小行星天基光学监测信噪比分析

小行星天基光学监测信噪比分析 Analysis of signal-to-noise ratio for space-based optical surveillance of asteroids 1 2 first-author 王新涛（1995-），男，山东青岛人，博士研究生，2017年于北方工业大学获得学士学位，主要研究方向为小行星监测预警、轨道确定和卫星轨道动力学。E-mail：wangxintao17@mails.uca s.ac.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23628929&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23628936&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23628932&type= 22.01332855 32.00401306 王新涛 （1995-），男，山东青岛人，博士研究生，2017年于北方工业大学获得学士学位，主要研究方向为小行星监测预警、轨道确定和卫星轨道动力学。E-mail： wangxintao17@mails.uca s.ac.cn wangxintao17@mails.uca 1 2 1 2 corresp E-mail： limingtao@nssc.ac.cn E-mail： limingtao@nssc.ac.cn 李明涛（1982-），男，河南商丘人，研究员，博士生导师，2005年于西北工业大学获得学士学位，2010年于中国科学院研究生院获得博士学位，主要从事航天动力学与控制、行星防御与利用方面的研究。E-mail：limingtao@nssc.ac.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23628942&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23628951&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23628945&type= 22.01333618 32.00382996 李明涛 （1982-），男，河南商丘人，研究员，博士生导师，2005年于西北工业大学获得学士学位，2010年于中国科学院研究生院获得博士学位，主要从事航天动力学与控制、行星防御与利用方面的研究。E-mail： limingtao@nssc.ac.cn limingtao@nssc.ac.cn 中国科学院 国家空间科学中心，北京 101499 中国科学院 国家空间科学中心，北京 101499 National Space Science Center， Chinese Academy of Sciences， Beijing 101499， China National Space Science Center， Chinese Academy of Sciences， Beijing 101499， China 中国科学院大学，北京 100049 中国科学院大学，北京 100049 University of Chinese Academy of Sciences， Beijing 100049， China University of Chinese Academy of Sciences， Beijing 100049， China 针对小行星天基光学监测中的可见性问题，基于辐射度传输理论，引入黄道光和银道光亮度模型，建立了小行星天基光学监测的信噪比模型。结合设计的可见光传感器和望远镜的性能参数，计算了目标小行星相对望远镜的信噪比和视星等随时间的变化；考虑极限信噪比和太阳规避角，计算了地球领航轨道可见光望远镜对不同直径小行星的极限监测距离；改变小行星的物理参数，研究了信噪比和视星等对小行星物理参数的敏感性。结果表明：目标小行星在给定的场景时间内相对望远镜距离增大了约0.09AU，视星等增大了约0.51，目标小行星在口径为0.7 m的望远镜中信噪比减小了约3.78；由于观测天区覆盖地球周围0.05AU的空间，在地球领航轨道部署可见光望远镜预警来自太阳方向的小行星具有明显的优势，能为直径20 m的小行星提供750万公里的预警距离；在可见光波段，小行星的信噪比和视星等对反照率敏感，对温度和发射率不敏感。所建立的小行星天基光学监测信噪比模型可为小行星天基光学监测的信噪比动态分析和预警距离评估提供理论依据。 To investigate asteroid visibility in space-based optical surveillance using the radiative transfer theory， a brightness model of zodiacal and galactic light was introduced， and a signal-to-noise ratio （SNR） model for space-based asteroid surveillance was established. Combining the designed visible sensor with telescope performance parameters， the SNR and apparent magnitude of the target asteroid relative to the telescope over time were simulated. By considering the minimum SNR and solar avoidance angle， the maximum surveillance distance of a visible telescope deployed on an Earth-leading heliocentric orbit for asteroids of different diameters was calculated. The sensitivity of the SNR and apparent magnitude to the physical parameters of the asteroids was studied by changing the physical parameters of the asteroids. The results show that the distance of the asteroid relative to the telescope increased by approximately 0.09 AU， the apparent magnitude of the asteroid increased by approximately 0.51， and the SNR of the asteroid relative to a telescope with an aperture of 0.7 m decreased by approximately 3.78 during the period under observation. Owing to good coverage of the vicinity around the Earth with a radius of 0.05 AU， there is a unique advantage for visible telescopes deployed on Earth-leading heliocentric orbits to warn of asteroids approaching from the sunward direction； they can provide an early warning distance of 7.5 million km for asteroids with a diameter of 20 m. In the visible light band， the SNR and apparent magnitude of asteroids are sensitive to the visual albedo but not to the temperature and visual emissivity. The model established in this study provides a theoretical basis for the dynamic analysis of the SNR and the assessment of early warning distances for space-based optical surveillance of asteroids. 空间光学 小行星天基监测 天空背景 可见光传感器 信噪比模型 视星等 space optics space-based asteroid surveillance zodiacal and galactic light visible sensor signal-to-noise ratio model apparent magnitude 1 引　言 1 小行星轨道与地球轨道存在交叉，存在与地球碰撞的可能，地球历史上曾多次遭到小行星撞击，因此对小行星进行监测具有重要意义。小行星监测借助地基和天基平台，可以采用雷达技术（测距、测速和测角）、光电监测技术（光电测角和激光测距）等多种监测手段 ［ 1 1 ］ 。地基平台由于不受体积和质量等因素限制，可采用大发射功率获得很远的监测距离、采用大口径天线得到很高的空间分辨率，是目前小行星监测的主要手段。但小行星地基监测存在两个不利因素：一是监测站的有效覆盖范围无法达到对空域和时域的无缝覆盖，建立更多的监测站受到政治和地理等因素的制约；二是在现有的监测手段中，地基雷达的监测距离有限，地基光电监测易受天气影响，一般只能在晴夜工作，不能达到全天候和全天时的要求 ［ 2 2 ］ 。天基平台位于地球大气层外的空间轨道，具有布置灵活、不受大气和天气影响、监测范围大等优点，大大增强了对小行星的监测能力。此外，雷达技术的监测能力与距离的四次方成反比；光电监测技术的监测能力与距离的平方成反比 ［ 3 3 ］ ，小行星一般距离较远，因此光电监测技术更有利于小行星的监测，并受到越来越多的关注。 信噪比是决定目标能否被成功识别的一个重要因素，也是衡量辐射性能和探测器质量的重要指标。对于空间目标的信噪比分析，学者们已经做了较多的研究工作。金伟其、郝允祥等详细概述了空间目标监测信号的计算方法 ［ 4 4 - 5 5 4-5 ］ 。Joshua、王书宏、胡琸悦等在不考虑天空背景噪声的条件下，结合探测器噪声得出了小行星天基监测信噪比的一般表达式 ［ 6 6 - 8 8 6-8 ］ 。孙成明、张科科等假设天空背景亮度固定建立了空间目标监测信噪比的数学模型 ［ 9 9 - 10 10 9-10 ］ 。李克新、夏盛夫等结合恒星星表分别生成了红外和可见光波段的空间目标天基监测图像 ［ 11 11 - 12 12 11-12 ］ 。杨帆、万志、陈新锦等利用大气辐射传输软件MODTRAN得到随波长变化的天空背景辐射亮度，并分析了空间目标红外地基监测以及可见光遥感相机对地面目标监测的信噪比 ［ 13 13 - 15 15 13-15 ］ 。然而，基于固定天空背景亮度和基于MOTRAN软件得到的天空背景亮度不能反应天空背景亮度随太阳-小行星-望远镜相位的变化关系，致使小行星天基信噪比的动态分析与真实情况存在出入。 小行星天基监测的天空背景噪声主要为黄道光和银道光亮度 ［ 16 16 - 17 17 16-17 ］ 。本文基于辐射度传输理论，引入随方位和波长变化的黄道光和银道光亮度模型建立了天空背景亮度模型。首先推导了小行星天基监测的信噪比表达式，结合设计的可见光传感器和望远镜参数，计算了目标小行星相对望远镜的信噪比和视星等随时间的变化；然后考虑极限信噪比和太阳规避角，计算了地球领航轨道（地球前方1 058万公里）可见光望远镜对不同直径小行星的极限监测距离，论证了在地球领航轨道部署可见光望远镜预警来自太阳方向小行星的可行性 ［ 18 18 ］ ；最后改变小行星的物理参数，研究了信噪比和视星等对小行星物理参数的敏感性，验证了方法的可靠性。本文建立了考虑黄道光和银道光的小行星天基监测信噪比模型，为小行星天基监测的信噪比动态分析和预警距离评估提供了理论依据，同时可为天基光学载荷的方案设计及效能评估提供技术支撑。 2 信噪比模型 1 小行星在望远镜处产生的能量来源于反射的太阳辐射以及自身辐射，结合目标小行星的物理特性、相对太阳和望远镜的位置以及探测器和望远镜的性能参数，计算小行星的监测信号；结合天空背景亮度和探测器噪声，确定小行星天基监测的信噪比；以太阳为参考星，确定小行星相对望远镜的视星等。 2.1　信　号 2 2 . 1 . 1　反射太阳辐射 3 普朗克定律描述了黑体处于热力学温度 <math id="M1"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23628959&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23628956&type= 2.53999996 2.79399991 时，在波长为 <math id="M2"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630392&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630389&type= 1.77800000 2.79399991 处的单色辐射出射度，具体公式为： <math display="block" id="M3"><msub><mrow><mi>M</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi>h</mi><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msup></mrow></mfrac><mo>×</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>h</mi><mi>c</mi><mo>/</mo><mo stretchy="false">(</mo><msub><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub><mi>T</mi><mi>λ</mi><mo stretchy="false">)</mo></mrow></msup><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23628976&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23628972&type= 51.22333527 8.97466660 ， 其中： <math id="M4"><msub><mrow><mi>M</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23628984&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23628980&type= 