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Article(id=1322566910705775293, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322566845215912204, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2025.03.075, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1741536000000, receivedDateStr=2025-03-10, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793254123616, onlineDateStr=2026-10-29, pubDate=1771084800000, pubDateStr=2026-02-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793254123616, onlineIssueDateStr=2026-10-29, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793254123616, creator=13701087609, updateTime=1793254123616, updator=13701087609, issue=Issue{id=1322566845215912204, tenantId=1146029695717560320, journalId=1281212831689347082, year='2026', volume='46', issue='2', pageStart='139', pageEnd='261', issueExtLink='null', onlineDate='null', pubDate='1771084800000', pubDateStr='2026-02-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793254108003, creator='13701087609', updateTime=1793254108003, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=207, endPage=212, ext={EN=ArticleExt(id=1322566910915490494, articleId=1322566910705775293, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=The P03 Precession Model Considering Frame Bias, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

This paper presents the theoretical relationships between two sets of precession models relative to MERS and GCRS, as well as a complete precession model that accounts for frame bias. The theoretical relationships between different precession angles are derived using vector methods, and two new expressions for the general precession in longitude are provided, one of which is suitable for long-term precession studies. Through numerical calculations, the spherical angle expressions between the north ecliptic pole at the reference and observation epochs and the GCRS polar axis are revised. It is pointed out that the P03 or IAU2006 precession models contain errors on the order of microarcseconds, indicating inconsistencies among different precession angles at the level of several tens of microarcseconds, necessitating a re-examination of the numerical calculations of various precession angles.

, authors=Hanwei ZHANG1, Zhixiang LU1, Na SUN1, *, authorsList=Hanwei ZHANG, Zhixiang LU, Na SUN, authorCompany=null, correspAuthors=Na SUN, authorNote=null, correspAuthorsNote=null, copyrightStatement=All rights reserved. Unauthorized reproduction is prohibited., copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, fund=null), CN=ArticleExt(id=1322566913771811523, articleId=1322566910705775293, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=顾及框架偏差的P03岁差模型, columnId=1281909276545356011, journalTitle=大地测量与地球动力学, columnName=大地测量学, runingTitle=null, highlight=null, articleAbstract=

给出相对MERS和GCRS的2组岁差模型之间的理论关系及顾及框架偏差的完整岁差模型, 利用向量方法给出不同岁差角之间的理论关系; 并给出2个新的黄经总岁差表达式, 其中一个适合长期岁差研究。通过数值计算, 修正了参考历元、观测历元北黄极与GCRS极轴之间的球面角数值表达式。指出P03或IAU2006岁差模型存在μas量级的误差, 即在几十个μas量级上不同岁差角之间是不自洽的, 需要重新审视不同岁差角的数值计算。

, authors=张捍卫1, 逯智翔1, 孙娜1, *, authorsList=张捍卫, 逯智翔, 孙娜, authorCompany=null, correspAuthors=孙娜, authorNote=

张捍卫, 博士, 教授, 主要研究方向为大地测量学, E-mail:

