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Article(id=1322506584094261599, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322506501562937604, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2024.12.562, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1734364800000, receivedDateStr=2024-12-17, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793239740632, onlineDateStr=2026-10-29, pubDate=1765728000000, pubDateStr=2025-12-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793239740632, onlineIssueDateStr=2026-10-29, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793239740632, creator=13701087609, updateTime=1793239740632, updator=13701087609, issue=Issue{id=1322506501562937604, tenantId=1146029695717560320, journalId=1281212831689347082, year='2025', volume='45', issue='12', pageStart='1211', pageEnd='1316', issueExtLink='null', onlineDate='null', pubDate='1765728000000', pubDateStr='2025-12-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793239720954, creator='13701087609', updateTime=1793239720954, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=1221, endPage=1225, 1233, ext={EN=ArticleExt(id=1322506584266228064, articleId=1322506584094261599, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Construction of Gravity Field Order Selection Models for Dynamic Orbit Determination of Highly Elliptical Orbit Spacecraft, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

This study investigates the issue of using different gravity field model orders for the varying altitudes of highly elliptical orbits(HEOs). Using the gravity field optimization recursive algorithm, 54 circular orbits with different orbital altitudes were integrated with gravity field models of different orders for 6 hours and their errors were statistically analyzed. Four gravity field order selection models, including polynomial, power function, rational number approximation and piecewise selection, were constructed.The four HEOs in the lunar exploration project were simulated for experimental verification. The integration time for the four models decreased from 13-15 seconds to 2-3 seconds, significantly enhancing computational efficiency, with the integration times of the four selection models being close to each other.The polynomial fitting model achieved the highest integration accuracy among the four selection models, with the accuracy of the four HEOs using the polynomial fitting model being 55.25%, 95.88%, 77.33%, and 76.23% higher than that of the piecewise selection model, respectively.

, authors=Xiangtai MA1, Yanfeng HU2, Fan ZHANG3, Shaopeng MA4, authorsList=Xiangtai MA, Yanfeng HU, Fan ZHANG, Shaopeng MA, authorCompany=null, correspAuthors=null, authorNote=

MA Xiangtai, PhD, lecturer, majors in orbit determination of high-orbit spacecraft, E-mail:

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针对高椭圆轨道(HEO)高度变化应采取不同阶次的重力场模型这一问题展开研究。使用重力场优化递推算法，对54个轨道高度不同的圆轨道采用不同阶次的重力场模型进行6 h轨道积分并统计其误差，构建多项式、幂函数、有理数逼近和分段取舍4种重力场阶次选择模型。利用仿真探月工程中4个HEO进行实验验证，结果显示，4种模型积分时间由13~15 s下降到2~3 s，极大提高了计算效率，且4种选择模型积分时间接近；在4种选择模型中多项式拟合模型积分精度最高，4个HEO采用多项式拟合模型比分段取舍模型的积分精度分别提高55.25%、95.88%、77.33%、76.23%。

, authors=马祥泰1, 胡彦逢2, 张帆3, 马少鹏4, authorsList=马祥泰, 胡彦逢, 张帆, 马少鹏, authorCompany=null, correspAuthors=null, authorNote=

马祥泰，博士，讲师，主要研究方向为高轨道航天器定轨，E-mail：

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MA Xiangtai, PhD, lecturer, majors in orbit determination of high-orbit spacecraft, E-mail:

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MA Xiangtai, PhD, lecturer, majors in orbit determination of high-orbit spacecraft, E-mail:

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马祥泰，博士，讲师，主要研究方向为高轨道航天器定轨，E-mail：

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马祥泰，博士，讲师，主要研究方向为高轨道航天器定轨，E-mail：

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ArticleFig(id=1322506592042467736, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506584094261599, language=EN, label=Tab. 1, caption=

Gravity field order and orbit errors of 54 orbits with 6-hour orbital integration errors less than 1 mm