14.81666565 4.48733330 为光谱辐射出射度； <math id="M5"><mi>h</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23628991&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23628987&type= 1.94733346 2.79399991 为普朗克常数， <math id="M6"><mi>h</mi><mo>=</mo><mn mathvariant="normal">6.626</mn><mtext> </mtext><mn mathvariant="normal">068</mn><mtext> </mtext><mn mathvariant="normal">96</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">34</mn></mrow></msup><mtext> </mtext><mi mathvariant="normal">J</mi><mo>⋅</mo><mi mathvariant="normal">s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629000&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23628995&type= 44.45000076 3.13266683 ； <math id="M7"><mi>c</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629008&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629004&type= 1.52400005 2.79399991 为真空中的光速， <math id="M8"><mi>c</mi><mo>=</mo><mn mathvariant="normal">2.997</mn><mtext> </mtext><mn mathvariant="normal">924</mn><mtext> </mtext><mn mathvariant="normal">58</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">8</mn></mrow></msup><mrow><mrow><mtext> </mtext><mi mathvariant="normal">m</mi></mrow><mo>/</mo><mrow><mi mathvariant="normal">s</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629018&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629012&type= 42.07933044 3.55599999 ； <math id="M9"><msub><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629022&type= 3.13266683 3.72533321 为玻尔兹曼常数， <math id="M10"><msub><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1.380</mn><mtext> </mtext><mn mathvariant="normal">648</mn><mtext> </mtext><mn mathvariant="normal">52</mn><mo>×</mo><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">23</mn></mrow></msup><mtext> </mtext><mrow><mrow><mi mathvariant="normal">J</mi></mrow><mo>/</mo><mrow><mi mathvariant="normal">K</mi></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629034&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629030&type= 46.82066727 3.89466691 。温度为5 700~6 100 K的黑体光谱辐射出射度 <math id="M11"><msub><mrow><mi>M</mi></mrow><mrow><mi>λ</mi></mrow></msub><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629045&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629040&type= 14.81666565 4.48733330 随波长的变化曲线如 图1 图1 所示，当黑体的温度升高时，其光谱辐射的峰值波长向短波方向移动。 太阳可以近似看成一个温度 <math id="M12"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">n</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">5</mn><mtext> </mtext><mn mathvariant="normal">900</mn><mtext> </mtext><mi mathvariant="normal">K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629066&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629062&type= 22.01333237 3.80999994 的黑体，它在波长区间 <math id="M13"><mo stretchy="false">[</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629070&type= 11.17599964 4.48733330 的总辐射出射度 <math id="M14"><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629084&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629080&type= 6.34999990 3.72533321 为： <math display="block" id="M15"><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">n</mi></mrow></msub><mo>=</mo><msubsup><mo>∫</mo><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msubsup><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>λ</mi></mrow></msub><mfenced separators="|"><mrow><mi>λ</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">n</mi></mrow></msub></mrow></mfenced><mi mathvariant="normal">d</mi><mi>λ</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629095&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629089&type= 36.83000183 8.29733276 . 对于某些自身发射辐射的辐射源，其辐射亮度与方向无关，这类辐射源称为朗伯辐射体。绝对黑体是典型的朗伯辐射体，因此可由太阳的辐射亮度 <math id="M16"><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629100&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629097&type= 5.33400011 3.72533321 得到辐射出射度 <math id="M17"><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629104&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629102&type= 6.34999990 3.72533321 ，计算公式如下： <math display="block" id="M18"><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">n</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">n</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">π</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629106&type= 17.27199936 7.95866632 . 太阳对小行星的辐射照度可认为是朗伯圆盘辐射体对空间点的辐射照度，如 图2 图2 所示。根据距离平方反比定律，辐射照度与辐射强度成正比，与距离的平方成反比。太阳对小行星的辐射照度 <math id="M19"><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630013&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630009&type= 5.50333309 3.72533321 的计算公式如下： <math display="block" id="M20"><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">n</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">n</mi></mrow></msub><mo>×</mo><mi mathvariant="normal">π</mi><msubsup><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msubsup><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">s</mi><mo>/</mo><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo>×</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></mrow><mo>/</mo><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">s</mi><mo>/</mo><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">j</mi></mrow></msub></mrow></mrow></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629117&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629115&type= 53.67866135 12.10733318 ， 式中： <math id="M21"><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">n</mi></mrow></msub><mo>×</mo><mi mathvariant="normal">π</mi><msubsup><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629122&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629120&type= 15.15533447 3.97933316 为太阳的辐射强度； <math id="M22"><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">s</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629127&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629124&type= 2.45533323 3.72533321 是太阳半径； <math id="M23"><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">s</mi><mo>/</mo><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629134&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629129&type= 6.51933336 3.72533321 是太阳与小行星质心之间的距离。 太阳辐射入射到不透明的小行星表面时，一部分能量被小行星吸收，一部分能量从小行星表面反射。被反射的能量与入射能量之比称为物体的反射本领，小行星的反射本领采用反照率 <math id="M24"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629149&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629146&type= 1.94733346 2.79399991 表示。对于未知小行星， α 通常取0.142 ［ 19 19 ］ 。当太阳-小行星-望远镜呈不同的相位关系时，望远镜看到小行星被太阳照亮的面积是不同的。 <math id="M25"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630396&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630393&type= 1.94733346 3.64066648 定义为小行星被太阳照亮且被望远镜看到的面积与小行星被太阳照亮面积之比。 <math id="M26"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630396&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630393&type= 1.94733346 3.64066648 的准确值需要考虑小行星的真实形状和自转轴指向。 <math id="M27"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630396&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630393&type= 1.94733346 3.64066648 随时间的变化曲线（即光变曲线）对确定小行星形状、自转轴指向和自转周期具有重要意义 ［ 20 20 - 21 21 20-21 ］ 。对于球形小行星可采用 <math id="M28"><mi>β</mi><mo stretchy="false">(</mo><mi>κ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi mathvariant="normal">π</mi><mo>-</mo><mi>κ</mi></mrow><mrow><mi mathvariant="normal">π</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629169&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629167&type= 21.59000015 7.95866632 近似，其中 <math id="M29"><mi>κ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629173&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629171&type= 1.86266661 2.79399991 为太阳-小行星-望远镜的夹角，即太阳角距。因此小行星的辐射出射度 <math id="M30"><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629177&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629175&type= 10.41399956 3.72533321 为： <math display="block" id="M31"><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mi>α</mi><mi>β</mi><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629182&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629180&type= 24.55333328 4.57200003 . 与太阳的辐射出射度类似，小行星反射的辐射亮度 <math id="M32"><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629184&type= 9.39799976 3.72533321 为： <math display="block" id="M33"><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">π</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629191&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629189&type= 25.39999962 7.95866632 . 再由照度的距离平方反比定律得到小行星反射的太阳辐射对望远镜的辐射照度 <math id="M34"><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629195&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629193&type= 9.56733322 3.72533321 ，即： <math display="block" id="M35"><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>×</mo><mi mathvariant="normal">π</mi><msubsup><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msubsup><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">j</mi><mo>/</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo>×</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">j</mi></mrow></msub></mrow><mo>/</mo><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">j</mi><mo>/</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">c</mi></mrow></msub></mrow></mrow></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629200&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629198&type= 65.44733429 12.10733318 ， 式中：分子为小行星反射太阳辐射的辐射强度； <math id="M36"><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629204&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629202&type= 4.14866638 3.72533321 是小行星半径； <math id="M37"><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">j</mi><mo>/</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629262&type= 7.36600018 3.72533321 是小行星质心与望远镜之间的距离。 2 . 1 . 2　自身辐射 3 小行星自身具有一定的温度，也会辐射出一部分能量。由于小行星并非黑体，为描述非黑体的辐射，引入辐射发射率的概念。辐射发射率 <math id="M38"><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629212&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629210&type= 1.43933344 2.79399991 定义为：在相同温度下，辐射体的辐射出射度与黑体的辐射出射度之比， ε 常取0.9 ［ 22 22 ］ ，温度为 <math id="M39"><mn mathvariant="normal">280</mn><mo>~</mo><mn mathvariant="normal">320</mn><mtext> </mtext><mi mathvariant="normal">K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629216&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629214&type= 17.86466599 2.87866688 的黑体光谱辐射出射度如 图3 图3 所示。其中，灰体的辐射发射率与波长无关，小行星一般可认为是灰体。因此热力学温度为 <math id="M40"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629221&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629219&type= 5.24933338 3.72533321 的小行星在波长区间 <math id="M41"><mo stretchy="false">[</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629226&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629224&type= 11.17599964 4.48733330 的总辐射出射度 <math id="M42"><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629228&type= 9.56733322 3.72533321 为： <math display="block" id="M43"><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mi>ε</mi><msubsup><mo>∫</mo><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msubsup><mrow><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi>h</mi><msup><mrow><mi>c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">5</mn></mrow></msup></mrow></mfrac><mo>×</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mi>h</mi><mi>c</mi><mo>/</mo><mo stretchy="false">(</mo><msub><mrow><mi>k</mi></mrow><mrow><mi mathvariant="normal">B</mi></mrow></msub><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">j</mi></mrow></msub><mi>λ</mi><mo stretchy="false">)</mo></mrow></msup><mo>-</mo><mn mathvariant="normal">1</mn></mrow></mfrac><mi mathvariant="normal">d</mi><mi>λ</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629235&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629232&type= 58.92800140 8.97466660 . 小行星自身的辐射亮度 <math id="M44"><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629244&type= 8.55133343 3.72533321 及小行星自身对望远镜的辐射照度 <math id="M45"><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629251&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629249&type= 8.80533314 3.72533321 （与前面类似，小行星对望远镜的辐射照度仍可认为是朗伯圆盘辐射体对空间中某点的辐射照度）分别为： <math display="block" id="M46"><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">π</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629255&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629253&type= 23.70666695 7.95866632 ， <math display="block" id="M47"><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>×</mo><mi mathvariant="normal">π</mi><msubsup><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><msubsup><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">j</mi><mo>/</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">c</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629260&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629258&type= 33.78200150 9.39799976 ， 其中 <math id="M48"><msub><mrow><mi>d</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">j</mi><mo>/</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">c</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629262&type= 7.36600018 3.72533321 为小行星质心与望远镜之间的距离。 