, correspAuthorsNote=
孙娜, 副教授, 主要研究方向为大地测量学, E-mail:
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Journal of Geophysical Research: Solid Earth, 2002, 107 (B4), articleTitle=Modeling of Nutation-Precession: Very Long Baseline Interferometry Results, refAbstract=null)], funds=[Fund(id=1322566918620426991, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566910705775293, awardId=12473068, language=CN, fundingSource=国家自然科学基金(12473068), fundOrder=null, country=null), Fund(id=1322566918683341552, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566910705775293, awardId=42074002, language=CN, fundingSource=国家自然科学基金(42074002), fundOrder=null, country=null), Fund(id=1322566918763033329, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566910705775293, awardId=41931075, language=CN, fundingSource=国家自然科学基金(41931075), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1322566914308682436, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566910705775293, xref=null, ext=[AuthorCompanyExt(id=1322566914317071045, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566910705775293, companyId=1322566914308682436, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1 School of Civil Engineering and Geomatics, Shandong University of Technology, Zibo 255000, China), AuthorCompanyExt(id=1322566914321265350, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566910705775293, companyId=1322566914308682436, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1 山东理工大学建筑工程与空间信息学院, 淄博市, 255000)])], figs=[ArticleFig(id=1322566917861257959, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566910705775293, language=EN, label=Fig. 1, caption=Ecliptic and mean equatorial reference system of reference epoch, figureFileSmall=aWyd4XRRnFl5hpd09gEJQQ==, figureFileBig=+/omUAT9D/SeSboGyxeklw==, tableContent=null), ArticleFig(id=1322566917928366824, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566910705775293, language=CN, label=图1, caption=参考历元黄道和平赤道坐标系, figureFileSmall=aWyd4XRRnFl5hpd09gEJQQ==, figureFileBig=+/omUAT9D/SeSboGyxeklw==, tableContent=null), ArticleFig(id=1322566918129693417, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566910705775293, language=EN, label=Fig. 2, caption=The total precession diagram relative to MERS in thenorthern celestial pole region, figureFileSmall=4eo/vq3M07ZyBEFVGDJe/g==, figureFileBig=SfogfP2xg/yiAt8VcqSMkQ==, tableContent=null), ArticleFig(id=1322566918196802282, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566910705775293, language=CN, label=图2, caption=相对于MERS的北天极区域总岁差示意图, figureFileSmall=4eo/vq3M07ZyBEFVGDJe/g==, figureFileBig=SfogfP2xg/yiAt8VcqSMkQ==, tableContent=null), ArticleFig(id=1322566918272299755, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566910705775293, language=EN, label=Tab. 1, caption=

IAU2006 precession model

, figureFileSmall=null, figureFileBig=null, tableContent=
序号岁差角xA0/(″)xA1/(″)xA2/(″)xA3/10-3″xA4/10-6″xA5/10-7″
1ωA84 381.406 000-0.025 7540.051 262 3-7.725 03-0.4673.337
2ψA0.000 0005 038.481 507-1.079 006 9-1.140 45132.851-0.951
3θA0.000 0002 004.191 903-0.429 493 4-41.822 64-7.089-1.274
4ζA2.650 5452 306.083 2270.298 849 918.018 28-5.971-3.173
5πA0.000 00046.998 973-0.033 492 6-0.125 590.113-0.022
6ΠA629 546.793 600-867.957 5800.157 992 0-0.537 10-47.9700.720
7фA84 381.406 000-46.811 0150.051 126 90.532 89-0.440-0.176
8γA0.000 00010.556 4030.493 204 4-0.312 38-2.7880.260
9εA84 381.406 000-46.836 769-0.000 183 12.003 40-0.576-0.434
10χA0.000 00010.556 403-2.381 429 2-1.211 97170.663-0.560
11τA0.000 0005 038.481 5071.558 417 6-0.185 22-26.452-0.148
12zA-2.650 5452 306.077 1811.092 734 818.268 37-28.596-2.904
13ΛA629 546.793 6004 160.838 6161.263 426 9-0.457 46-71.8200.337
14ε084 381.406 000
15PA0.000 0004.199 0940.193 987 3-0.224 66-0.9120.120
16QA0.000 000-46.811 0150.051 028 30.524 13-0.646-0.172
17pA0.000 0005 028.796 1951.105 434 80.079 64-23.8570.383
), ArticleFig(id=1322566918356185836, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566910705775293, language=CN, label=表1, caption=

IAU2006岁差模型

, figureFileSmall=null, figureFileBig=null, tableContent=
序号岁差角xA0/(″)xA1/(″)xA2/(″)xA3/10-3″xA4/10-6″xA5/10-7″
1ωA84 381.406 000-0.025 7540.051 262 3-7.725 03-0.4673.337
2ψA0.000 0005 038.481 507-1.079 006 9-1.140 45132.851-0.951
3θA0.000 0002 004.191 903-0.429 493 4-41.822 64-7.089-1.274
4ζA2.650 5452 306.083 2270.298 849 918.018 28-5.971-3.173
5πA0.000 00046.998 973-0.033 492 6-0.125 590.113-0.022
6ΠA629 546.793 600-867.957 5800.157 992 0-0.537 10-47.9700.720
7фA84 381.406 000-46.811 0150.051 126 90.532 89-0.440-0.176
8γA0.000 00010.556 4030.493 204 4-0.312 38-2.7880.260
9εA84 381.406 000-46.836 769-0.000 183 12.003 40-0.576-0.434
10χA0.000 00010.556 403-2.381 429 2-1.211 97170.663-0.560
11τA0.000 0005 038.481 5071.558 417 6-0.185 22-26.452-0.148
12zA-2.650 5452 306.077 1811.092 734 818.268 37-28.596-2.904
13ΛA629 546.793 6004 160.838 6161.263 426 9-0.457 46-71.8200.337
14ε084 381.406 000
15PA0.000 0004.199 0940.193 987 3-0.224 66-0.9120.120
16QA0.000 000-46.811 0150.051 028 30.524 13-0.646-0.172
17pA0.000 0005 028.796 1951.105 434 80.079 64-23.8570.383
), ArticleFig(id=1322566918427489005, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566910705775293, language=EN, label=Tab. 2, caption=