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轨道高度/km 阶次 X方向误差/mm Y方向误差/mm Z方向误差/mm 三维误差/mm
1 000 68 -0.867 -0.381 0.478 1.061
1 500 51 -0.508 -0.161 0.428 0.684
2 000 40 -0.581 -0.447 -0.178 0.755
2 500 32 -0.430 0.665 0.012 0.792
3 000 29 0.213 -0.374 0.013 0.430
3 500 23 -0.143 0.994 0.299 1.048
4 000 22 0.472 -0.110 0.968 1.083
4 500 20 -0.346 0.969 0.824 1.318
5 000 17 -0.910 0.641 -0.038 1.114
5 550 20 -0.218 0.091 0.189 0.302
6 000 16 -0.586 0.591 -0.134 0.843
6 500 16 -0.038 0.095 0.220 0.243
7 000 14 0.876 -0.254 -0.555 1.068
7 500 15 -0.435 -0.274 -0.217 0.558
8 000 13 0.542 0.652 -0.636 1.060
8 500 13 -0.972 0.208 0.062 0.996
9 000 12 -0.891 -0.952 -0.455 1.381
9 500 11 -0.625 -0.090 0.349 0.722
10 000 12 -0.297 0.107 0.159 0.353
11 000 10 0.931 -0.558 -0.588 1.235
12 000 10 0.431 0.436 0.078 0.618
13 000 10 0.140 0.068 0.094 0.182
14 000 10 0.069 0.266 0.181 0.329
15 000 8 0.412 0.819 0.247 0.949
16 000 10 0.030 0.104 0.041 0.115
17 000 10 -0.018 0.005 0.014 0.024
18 000 9 0.647 0.305 0.106 0.723
19 000 7 -0.492 0.731 0.907 1.264
20 000 8 -0.042 0.002 -0.140 0.147
21 000 8 0.201 -0.022 -0.024 0.203
22 000 7 0.124 0.182 0.588 0.628
23 000 7 0.550 0.196 0.670 0.888
24 000 6 0.594 -0.793 0.161 1.003
25 000 7 0.718 0.450 0.780 1.152
26 000 7 -0.197 0.221 0.264 0.397
27 000 7 -0.035 -0.165 -0.277 0.324
28 000 7 0.038 0.101 -0.099 0.146
29 000 7 -0.113 -0.100 -0.283 0.321
30 000 7 0.016 0.110 0.159 0.193
31 000 6 0.553 -0.544 -0.492 0.919
32 000 6 0.754 -0.354 0.031 0.833
33 000 7 0.034 0.006 0.085 0.092
34 000 6 0.204 -0.076 0.073 0.230
35 000 6 0.367 -0.091 0.023 0.379
36 000 6 -0.580 -0.316 -0.439 0.794
37 000 6 0.095 -0.006 0.030 0.100
38 000 6 -0.256 -0.137 -0.042 0.293
39 000 5 -0.796 -0.037 0.885 1.191
40 000 5 -0.893 -0.767 0.234 1.200
45 000 5 0.247 0.537 -0.403 0.715
50 000 5 0.170 -0.075 -0.220 0.288
55 000 5 0.108 -0.061 -0.114 0.168
60 000 4 -0.131 0.707 0.117 0.728
), ArticleFig(id=1322506592134742425, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506584094261599, language=CN, label=表1, caption=