望远镜接收到的辐射来源有两部分，一部分是小行星反射的太阳辐射，另一部分是小行星的自身辐射，因此望远镜的总辐射照度 <math id="M49"><mi>E</mi><mo>=</mo><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">f</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub><mo>+</mo><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629268&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629265&type= 30.64933395 3.80999994 。 单个光子的能量为 hc / <math id="M50"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629273&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629270&type= 1.77800000 2.79399991 ，望远镜单位时间单位面积接收到的光子数量 <math id="M51"><mi>Φ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629276&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629274&type= 2.87866688 2.79399991 为： <math display="block" id="M52"><mi>Φ</mi><mo>=</mo><mfrac><mrow><mi>E</mi></mrow><mrow><mi>h</mi><mi>c</mi><mo>/</mo><mi>λ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629280&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629277&type= 15.32466698 8.38199997 . 在典型的空间目标光学相机系统中，由于小行星距离望远镜较远，像点一般小于一个CCD像元面积；但是当光学系统存在一定相差或系统不完善时，信号能量会分布在几个CCD像元上，此时 式（11） 式（11） 应乘一个小于1的系数，称为“偏落因子”，表征落在单个像元面积中的信号能量占总能量的百分比。本文假设小行星的像点落在一个CCD像元上，因此对于口径为 <math id="M53"><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629284&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629282&type= 3.97933316 3.72533321 ，光学透过率为 <math id="M54"><msub><mrow><mi>τ</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629289&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629287&type= 4.31799984 3.72533321 ，量子效率为 <math id="M55"><mi>η</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629293&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629291&type= 1.86266661 3.64066648 的光学望远镜，积分时间为 <math id="M56"><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629298&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629296&type= 3.38666677 3.72533321 时，望远镜单像元接收到小行星的光子数 <math id="M57"><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">l</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629302&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629300&type= 7.02733374 3.72533321 为： <math display="block" id="M58"><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">l</mi></mrow></msub><mo>=</mo><mi>Φ</mi><mo>×</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">t</mi></mrow></msub><mo>×</mo><mi>η</mi><mo>×</mo><mi>G</mi><mo>×</mo><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">t</mi></mrow></msub><mo>×</mo><mi>N</mi><mo>×</mo><mfrac><mrow><mi mathvariant="normal">π</mi><msubsup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629306&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629304&type= 65.44733429 8.12800026 ， 其中： <math id="M59"><mi>G</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629311&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629309&type= 2.79399991 2.79399991 为光电子增益， <math id="M60"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629316&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629314&type= 2.87866688 2.87866688 为在积分时间内获得的图像帧数。 2.2　噪　声 2 2 . 2 . 1　天空背景噪声 3 天空背景的光源主要为黄道光和银道光。黄道光的起因主要是行星际尘埃对太阳光的散射，黄道光大致沿着黄道面伸展，它从黄道光光锥的顶部起朝着背日方向延伸，亮度不断下降，直到离太阳 <math id="M61"><mn mathvariant="normal">135</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629318&type= 6.68866634 3.04800010 左右的地方，此后亮度重新上升，到反日点附近又达到极大，在反日点附近有一个大约 <math id="M62"><mn mathvariant="normal">20</mn><mo>°</mo><mo>×</mo><mn mathvariant="normal">10</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629326&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629324&type= 14.39333248 3.04800010 的区域显得比周围更亮，称为对日照 ［ 23 23 ］ ；银道光的亮度来源主要为恒星。根据Roach等人的模型 ［ 24 24 ］ ，黄道光亮度如 图4 图4 所示，其中横坐标为相对太阳黄经，纵坐标为黄纬；银道光亮度如 图5 图5 所示，其中横坐标为银经，纵坐标为银纬。 图4 图4 和 图5 图5 中亮度单位均为 <math id="M63"><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msub><mfenced separators="|"><mrow><mi>v</mi><mi>i</mi><mi>s</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629331&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629328&type= 11.93799973 4.06400013 ， <math id="M64"><mn mathvariant="normal">205</mn><mo> </mo><mo> </mo><mo> </mo><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msub><mfenced separators="|"><mrow><mi>v</mi><mi>i</mi><mi>s</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629335&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629333&type= 18.37266731 4.06400013 对应的天空背景亮度为 <math id="M65"><mn mathvariant="normal">22</mn><mo> </mo><mo> </mo><mo> </mo><mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">g</mi></mrow><mo>/</mo><mrow><msup><mrow><mo stretchy="false">(</mo><mo>″</mo><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629339&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629337&type= 18.79599953 4.57200003 。 以 <math id="M66"><msub><mrow><mi>S</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msub><mfenced separators="|"><mrow><mi>v</mi><mi>i</mi><mi>s</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629364&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629359&type= 11.93799973 4.06400013 为单位表示的天空亮度为： <math display="block" id="M67"><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">k</mi><msub><mrow><mi mathvariant="normal">y</mi></mrow><mrow><mi>λ</mi></mrow></msub></mrow></msub><mo>=</mo><mi>c</mi><mi>f</mi><mo>×</mo><mi>B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629366&type= 20.82799911 4.06400013 ， <math display="block" id="M68"><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">k</mi><mi mathvariant="normal">y</mi></mrow></msub><mo>=</mo><msubsup><mo>∫</mo><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msubsup><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">k</mi><msub><mrow><mi mathvariant="normal">y</mi></mrow><mrow><mi>λ</mi></mrow></msub></mrow></msub><mi mathvariant="normal">d</mi><mi>λ</mi></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629374&type= 24.46866608 8.29733276 ， 其中： B 为亮度， <math id="M69"><mi>c</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629381&type= 2.62466669 3.64066648 是与波长有关的转换系数，具体数值如 表1 表1 所示 ［ 25 25 ］ 。 