式为[16]:
$$\sigma _{{l}}^2 = 2{\lambda ^2}{r_{0{{l}}}}^{ - 5/3}(0.179{D^{ - 1/3}} - 0.096\;8{d^{ - 1/3}})$$ (1a) $$\sigma _{{t}}^2 = 2{\lambda ^2}{r_{0{{t}}}}^{ - 5/3}(0.179{D^{ - 1/3}} - 0.145{d^{ - 1/3}})$$ (1b) 式中:σl2、σt2为两子瞳波前差分到达角的纵向、横向方差(纵向是指两孔中心连线方向,横向为其垂直方向);D为子瞳直径;d为子瞳间距;λ为计算r0的取值波长,r0l对应于纵向方差算出的r0,r0t对应于横向方差算出的r0。在大气湍流局地均匀各向同性假设下,r0l、r0t统计上应该相一致,可以理解为变化趋势一致,数值大小相差不大;单次r0一般取两者的均值。文中将从实验中r0l、r0t统计上的趋势变化来验证计算公式的正确性。
上述Sarazin & Roddier计算公式在文中也称传统计算公式。对该公式进行推导分析可知[15-16],公式中系数0.179应该为0.182,公式前一项0.364λ2D−1/3r0−5/3为单孔上的Z-tilt到达角起伏方差(若分解为纵向和横向两个方向的到达角起伏方差,则单个方向上的到达角起伏方差为总方差的一半0.182λ2D−1/3r0−5/3),也即两孔径上的某一方向到达角方差之和。公式后一项为减去的两子孔中心点的到达角起伏相关函数,对于到达角的纵向与横向,其值分别为2λ2r0l−5/3×0.0968d −1/3、2λ2r0t−5/3×0.145d −1/3。应该注意,这里用两点的到达角起伏相关来近似成两孔径上的到达角起伏相关。该近似只有在子孔间距要求d≥2D下才成立,可以预见,由于孔径效应,d/D越小,这种近似程度越低。Sasiela[17]利用横向谱滤波及梅林变换技术研究倾斜非等晕性时给出了更准确的Z-tilt差分到达角的纵向、横向方差公式,为叙述方便,文中也称新计算公式:
$$\left[ \begin{array}{l} \sigma _l^2 \\ \sigma _t^2 \\ \end{array} \right] = \dfrac{{0.364{\lambda ^2}{D^{ - 1/3}}}}{{{r_0}^{5/3}}}\left\{ {\left[ \begin{array}{l} 1 \\ 1 \\ \end{array} \right] - \left[ \begin{array}{l} 0.531{\left( {\dfrac{D}{d}} \right)^{1/3}} \\ 0.799{\left( {\dfrac{D}{d}} \right)^{1/3}} \\ \end{array} \right.} \right.\left. {\left. \begin{array}{l} {}_4{F_3}\left. {\left[ { - \dfrac{5}{6},\dfrac{5}{2},\dfrac{1}{6},\dfrac{2}{3};} \right.5,3, - \dfrac{1}{3};{{\left( {\dfrac{D}{d}} \right)}^2}} \right] \\ {}_{\rm{3}}{F_2}\left[ { - \dfrac{5}{6},\dfrac{5}{2},\dfrac{1}{6};} \right.5,3;\left. {{{\left( {\dfrac{D}{d}} \right)}^2}} \right] \\ \end{array} \right]} \right\}\;\;d > D{\rm{ }}$$ (2) 式中:qFp为广义超几何函数,可以通过数值软件计算得出。为了方便比较传统计算公式(其中系数0.179已校正为0.182)与新计算公式的差异,可以将两式改写为规格化模式:
$$\left[ \begin{array}{l} \sigma _l^2 \\ \sigma _t^2 \\ \end{array} \right] = 0.364{\lambda ^2}\frac{{{D^{ - 1/3}}}}{{{r_0}^{5/3}}}\left[ \begin{array}{l} {K_l} \\ {K_t} \\ \end{array} \right]{\rm{ }}$$ (3) 如前所述,0.364λ2D−1/3r0−5/3是单孔上的到达角起伏方差,系数Kl、Kt称为归一化差分系数,为1减去两孔径上到达角起伏相关系数,分别代表两孔径上到达角纵向或横向差分后残余的起伏,物理意义明确。分别将传统计算公式(1)、(2)和新计算公式(3)对应的归一化差分系数设为KlO、KtO和KlN、KtN,计算出它们随d/D的变化关系如图1所示。从图中可以看出,随着d/D的增大,两种公式对应的系数越来越接近;当d/D≥2时,可以用传统计算公式近似Sasiela公式,但当d/D<2时,随着d/D的进一步减小,两公式系数差异越来越明显。
图 1 归一化差分系数的对比。虚线对应于传统计算公式的KlO、KtO,实线对应于新计算公式的KlN、KtN
Figure 1. Comparison of the normalized differential coefficients. Dashed line: KlO and KtO from the traditional formulae, solid line: KlN and KtN from the new formulae