54个轨道6 h轨道积分误差小于1 mm的重力场阶次及轨道误差

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轨道高度/km 阶次 X方向误差/mm Y方向误差/mm Z方向误差/mm 三维误差/mm
1 000 68 -0.867 -0.381 0.478 1.061
1 500 51 -0.508 -0.161 0.428 0.684
2 000 40 -0.581 -0.447 -0.178 0.755
2 500 32 -0.430 0.665 0.012 0.792
3 000 29 0.213 -0.374 0.013 0.430
3 500 23 -0.143 0.994 0.299 1.048
4 000 22 0.472 -0.110 0.968 1.083
4 500 20 -0.346 0.969 0.824 1.318
5 000 17 -0.910 0.641 -0.038 1.114
5 550 20 -0.218 0.091 0.189 0.302
6 000 16 -0.586 0.591 -0.134 0.843
6 500 16 -0.038 0.095 0.220 0.243
7 000 14 0.876 -0.254 -0.555 1.068
7 500 15 -0.435 -0.274 -0.217 0.558
8 000 13 0.542 0.652 -0.636 1.060
8 500 13 -0.972 0.208 0.062 0.996
9 000 12 -0.891 -0.952 -0.455 1.381
9 500 11 -0.625 -0.090 0.349 0.722
10 000 12 -0.297 0.107 0.159 0.353
11 000 10 0.931 -0.558 -0.588 1.235
12 000 10 0.431 0.436 0.078 0.618
13 000 10 0.140 0.068 0.094 0.182
14 000 10 0.069 0.266 0.181 0.329
15 000 8 0.412 0.819 0.247 0.949
16 000 10 0.030 0.104 0.041 0.115
17 000 10 -0.018 0.005 0.014 0.024
18 000 9 0.647 0.305 0.106 0.723
19 000 7 -0.492 0.731 0.907 1.264
20 000 8 -0.042 0.002 -0.140 0.147
21 000 8 0.201 -0.022 -0.024 0.203
22 000 7 0.124 0.182 0.588 0.628
23 000 7 0.550 0.196 0.670 0.888
24 000 6 0.594 -0.793 0.161 1.003
25 000 7 0.718 0.450 0.780 1.152
26 000 7 -0.197 0.221 0.264 0.397
27 000 7 -0.035 -0.165 -0.277 0.324
28 000 7 0.038 0.101 -0.099 0.146
29 000 7 -0.113 -0.100 -0.283 0.321
30 000 7 0.016 0.110 0.159 0.193
31 000 6 0.553 -0.544 -0.492 0.919
32 000 6 0.754 -0.354 0.031 0.833
33 000 7 0.034 0.006 0.085 0.092
34 000 6 0.204 -0.076 0.073 0.230
35 000 6 0.367 -0.091 0.023 0.379
36 000 6 -0.580 -0.316 -0.439 0.794
37 000 6 0.095 -0.006 0.030 0.100
38 000 6 -0.256 -0.137 -0.042 0.293
39 000 5 -0.796 -0.037 0.885 1.191
40 000 5 -0.893 -0.767 0.234 1.200
45 000 5 0.247 0.537 -0.403 0.715
50 000 5 0.170 -0.075 -0.220 0.288
55 000 5 0.108 -0.061 -0.114 0.168
60 000 4 -0.131 0.707 0.117 0.728
), ArticleFig(id=1322506592243794330, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506584094261599, language=EN, label=Tab. 2, caption=

Parameters of four simulated large elliptical Earth-Moon transfer orbits

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仿真轨道 半长轴/km 轨道倾角/(°) 升交点赤经/(°) 偏心率 近地点幅角/(°) 平近点角/(°)
HEO-1 32 037.2 30.900 181.064 0.78 179.093 180
HEO-2 42 385.9 31.000 180.866 0.83 179.401 0
HEO-3 67 509.6 30.999 180.618 0.89 179.789 0
HEO-4 212 857.3 30.983 180.485 0.96 179.983 0
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4个仿真大椭圆地月转移轨道的参数

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仿真轨道 半长轴/km 轨道倾角/(°) 升交点赤经/(°) 偏心率 近地点幅角/(°) 平近点角/(°)
HEO-1 32 037.2 30.900 181.064 0.78 179.093 180
HEO-2 42 385.9 31.000 180.866 0.83 179.401 0
HEO-3 67 509.6 30.999 180.618 0.89 179.789 0
HEO-4 212 857.3 30.983 180.485 0.96 179.983 0
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RMS values in different directions of different gravity field order selection models