λ/ <math id="M72"><mi mathvariant="normal">μ</mi><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629404&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629401&type= 4.23333359 3.13266683 <math id="M73"><mi>c</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629410&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629407&type= 2.20133328 3.13266683 <math id="M74"><mn>0.36</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629416&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629413&type= 5.67266655 2.37066650 <math id="M75"><mn>1.37</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>8</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629423&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629419&type= 15.57866764 2.62466669 <math id="M76"><mn>0.44</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629427&type= 5.67266655 2.37066650 <math id="M77"><mn>2.17</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>8</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629437&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629434&type= 15.57866764 2.62466669 <math id="M78"><mn>0.502</mn><mo>±</mo><mn>0.012</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629443&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629440&type= 18.71133423 2.45533323 <math id="M79"><mn>1.55</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>8</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629450&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629446&type= 15.57866764 2.62466669 <math id="M80"><mn>0.530</mn><mo>±</mo><mn>0.003</mn><mtext> </mtext><mn>5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629456&type= 20.91266632 2.53999996 <math id="M81"><mn>1.33</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>8</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629469&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629464&type= 15.57866764 2.62466669 <math id="M82"><mn>0.55</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629478&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629474&type= 5.67266655 2.37066650 <math id="M83"><mn>1.18</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>8</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629486&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629482&type= 15.57866764 2.62466669 <math id="M84"><mn>0.64</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629498&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629490&type= 5.67266655 2.37066650 <math id="M85"><mn>7.40</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>9</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629509&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629503&type= 15.57866764 2.62466669 <math id="M86"><mn>0.70</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629518&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629514&type= 5.67266655 2.37066650 <math id="M87"><mn>5.70</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>9</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629529&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629524&type= 15.57866764 2.62466669 <math id="M88"><mn>0.79</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629540&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629536&type= 5.67266655 2.37066650 <math id="M89"><mn>4.02</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>9</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629551&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629545&type= 15.57866764 2.62466669 <math id="M90"><mn>0.90</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629562&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629555&type= 5.67266655 2.37066650 <math id="M91"><mn>2.73</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>9</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629573&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629567&type= 15.57866764 2.62466669 <math id="M92"><mn>1.25</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629584&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629579&type= 5.75733376 2.37066650 <math id="M93"><mn>9.89</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>10</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629595&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629590&type= 16.42533302 2.62466669 <math id="M94"><mn>1.65</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629606&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629600&type= 5.67266655 2.37066650 <math id="M95"><mn>3.69</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>10</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629618&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629613&type= 16.42533302 2.62466669 <math id="M96"><mn>2.2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629631&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629625&type= 4.14866638 2.37066650 <math id="M97"><mn>1.29</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>10</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629642&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629637&type= 16.42533302 2.62466669 <math id="M98"><mn>3.5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629653&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629647&type= 4.14866638 2.37066650 <math id="M99"><mn>2.26</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>11</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629662&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629657&type= 16.51000023 2.62466669 <math id="M100"><mn>3.8</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629674&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629668&type= 4.14866638 2.37066650 <math id="M101"><mn>1.60</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>11</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629678&type= 16.42533302 2.62466669 <math id="M102"><mn>4.8</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629696&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629691&type= 4.14866638 2.37066650 <math id="M103"><mn>6.50</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>12</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629706&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629702&type= 16.42533302 2.62466669 <math id="M104"><mn>8.4</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629717&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629711&type= 4.14866638 2.37066650 <math id="M105"><mn>8.09</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>13</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629728&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629724&type= 16.42533302 2.62466669 <math id="M106"><mn>10</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629739&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629733&type= 3.21733332 2.37066650 <math id="M107"><mn>3.94</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>13</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629749&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629743&type= 16.42533302 2.62466669 <math id="M108"><mn>10.6</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629760&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629755&type= 5.67266655 2.37066650 <math