显然,减小子孔直径D与间距d的值可以使DIMM小型化。对于整层大气观测,D大于3 cm就可以满足近场条件[15],由于观测信噪比要求,D取6 cm左右。采用新计算公式,此时子孔间距d最小可取为D,即两子孔紧挨。那么,口径12 cm左右的小望远镜就可以用来作为DIMM的光学接收主体,这种小型DIMM将会比传统DIMM (口径一般在30 cm左右)缩小近2倍,从而更轻量便携。
Analysis of formulae in DIMM and the verified experiment
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摘要: 差分像运动监测仪(DIMM)是目前被广泛应用的大气相干长度测量仪,其传统计算公式中对接收子孔的要求限制了仪器的小型化。通过理论分析与数值计算,文中明确了计算公式中的物理意义,并提出采用新的计算公式能够降低对仪器几何结构的限制。用两个紧挨着的6 cm子孔替换原来标准DIMM中的子孔面罩,得到小型化DIMM测试原型,利用该小型化DIMM原型与一台标准DIMM放在一起同时观测相同的恒星开展大气相干长度测量对比实验。对于小型DIMM,实验结果显示,采用传统计算公式得到的纵向大气相干长度明显大于横向,而采用新计算公式得到的纵向与横向大气相干长度统计上更相一致。验证了文中关于DIMM中计算公式的论述。Abstract: The Differential Image Motion Monitor (DIMM) is a widely used instrument for measuring atmospheric coherence length. The miniaturization of the instrument is limited by the requirement of receiving sub-aperture in the traditional formulae. Through theoretical analysis and numerical calculation, the physical meaning of formulae was clarified, and the new calculation formulae were proposed to reduce the limitation on the geometrical structure of the instrument. A miniaturized DIMM test prototype was obtained by replacing the mask in the original standard DIMM with two adjacent 6 cm sub-aperture mask. The miniaturized DIMM prototype was used together with a standard DIMM to simultaneously observe the same star for the comparison experiment of atmospheric coherence length measurement. For miniaturized DIMM, the experimental results show that the longitudinal atmospheric coherence length obtained by the traditional formulae is significantly larger than the transverse, and the longitudinal and transverse atmospheric coherence length obtained by the new formulae is more statistically consistent. Thus, the discussion on the calculation formula in DIMM is verified.
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图 3 纵向与横向大气相干长度对比。 (a) 小型DIMM中利用传统公式计算得出的r0l、r0t;(b) 小型DIMM中利用新公式计算得出的r0l、r0t。标准DIMM中得出的r0l、r0t整体减去4显示在图中(底部的线)以便比较
Figure 3. Comparison of the longitudinal and transverse atmospheric coherence length. (a) r0l、r0t calculated by the traditional formulae from miniDIMM; (b) r0l、r0t calculated by the new formulae from miniDIMM. r0l、r0t from standard DIMM minus 4 are also shown in the figure (the bottom lines) for comparison
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