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轨道 阶次选择模型 X方向/mm Y方向/mm Z方向/mm 3D/mm 积分耗时/s
HEO-1 多项式拟合 139.01 97.92 58.62 179.86 2.539
幂函数拟合 42.06 29.61 17.85 54.45 2.573
有理数逼近 66.98 47.00 28.30 86.58 2.612
分段取舍 309.81 219.36 132.05 401.91 2.927
70阶×70阶 13.872
HEO-2 多项式拟合 5.51 4.44 3.13 7.73 3.27
幂函数拟合 80.80 57.63 34.26 104.99 2.503
有理数逼近 79.05 55.70 33.06 102.20 2.729
分段取舍 142.67 104.80 62.77 187.83 2.521
70阶×70阶 15.145
HEO-3 多项式拟合 93.28 56.86 34.15 114.46 2.507
幂函数拟合 132.01 80.53 48.21 161.98 2.395
有理数逼近 111.90 68.19 40.86 137.27 2.393
分段取舍 411.59 250.97 149.98 504.86 2.436
70阶×70阶 14.569
HEO-4 多项式拟合 4.30 4.10 3.15 6.73 2.312
幂函数拟合 8.64 7.63 4.38 12.33 2.307
有理数逼近 17.28 9.97 5.83 17.28 2.392
分段取舍 28.32 13.91 8.41 28.32 2.349
70阶×70阶 14.476
), ArticleFig(id=1322506592491258269, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322506584094261599, language=CN, label=表3, caption=

不同重力场阶次选择模型各方向RMS值

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轨道 阶次选择模型 X方向/mm Y方向/mm Z方向/mm 3D/mm 积分耗时/s
HEO-1 多项式拟合 139.01 97.92 58.62 179.86 2.539
幂函数拟合 42.06 29.61 17.85 54.45 2.573
有理数逼近 66.98 47.00 28.30 86.58 2.612
分段取舍 309.81 219.36 132.05 401.91 2.927
70阶×70阶 13.872
HEO-2 多项式拟合 5.51 4.44 3.13 7.73 3.27
幂函数拟合 80.80 57.63 34.26 104.99 2.503
有理数逼近 79.05 55.70 33.06 102.20 2.729
分段取舍 142.67 104.80 62.77 187.83 2.521
70阶×70阶 15.145
HEO-3 多项式拟合 93.28 56.86 34.15 114.46 2.507
幂函数拟合 132.01 80.53 48.21 161.98 2.395
有理数逼近 111.90 68.19 40.86 137.27 2.393
分段取舍 411.59 250.97 149.98 504.86 2.436
70阶×70阶 14.569
HEO-4 多项式拟合 4.30 4.10 3.15 6.73 2.312
幂函数拟合 8.64 7.63 4.38 12.33 2.307
有理数逼近 17.28 9.97 5.83 17.28 2.392
分段取舍 28.32 13.91 8.41 28.32 2.349
70阶×70阶 14.476
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高椭圆轨道航天器动力学定轨重力场阶次选择模型构建
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马祥泰 1 , 胡彦逢 2 , 张帆 3 , 马少鹏 4
大地测量与地球动力学 | 现代大地测量 2025,45(12): 1221-1225, 1233
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大地测量与地球动力学 |现代大地测量 2025 , 45 (12) : 1221 -1225, 1233
高椭圆轨道航天器动力学定轨重力场阶次选择模型构建
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MA Xiangtai, PhD, lecturer, majors in orbit determination of high-orbit spacecraft, E-mail:

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MA Xiangtai, PhD, lecturer, majors in orbit determination of high-orbit spacecraft, E-mail:

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马祥泰，博士，讲师，主要研究方向为高轨道航天器定轨，E-mail：

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马祥泰1 , 胡彦逢2, 张帆3, 马少鹏4
作者信息
  • 1 宿迁学院建筑工程学院, 江苏省宿迁市黄河南路399号, 223800
  • 2 航天工程大学航天信息学院, 北京市八一路1号, 101416
  • 3 山东省地质测绘院, 济南市二环东路 11101号, 250002
  • 4 河北省科学院地理科学研究所/河北省地理信息开发应用技术创新中心, 石家庄市范西路80号, 050011
作者简介:

马祥泰，博士，讲师，主要研究方向为高轨道航天器定轨，E-mail：

Construction of Gravity Field Order Selection Models for Dynamic Orbit Determination of Highly Elliptical Orbit Spacecraft
Xiangtai MA1 , Yanfeng HU2, Fan ZHANG3, Shaopeng MA4
Affiliations
  • 1 School of Civil Engineering and Architecture, Suqian University, 399 South-Huanghe Road, Suqian 223800, China
  • 2 School of Astronautics, Space Engineering University, 1 Bayi Road, Beijing 101416, China
  • 3 Shandong GEO-Surveying and Mapping Institute, 11101 East-Erhuan Road, Jinan 250002, China
  • 4 Institute of Geographical Sciences, Hebei Academy of Sciences/Hebei Technology Innovation Center for Geographic Information Application, 80 Fanxi Road, Shijiazhuang 050011, China

    • MA Xiangtai, PhD, lecturer, majors in orbit determination of high-orbit spacecraft, E-mail:

出版时间: 2025-12-15 doi: 10.14075/j.jgg.2024.12.562
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摘要
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针对高椭圆轨道(HEO)高度变化应采取不同阶次的重力场模型这一问题展开研究。使用重力场优化递推算法，对54个轨道高度不同的圆轨道采用不同阶次的重力场模型进行6 h轨道积分并统计其误差，构建多项式、幂函数、有理数逼近和分段取舍4种重力场阶次选择模型。利用仿真探月工程中4个HEO进行实验验证，结果显示，4种模型积分时间由13~15 s下降到2~3 s，极大提高了计算效率，且4种选择模型积分时间接近；在4种选择模型中多项式拟合模型积分精度最高，4个HEO采用多项式拟合模型比分段取舍模型的积分精度分别提高55.25%、95.88%、77.33%、76.23%。

高椭圆轨道  /  动力学定轨  /  重力场阶次  /  重力场模型

This study investigates the issue of using different gravity field model orders for the varying altitudes of highly elliptical orbits(HEOs). Using the gravity field optimization recursive algorithm, 54 circular orbits with different orbital altitudes were integrated with gravity field models of different orders for 6 hours and their errors were statistically analyzed. Four gravity field order selection models, including polynomial, power function, rational number approximation and piecewise selection, were constructed.The four HEOs in the lunar exploration project were simulated for experimental verification. The integration time for the four models decreased from 13-15 seconds to 2-3 seconds, significantly enhancing computational efficiency, with the integration times of the four selection models being close to each other.The polynomial fitting model achieved the highest integration accuracy among the four selection models, with the accuracy of the four HEOs using the polynomial fitting model being 55.25%, 95.88%, 77.33%, and 76.23% higher than that of the piecewise selection model, respectively.