id="M109"><mn>3.15</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>13</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629769&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629764&type= 16.42533302 2.62466669 <math id="M110"><mn>12</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629781&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629774&type= 3.21733332 2.37066650 <math id="M111"><mn>1.91</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>13</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629792&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629788&type= 16.42533302 2.62466669 <math id="M112"><mn>20</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629803&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629798&type= 3.21733332 2.37066650 <math id="M113"><mn>2.56</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>14</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629811&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629808&type= 16.42533302 2.62466669 <math id="M114"><mn>21</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629820&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629814&type= 3.21733332 2.37066650 <math id="M115"><mn>2.10</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>14</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629826&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629821&type= 16.42533302 2.62466669 <math id="M116"><mn>25</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629838&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629829&type= 3.21733332 2.37066650 <math id="M117"><mn>1.06</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>14</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629847&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629842&type= 16.42533302 2.62466669 与小行星信号的计算方法类似，由辐射亮度计算出望远镜单位时间单位面积接收到天空背景的光子数 <math id="M118"><msub><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">k</mi><mi mathvariant="normal">y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629863&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629859&type= 5.75733376 3.72533321 ，即： <math display="block" id="M119"><msub><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">k</mi><mi mathvariant="normal">y</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">k</mi><mi mathvariant="normal">y</mi></mrow></msub><mo>×</mo><mi>Ω</mi></mrow><mrow><mi>h</mi><mi>c</mi><mo>/</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">d</mi></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629871&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629866&type= 23.96066666 9.05933285 ， 其中 <math id="M120"><mi>Ω</mi><mo>=</mo><mfrac><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629881&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629876&type= 11.00666618 9.22866726 ，为望远镜单像元的立体角。进而得到望远镜单像元接收到天空背景的光子数 <math id="M121"><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">k</mi><mi mathvariant="normal">y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629892&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629885&type= 5.07999992 3.72533321 ，即： <math display="block" id="M122"><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">k</mi><mi mathvariant="normal">y</mi></mrow></msub><mo>=</mo><msub><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">k</mi><mi mathvariant="normal">y</mi></mrow></msub><mo>×</mo><msub><mrow><mi>τ</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">t</mi></mrow></msub><mo>×</mo><mi>η</mi><mo>×</mo><mi>G</mi><mo>×</mo><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">t</mi></mrow></msub><mo>×</mo><mi>N</mi><mo>×</mo><mfrac><mrow><mi mathvariant="normal">π</mi><msubsup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629900&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629896&type= 66.46333313 8.12800026 . 其中 <math id="M123"><msub><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">d</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629914&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629910&type= 4.91066647 3.72533321 为望远镜工作波段的中间值。 2 . 2 . 2　散粒噪声 3 进入CCD光敏区的光子产生光电子的过程可看作是独立、均匀、连续发生的随机过程。通过望远镜测量到的光强度能给出收集到光子的平均数量，但是无法得知任意时刻实际收集到的光子数量，光子在给定时间内到达的概率由泊松分布控制。泊松分布在大量粒子数时趋向于正态分布，此时散粒噪声的标准差等于平均粒子数的平方根，从而散粒噪声为： <math display="block" id="M124"><msub><mrow><mi>δ</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi></mrow></msub><mo>=</mo><mroot><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">l</mi></mrow></msub></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629918&type= 20.65866661 4.99533367 . 2 . 2 . 3　暗电流噪声 3 暗电流是由CCD像元中积累的热生电子引起的。暗电流积累的速率取决于CCD的温度；天文CCD相机一般工作在制冷状态下，以减小暗电流噪声，暗电流是随时间累积的，因此暗电流噪声的平均值取决于温度和积分时间。这一过程也是泊松随机分布过程，因此这种噪声（或暗信号）也称为暗电流散粒噪声，其标准差等于平均暗电流粒子数的平方根，从而暗电流噪声为： <math display="block" id="M125"><msub><mrow><mi>δ</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">k</mi></mrow></msub><mo>=</mo><mroot><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub><mo>×</mo><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">t</mi></mrow></msub></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629929&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629924&type= 26.50066566 4.99533367 ， 其中： <math id="M126"><msub><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629937&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629935&type= 4.48733330 3.80999994 为暗电流积累的速度，其大小与CCD温度有关。 4 . 2 . 4　读出噪声 3 读出噪声是将每个像元的电荷转换为信号，并将其转换为数字值（ADU或灰度值）时系统组件产生的所有噪声的组合。读出频率越高，读出噪声越高。读出噪声的降低，可以通过改进相机的电子设计或降低读出速度来实现 ［ 26 26 ］ 。读出噪声是在读出像元信号时添加到像元的，各个像元的读出噪声是不均匀的，因此通常将读出噪声表示为一定数量的电子 <math id="M127"><mi mathvariant="normal">R</mi><mi mathvariant="normal">M</mi><mi mathvariant="normal">S</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629942&type= 8.29733276 2.87866688 ，如 <math id="M128"><msub><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">R</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629955&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629951&type= 4.40266657 3.80999994 =0.35e - RMS@500 kHz。 4 . 2 . 5　逆增益 3 逆增益通常为一常量，即： <math display="block" id="M129"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub><mo>=</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">t</mi><mo>.</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629963&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629959&type= 17.52599907 3.72533321 2.3　信噪比 2 由上述公式计算出的信号和各项噪声，小行星在望远镜内单像元的信噪比为： <math display="block" id="M130"><mi>S</mi><mi>N</mi><mi>R</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">l</mi></mrow></msub></mrow><mrow><mroot><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">k</mi><mi mathvariant="normal">y</mi></mrow></msub><mo>+</mo><msubsup><mrow><mi>δ</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">t</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>δ</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">k</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">R</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629973&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629969&type= 65.61666107 16.34066582 . 