highly elliptical orbit  /  dynamic orbit determination  /  gravity field order  /  gravity field model
马祥泰, 胡彦逢, 张帆, 马少鹏. 高椭圆轨道航天器动力学定轨重力场阶次选择模型构建. 大地测量与地球动力学, 2025 , 45 (12) : 1221 -1225, 1233 . DOI: 10.14075/j.jgg.2024.12.562
Xiangtai MA, Yanfeng HU, Fan ZHANG, Shaopeng MA. Construction of Gravity Field Order Selection Models for Dynamic Orbit Determination of Highly Elliptical Orbit Spacecraft[J]. Journal of Geodesy and Geodynamics, 2025 , 45 (12) : 1221 -1225, 1233 . DOI: 10.14075/j.jgg.2024.12.562
正文
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随着卫星重力测量分辨率和精度的不断提高，地球重力场数据不断精细化，目前的重力场模型已经能够满足动力学定轨的需求[1-2]。重力场模型的阶次越高，轨道积分的精度越高，但计算量也随之增大[3-5]。
对于GRACE-A(500 km)卫星和CHAMP(360 km)卫星的定轨，选择50阶×50阶的重力场模型较为合适。对于SAC-C(705 km)卫星，选择30阶×30阶的重力场模型即可满足在2~4个轨道周期内积分误差在100 m以内的需求[6]。Swarm(500 km)卫星采用30阶×30阶重力场模型时，轨道精度可达dm级；采用100阶×100阶重力场模型时，精度可优于3 cm[7]。Li等[8]和李朵朵等[9]在研究FY-3C(836 km)和HY-2B(971 km)卫星定轨时，采用120阶×120阶重力场模型，精度可达到cm级。赖文等[10]评估了BDS-3 B1C和B2a信号的定轨服务水平，结合12阶×12阶的重力场模型，轨道精度可优于6 cm。Wang等[11]基于TJS-2(36 000 km)卫星的GNSS数据进行定轨分析，采用50阶×50阶的重力场模型，定轨精度可达16.9 m。综上所述，对于轨道高度低于2 000 km的低轨卫星，动力学定轨一般取70~120阶的重力场模型；对于轨道高度约20 000 km的中轨卫星，大约取12阶的重力场模型即可满足精度需求。对于n阶重力场模型，每增加一个阶次，计算地球重力加速度时至少增加n次循环。在航天器自主定轨中，既要考虑星载处理器轨道积分的解算效率，也要满足轨道积分的定轨精度要求。
在高轨道航天器定轨中，还存在高椭圆轨道以及霍曼转移轨道。已有学者将GNSS技术应用到高轨道航天器定轨中[12-13]。这类轨道高度随时间发生较大变化，为了保证高椭圆轨道积分精度和提高计算效率，需要构建合理的随高度变化的重力场阶次选择模型。基于这一需求，本文提出4种随轨道高度变化的重力场阶次选择模型，并进行实验验证。
1 动力学定轨
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1.1 动力学模型
地球的引力加速度不仅受中心引力的影响，还受到地球内部密度分布不均所引发的非球形引力摄动的作用。地球对航天器的引力位用球谐函数在地固坐标系中展开为[14]：
$\begin{gather*}U(r, \lambda, \varphi)=\frac{G M}{r}\left[1+\sum\limits_{n=2}^{\infty} \sum\limits_{m=0}^{n}\left(\frac{a_{\mathrm{E}}}{r}\right)^{n} \cdot\right. \\\left.\bar{P}_{n m} \sin \varphi\left(\bar{C}_{n m} \cos m \lambda+\bar{S}_{n m} \sin m \lambda\right)\right]\end{gather*}$
式中，r表示卫星矢量长度；GM表示地球引力常数；φ、λ分别表示航天器在地固坐标系下的经度和纬度；aE表示地球的平均半径；$\bar{C}_{n m}$ 和$\bar{S}_{n m}$分别表示地球重力场球谐系数；$\bar{P}_{n m}$ 表示n阶m次伴随勒让德多项式。通过上述公式中给出的地球非球形引力位函数的梯度，可以精确计算出非球形引力摄动加速度：
$\ddot{\boldsymbol{r}}=\frac{\partial U}{\partial \boldsymbol{r}}\left(\frac{\partial r}{\partial \boldsymbol{r}}\right)+\frac{\partial U}{\partial \lambda}\left(\frac{\partial \lambda}{\partial \boldsymbol{r}}\right)+\frac{\partial U}{\partial \varphi}\left(\frac{\partial \varphi}{\partial \boldsymbol{r}}\right)$
1.2 重力场模型优化算法
在动力学轨道积分过程中，地球引力加速度的计算量最为庞大，为了编程计算的便捷性，本文引用一种高效的重力场优化递推算法[6]。
将式(1)地球非球形部分引力位改写为[15]：
$\left\{\begin{array}{l}U=\frac{G M}{a_{\mathrm{E}}} \sum\limits_{n=0}^{\infty} \sum\limits_{m=0}^{n}\left(C_{n m} V_{n m}+S_{n m} W_{n m}\right) \\V_{n m}=\left(\frac{a_{\mathrm{E}}}{r}\right)^{n+1} P_{n m} \sin \varphi \cos m \lambda \\W_{n m}=\left(\frac{a_{\mathrm{E}}}{r}\right)^{n+1} P_{n m} \sin \varphi \sin m \lambda\end{array}\right.$
根据缔合勒让德多项式的递推公式以及与地心经度有关的三角函数可以推导出Vnm和Wnm的递推关系，进而可以直接计算出地球引力加速度：
$\ddot{x}=\sum\limits_{n, m} \ddot{x}_{n m}, \ddot{y}=\sum\limits_{n, m} \ddot{y}_{n m}, \ddot{z}=\sum\limits_{n, m} \ddot{z}_{n m} $
2 重力场阶次选择模型
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2.1 重力场阶次选择模型建立
选取54个高度不同的圆轨道进行轨道积分实验，以此构建不同轨道高度重力场阶次选择模型。为降低其他摄动力模型对模拟计算结果的潜在干扰，在构建动力学模型时仅纳入地球引力的影响，以70阶×70阶的重力场模型积分轨道作为参考轨道，选择某时刻仿真数据作为各轨道的初值，使用RK8积分方法，积分时间为6 h，步长选择30 s。对54个轨道进行6 h轨道积分计算，重力场模型阶次从0阶×0阶依次计算至70阶×70阶，判断最后一个历元积分轨道与参考轨道各坐标分量的差异，当各方向积分误差均小于1 mm时停止计算并统计当前重力场阶次。各轨道分量6 h积分结果与参考轨道差异小于1 mm时的阶次及误差统计见表 1，实验计算流程见图 1。
高度低于1 000 km的航天器动力学定轨对重力场阶次要求较高，因此统一设置为70阶×70阶。当轨道高度高于1 000 km时，根据表 1所示轨道高度与重力场阶次的关系，基于最小二乘准则进行函数拟合，构建4种重力场阶次选择模型，其中x为轨道高度(单位为km)，f(x)为选择的重力场模型阶次，各拟合模型的轨道高度与重力场阶次关系见图 2，4种重力场阶次选择模型如下：
1) 多项式拟合模型。
$\left\{\begin{array}{l}f(x)=70, x \leqslant 1000 \\f(x)=-5.273 \times 10^{-18} x^{5}+1.947 \times 10^{-13} \cdot \\\quad x^{4}-2.793 \times 10^{-9} x^{3}+1.958 \times 10^{-5} x^{2}- \\\quad 0.06937 x+120, 1000<x \leqslant 10000 \\f(x)=2.572 \times 10^{-9} x^{2}-0.0002943 x+ \\\quad 13.01, 10000<x \leqslant 60000 \\f(x)=5, 60000<x\end{array}\right.$
2) 幂函数拟合模型。
$\left\{\begin{array}{l}f(x)=3.01 \times 10^{4} x^{-0.8862}+ \\\quad 3.072, x>1000 \\f(x)=70, x \leqslant 1000\end{array}\right.$
3) 有理数逼近模型。
$\left\{\begin{array}{l}f(x)=\frac{3.756 x+7.961 \times 10^{4}}{x+216}, x>1000 \\f(x)=70, x \leqslant 1000\end{array}\right.$
4) 分段取舍模型。
$\left\{\begin{array}{l}f(x)=70, x \leqslant 1000 \\f(x)=50, 1000<x \leqslant 3000 \\f(x)=30, 3000<x \leqslant 5000 \\f(x)=20, 5000<x \leqslant 10000 \\f(x)=10, 10000<x \leqslant 60000 \\f(x)=5, 60000 \leqslant x\end{array}\right.$
在实际计算中，根据轨道高度和函数关系解算出的f(x)为小数时，采取向上取整运算，即最终选择的重力场模型阶次为Ceil(f(x))。
2.2 HEO验证重力场阶次选择模型
GNSS空间服务域(space service volume, SSV)的范围为3 000~36 000 km，地月空间的范围从SSV定义的边缘延伸到距地心约52万km的地球周围空间。因此，利用探月工程中常用的4个HEO进行重力场阶次选择模型验证实验，4个轨道分别命名为HEO-1、HEO-2、HEO-3、HEO-4，轨道参数见表 2。