2.4　视星等 2 生理学得出，人眼的反应跟照度 <math id="M131"><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630021&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630017&type= 2.62466669 2.79399991 的对数成正比，即有： <math display="block" id="M132"><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">v</mi></mrow></msub><mo>=</mo><mi>K</mi><mo>×</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629989&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629985&type= 30.47999954 4.82600021 ， <math display="block" id="M133"><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">v</mi><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">v</mi><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>K</mi><mo>×</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629998&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23629994&type= 43.94200134 9.90600014 . 通过比较星等与光度计测出的照度，发现星等相差5的照度之比约为100，因此常数 <math id="M134"><mi>K</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn mathvariant="normal">5</mn></mrow><mrow><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">100</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mo>-</mo><mn mathvariant="normal">2.5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630006&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630001&type= 37.59199905 9.05933285 ；星等相差1，照度之比为2.512。特别地，天体的观测亮度和有效波段有关，不同波段观测的星等值有差别，因而有不同的星等系统。星等一般对应于星的观测（“视”）亮暗程度，因此常称视星等。取太阳为参考星，视星等为 - 26.74，对应的辐照度为 <math id="M135"><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630013&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630009&type= 5.50333309 3.72533321 ，则辐照度为 <math id="M136"><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630021&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630017&type= 2.62466669 2.79399991 的小行星的视星等 <math id="M137"><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">v</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630028&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630025&type= 4.57200003 3.72533321 为： <math display="block" id="M138"><msub><mrow><mi>M</mi></mrow><mrow><mi mathvariant="normal">v</mi></mrow></msub><mo>=</mo><mo>-</mo><mn mathvariant="normal">26.74</mn><mo>-</mo><mn mathvariant="normal">2.5</mn><mo>×</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><msub><mrow><mi mathvariant="normal">g</mi></mrow><mrow><mn mathvariant="normal">10</mn></mrow></msub><mfenced separators="|"><mrow><mfrac><mrow><mi>E</mi></mrow><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">n</mi></mrow></msub></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630036&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630033&type= 53.34000015 9.90600014 . 3 仿真计算 1 以给出的场景为例，仿真目标小行星相对地球领航轨道可见光望远镜的距离、信噪比和视星等随时间的动态变化；以日地L1点和地球领航轨道为例，计算望远镜对不同尺寸小行星的极限监测距离，分析在地球领航轨道部署可见光望远镜预警来自太阳方向小行星的可行性。 3.1　仿真参数 2 目标小行星在日心黄道J2000坐标系中的轨道根数如 表2 表2 所示。小行星轨道递推模型为在太阳中心引力下的二体模型。目标小行星的物理特性参数如 表3 表3 所示。望远镜和探测器的性能参数分别如 表4 表4 和 表5 表5 所示。 <math id="M139"><mi>a</mi><mo>/</mo><mi mathvariant="normal">A</mi><mi mathvariant="normal">U</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630042&type= 8.04333401 2.79399991 <math id="M140"><mi>e</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630053&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630050&type= 1.35466671 2.37066650 <math id="M141"><mi>i</mi><mo>/</mo><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630061&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630057&type= 6.26533365 3.04800010 <math id="M142"><mi>Ω</mi><mo>/</mo><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630067&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630064&type= 7.36600018 3.04800010 <math id="M143"><mi>ω</mi><mo>/</mo><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630073&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630070&type= 7.45066643 3.04800010 <math id="M144"><mi>F</mi><mo>/</mo><mo stretchy="false">(</mo><mo>°</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630076&type= 7.45066643 3.04800010 1 0.05 10 10 30 165 参　数 数　值 温度 <math id="M145"><msub><mrow><mi>T</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">b</mi><mi mathvariant="normal">j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630094&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630090&type= 4.57200003 3.21733332 /K <math id="M146"><mn>300</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630103&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630100&type= 4.74133301 2.37066650 发射率 <math id="M147"><mi>ε</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630106&type= 1.26999998 2.45533323 <math id="M148"><mn>0.9</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630114&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630112&type= 4.14866638 2.37066650 反照率 <math id="M149"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630121&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630119&type= 1.69333339 2.37066650 <math id="M150"><mn>0.142</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630129&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630126&type= 7.28133297 2.37066650 直径 <math id="M151"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630133&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630131&type= 2.45533323 2.37066650 /m <math id="M152"><mn>100</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630139&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630136&type= 4.74133301 2.37066650 参　数 数　值 口径 <math id="M153"><msub><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630159&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630156&type= 3.47133350 3.21733332 /m <math id="M154"><mn>0.7</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630166&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630162&type= 4.14866638 2.37066650 焦距 <math id="M155"><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630169&type= 0.93133330 3.13266683 /m <math id="M156"><mn>5</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630179&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630176&type= 1.60866666 2.37066650 光学透过率 <math id="M157"><msub><mrow><mi>τ</mi></mrow><mrow><mi mathvariant="normal">o</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630184&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630181&type= 