为比较各重力场阶次选择模型用于轨道积分的精度及计算时间，对4个HEO进行4 d的模拟计算。模拟计算时仅考虑地球引力，使用RK8积分方法，步长取30 s，以70阶×70阶的重力场模型积分轨道作为参考轨道。4个轨道采用不同重力场阶次的积分误差序列与轨道高度的关系见图 3，各轨道采用不同重力场阶次选择模型4 d的各方向积分误差统计及积分耗时见表 3。
由图 3和表 3可知，对于HEO-1，采用分段取舍模型4 d轨道积分误差最大，三维误差RMS值为401.91 mm，其次为多项式拟合模型，三维误差RMS值为179.86 mm，误差RMS值最小的是幂函数拟合模型，三维误差RMS值为54.45 mm。4种重力场阶次选择模型整个弧段积分误差均在2.5 m以内，积分耗时接近，在2.5 ~3 s之间，4 d中轨道绕地球运行6圈，轨道积分误差呈现一定周期性，近地点处积分误差最大。对于HEO-2，轨道积分误差RMS值按照大小排序分别为分段取舍模型、幂函数拟合模型、有理数逼近模型和多项式拟合模型，其中多项式拟合模型积分精度比其他3种模型积分精度高出2个数量级，与之对应的积分耗时同样高于其他3种模型，达到3.27 s，其他3种选择模型计算时间为2.5 s左右。对于HEO-3，采用分段取舍模型积分误差最大，整个弧段积分误差RMS值为504.86 mm，其他3种选择模型积分误差相当，多项式拟合模型误差RMS值最小，为114.46 mm。对于HEO-4，4种重力场阶次选择模型耗时接近，积分误差最小的为多项式拟合模型，误差最大的是分段取舍模型，4 d仿真中轨道高度最高接近3.84×105 km，未能完整地绕地运行1周，采用4种重力场阶次选择模型4 d积分误差最大值分别达到65.49 mm、41.17 mm、23.19 mm、12.38 mm。4个HEO采用70阶×70阶重力场模型轨道积分时间在13~16 s之间，而采用本文构建的4种重力场阶次选择模型积分时间下降至2~3 s，极大提高了计算效率。由于多项式拟合模型在轨道高度大于60 000 km时重力场阶次固定为5阶×5阶，因此高轨道航天器定轨中应优先选择多项式重力场阶次选择模型，4个仿真轨道多项式拟合模型比分段取舍模型精度分别提高55.25%、95.88%、77.33%、76.23%。轨道高度越高，轨道积分时对重力场阶次的要求越低，根据变化的轨道高度选择合理的重力场阶次有助于保证轨道积分精度和计算效率。
3 结语
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1) 在分析54个不同高度的圆轨道积分误差的基础上，给出多项式拟合重力场阶次选择模型、幂函数拟合重力场阶次选择模型、有理数逼近重力场阶次选择模型和分段取舍重力场阶次选择模型。
2) 利用仿真探月工程中的4个HEO对4种重力场阶次选择模型进行验证。4种模型积分时间由70阶×70阶的13~16 s下降到2~3 s，极大提高了积分效率，且4种模型计算效率接近。4个轨道在采用多项式拟合重力场阶次选择模型时积分精度最高，比分段取舍重力场阶次选择模型积分精度分别提高55.25%、95.88%、77.33%、76.23%。因此，在深空探测中进行动力学定轨时，可选择多项式拟合重力场阶次选择模型进行轨道积分运算。
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  • 国家自然科学基金(42174044)
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1
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2025年第45卷第12期
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doi: 10.14075/j.jgg.2024.12.562
  • 接收时间：2024-12-17
  • 首发时间：2026-10-29
  • 出版时间：2025-12-15
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  • 收稿日期：2024-12-17
基金
National Natural Science Foundation of China(42174044)
国家自然科学基金(42174044)
作者信息
    1 宿迁学院建筑工程学院, 江苏省宿迁市黄河南路399号, 223800
    2 航天工程大学航天信息学院, 北京市八一路1号, 101416
    3 山东省地质测绘院, 济南市二环东路 11101号, 250002
    4 河北省科学院地理科学研究所/河北省地理信息开发应用技术创新中心, 石家庄市范西路80号, 050011
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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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