3.80999994 3.21733332 <math id="M158"><mn>0.8</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630187&type= 4.14866638 2.37066650 参　数 数　值 光电增益 <math id="M159"><mi>G</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630203&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630200&type= 2.37066650 2.37066650 <math id="M160"><mn>1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630228&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630225&type= 1.60866666 2.37066650 量子效率 η <math id="M161"><mn>0.7</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630217&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630213&type= 4.14866638 2.37066650 图像帧数 <math id="M162"><mi>N</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630222&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630220&type= 2.45533323 2.45533323 <math id="M163"><mn>1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630228&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630225&type= 1.60866666 2.37066650 暗电流 <math id="M164"><msub><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">D</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630235&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630232&type= 3.89466691 3.21733332 /（e - ·s -1 ·pixel -1 ） <math id="M165"><mn>0.6</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630244&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630240&type= 4.14866638 2.37066650 读出噪声 <math id="M166"><msub><mrow><mi>N</mi></mrow><mrow><mi mathvariant="normal">R</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630249&type= 3.80999994 3.21733332 <math id="M167"><mn>0.35</mn><mo> </mo><mo> </mo><mo> </mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo></mrow></msup><mo> </mo><mi mathvariant="normal">R</mi><mi mathvariant="normal">M</mi><mi mathvariant="normal">S</mi><mo> </mo><mo>@</mo><mn>500</mn><mtext> </mtext><mi mathvariant="normal">k</mi><mi mathvariant="normal">H</mi><mi mathvariant="normal">z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630261&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630257&type= 30.64933395 2.96333337 逆增益 <math id="M168"><msub><mrow><mi>G</mi></mrow><mrow><mi mathvariant="normal">I</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630270&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630266&type= 3.21733332 3.21733332 <math id="M169"><mrow><mrow><msup><mrow><mn>10</mn></mrow><mrow><mn>5</mn></mrow></msup></mrow><mo>/</mo><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>14</mn></mrow></msup></mrow></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630279&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630273&type= 8.55133343 3.04800010 工作波段 <math id="M170"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630284&type= 1.52400005 2.37066650 /nm <math id="M171"><mn>380</mn><mo>-</mo><mn>800</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630295&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630292&type= 13.71599960 2.37066650 像元尺寸 <math id="M172"><mi>a</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630300&type= 1.60866666 2.37066650 /μm <math id="M173"><mn>10</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630312&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630308&type= 3.21733332 2.37066650 积分时间 <math id="M174"><msub><mrow><mi>t</mi></mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630315&type= 2.96333337 3.21733332 /s <math id="M175"><mn>10</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630324&type= 3.21733332 2.37066650 3.2　仿真结果 2 3 . 2 . 1　场景仿真 3 在2025-05-01 12：00：00至2025-07-04 12：00：00期间，小行星和地球领航轨道可见光望远镜的轨道以及小行星相对望远镜的距离分别如 图6 图6 和 图7 图7 所示，小行星相对望远镜的信噪比和视星等分别如 图8 图8 和 图9 图9 所示。在场景时间内，目标小行星相对望远镜的距离增大了约0.09AU，信噪比变化减小了约3.78，小行星的视星等增大了约0.51。 3 . 2 . 2　监测距离 3 一般认为可以成功识别目标所需的最低信噪比为6 ［ 27 27 - 29 29 27-29 ］ ，若直径为 <math id="M176"><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630386&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630383&type= 2.79399991 2.79399991 的小行星在位置黄经 <math id="M177"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630392&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630389&type= 1.77800000 2.79399991 、黄纬 <math id="M178"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630396&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630393&type= 1.94733346 3.64066648 处的信噪比恰好为6，此时小行星与望远镜之间的距离 <math id="M179"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630402&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630399&type= 2.20133328 2.79399991 为望远镜对此小行星在该情形下的极限监测距离。当望远镜部署于日地L1点或地球领航轨道时，对不同直径小行星的极限监测距离分别如 图10 图10 和 图11 图11 所示（彩图见期刊电子版），其中白色区域是望远镜对太阳的规避角（ <math id="M180"><mn mathvariant="normal">30</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630470&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630467&type= 4.82600021 3.04800010 ），红色虚线表示距离地球0.05AU的区域（进入此空间内的小行星认为对地球存在威胁）。显然，由于太阳规避角的存在，位于日地L1点的可见光望远镜可观测天区无法覆盖地球周围0.05AU中朝向太阳的区域；而地球领航轨道由于轨道优势，则可观测天区覆盖地球周围0.05AU的区域，能为来自太阳方向20 m级的小行星提供750万公里的预警距离，因此在地球领航轨道部署望远镜预警来自太阳方向的小行星具有明显的优势。 3.3　参数敏感性分析 2 假设望远镜与太阳的距离为1AU，小行星位于太阳和望远镜延长线上距望远镜1000万公里处；小行星的物理特性参数、望远镜和探测器性能参数与3.1节完全相同。在上述基础上改变小行星的物理特性参数，如反照率、温度和发射率，研究信噪比和视星等对小行星物理参数的敏感性，验证本文方法的可靠性。小行星的信噪比和视星等随小行星反照率、温度和发射率的变化曲线如 图12 图12 ~ 图14 图14 所示。结果表明：在可见光波段，小行星的信噪比和视星等对反照率敏感，对小行星温度和发射率不敏感。原因是小行星反照率描述小行星反射太阳辐射的能力，而小行星温度和发射率描述小行星自身的辐射能力；在可见光波段，望远镜接收到的光子几乎完全来自于小行星反射的太阳辐射，望远镜接收到来自小行星自身辐射的光子约为小行星反射太阳辐射的 <math id="M181"><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mo>-</mo><mn mathvariant="normal">20</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630430&type= 7.61999989 3.04800010 量级，可以忽略不计；因此小行星的信噪比和视星等在可见光波段对与反射太阳辐射相关的参数反照率敏感，而对与小行星自身辐射相关的参数温度和发射率不敏感。 4 结　论 1 本文针对小行星天基光学监测中的可见性问题，提出了考虑黄道光和银道光的小行星天基光学监测信噪比模型，并介绍了小行星辐射度传输理论。然后，结合设计的可见光传感器和望远镜参数，仿真了目标小行星相对地球领航轨道可见光望远镜的距离、信噪比和视星等随时间的动态变化。考虑太阳规避角，计算了在日地L1点和地球领航轨道部署可见光望远镜对不同直径小行星的极限监测距离，论证了在地球领航轨道预警来自太阳方向小行星的可行性。改变小行星的物理参数，研究了信噪比和视星等对小行星物理参数的敏感性。结果表明：目标小行星在给定的观测时间内相对望远镜距离增大了约0.09AU，视星等增大了约0.51星等，目标小行星在口径为0.7 m的望远镜中信噪比减小了约3.78；望远镜的太阳规避角为 <math id="M182"><mn mathvariant="normal">30</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630470&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=23630467&type= 4.82600021 3.04800010 时，由于观测天区覆盖地球周围0.05AU的空间，在地球领航轨道部署可见光望远镜预警来自太阳方向的小行星具有明显的优势，能为来自太阳方向20米级的小行星提供750万公里的预警距离；在可见光波段，望远镜接收到光子几乎完全来自于小行星反射的太阳辐射，望远镜接收到来自小行星自身辐射的光子可以忽略不计，小行星的信噪比和视星等对反照率敏感，而对温度和发射率不敏感。建立的考虑黄道光和银道光的小行星天基光学监测信噪比模型，为小行星天基光学监测的信噪比动态分析提供了理论依据，同时为天基可见光载荷的方案设计与论证及效能评估提供了技术支撑。 参考文献 李振伟 ， 张涛 ， 张楠 ， 等 . 暗弱空间目标的高精度定位 ［J］. 光学 精密工程 ， 2015 ， 23 （ 9 ）： 2627 - 2634 . doi: 10.3788/ope.20152309.2627 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