IAU2006 precession model relative to GCRS

, figureFileSmall=null, figureFileBig=null, tableContent=
序号岁差角xG0/(″)xG1/(″)xG2/(″)xG3/10-3″xG4/10-6″xG5/10-7″
1ωG84 381.406 000-0.025 7540.051 262 3-7.725 03-0.4673.337
2ψG-0.041 7755 038.481 507-1.079 006 9-1.140 45132.851-0.951
3θG-0.016 6172 004.191 979-0.429 492 4-41.822 64-7.089-1.274
4ζG2.612 2172 306.083 2270.298 849 918.018 28-5.971-3.173
5πG0.000 00046.998 973-0.033 492 6-0.125 590.113-0.022
6ΠG629 546.835 375-867.957 5800.157 992 0-0.537 10-47.9700.720
7фG84 381.412 819-46.811 0160.051 126 90.532 89-0.440-0.176
8γG0.000 00010.556 3780.493 204 4-0.312 38-2.7880.260
9εG84 381.406 000-46.836 769-0.000 183 12.003 40-0.576-0.434
10χG0.000 00010.556 403-2.381 429 2-1.211 97170.663-0.560
11τG-0.041 7755 038.481 4841.558 417 6-0.185 22-26.452-0.148
12zG-2.650 5452 306.077 1151.092 733 018.268 38-28.596-2.904
13ΛG629 546.793 6004 160.838 6161.263 426 9-0.457 46-71.8200.337
14ф084 381.412 819
), ArticleFig(id=1322566918507180782, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322566910705775293, language=CN, label=表2, caption=

相对GCRS的IAU2006岁差模型

, figureFileSmall=null, figureFileBig=null, tableContent=
序号岁差角xG0/(″)xG1/(″)xG2/(″)xG3/10-3″xG4/10-6″xG5/10-7″
1ωG84 381.406 000-0.025 7540.051 262 3-7.725 03-0.4673.337
2ψG-0.041 7755 038.481 507-1.079 006 9-1.140 45132.851-0.951
3θG-0.016 6172 004.191 979-0.429 492 4-41.822 64-7.089-1.274
4ζG2.612 2172 306.083 2270.298 849 918.018 28-5.971-3.173
5πG0.000 00046.998 973-0.033 492 6-0.125 590.113-0.022
6ΠG629 546.835 375-867.957 5800.157 992 0-0.537 10-47.9700.720
7фG84 381.412 819-46.811 0160.051 126 90.532 89-0.440-0.176
8γG0.000 00010.556 3780.493 204 4-0.312 38-2.7880.260
9εG84 381.406 000-46.836 769-0.000 183 12.003 40-0.576-0.434
10χG0.000 00010.556 403-2.381 429 2-1.211 97170.663-0.560
11τG-0.041 7755 038.481 4841.558 417 6-0.185 22-26.452-0.148
12zG-2.650 5452 306.077 1151.092 733 018.268 38-28.596-2.904
13ΛG629 546.793 6004 160.838 6161.263 426 9-0.457 46-71.8200.337
14ф084 381.412 819
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顾及框架偏差的P03岁差模型
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张捍卫 1 , 逯智翔 1 , 孙娜 1, *
大地测量与地球动力学 | 大地测量学 2026,46(2): 207-212
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大地测量与地球动力学 |大地测量学 2026 , 46 (2) : 207 -212
顾及框架偏差的P03岁差模型
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张捍卫1 , 逯智翔1, 孙娜1, *
作者信息
  • 1 山东理工大学建筑工程与空间信息学院, 淄博市, 255000
通讯作者:
孙娜, 副教授, 主要研究方向为大地测量学, E-mail:
作者简介:

张捍卫, 博士, 教授, 主要研究方向为大地测量学, E-mail:

The P03 Precession Model Considering Frame Bias
Hanwei ZHANG1 , Zhixiang LU1, Na SUN1, *
Affiliations
  • 1 School of Civil Engineering and Geomatics, Shandong University of Technology, Zibo 255000, China
出版时间: 2026-02-15 doi: 10.14075/j.jgg.2025.03.075
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摘要
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给出相对MERS和GCRS的2组岁差模型之间的理论关系及顾及框架偏差的完整岁差模型, 利用向量方法给出不同岁差角之间的理论关系; 并给出2个新的黄经总岁差表达式, 其中一个适合长期岁差研究。通过数值计算, 修正了参考历元、观测历元北黄极与GCRS极轴之间的球面角数值表达式。指出P03或IAU2006岁差模型存在μas量级的误差, 即在几十个μas量级上不同岁差角之间是不自洽的, 需要重新审视不同岁差角的数值计算。

北平天极  /  北黄极  /  参考历元平赤道参考系  /  地球质心时空参考系  /  岁差模型

This paper presents the theoretical relationships between two sets of precession models relative to MERS and GCRS, as well as a complete precession model that accounts for frame bias. The theoretical relationships between different precession angles are derived using vector methods, and two new expressions for the general precession in longitude are provided, one of which is suitable for long-term precession studies. Through numerical calculations, the spherical angle expressions between the north ecliptic pole at the reference and observation epochs and the GCRS polar axis are revised. It is pointed out that the P03 or IAU2006 precession models contain errors on the order of microarcseconds, indicating inconsistencies among different precession angles at the level of several tens of microarcseconds, necessitating a re-examination of the numerical calculations of various precession angles.

north mean celestial pole  /  north ecliptic pole  /  mean equatorial reference system of reference epoch  /  geocentric celestial reference system  /  precession model
张捍卫, 逯智翔, 孙娜. 顾及框架偏差的P03岁差模型. 大地测量与地球动力学, 2026 , 46 (2) : 207 -212 . DOI: 10.14075/j.jgg.2025.03.075
Hanwei ZHANG, Zhixiang LU, Na SUN. The P03 Precession Model Considering Frame Bias[J]. Journal of Geodesy and Geodynamics, 2026 , 46 (2) : 207 -212 . DOI: 10.14075/j.jgg.2025.03.075
正文
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人类对时空的认知和探索已进入精密(量子)测量时代，传统时空基准面临理论延拓与技术突破的需求。因此，需要继续发展空间测地技术，为未来自主时空基准建设提供必要的技术、理论和数据处理等全方位支撑[1]。时空基准不但是国民经济、国防建设、地球环境和空间科学研究的理论基础，也是国家安全和信息化战争的重要保障[2]。目前国际天文联合会已发布了第三代国际天球参考架ICRF3[3]，其轴稳定性为10~20 μas; 国际地球自转与参考系服务组织公布了最新的国际地球参考框架ITRF2020[4]，其核心站点的坐标残差为3.7 mm。地球与天球参考架之间通过地球定向参数连接，其中岁差是主要参数之一。主要的岁差模型有P03岁差模型[5]、F03岁差模型[6]和B03岁差模型[7]。Capitaine等[8]指出，F03、B03与P03相比，在一个世纪后其差别小于10 mas; Ghelichkhan等[9]指出，岁差模型所采用的地球动力学因子对时间的依赖性是复杂和非线性的，对赤道岁差会产生较大影响; Tang等[10]在相对论框架下给出了一个长期岁差模型，在距离参考历元前后的106 a内岁差模型是有效的; 也有学者研究了太阳系矮行星和小行星[11]及火星的自转和定向[12]。
本文利用向量方法，给出不同岁差角之间的理论关系，并给出2个新的黄经总岁差表达式，其中一个适合于长期岁差研究；同时，给出顾及框架偏差的所有岁差角的数值模型。通过数值计算，发现参考文献[5]中γGCRS的0阶项有误，且P03或IAU2006岁差模型存在着几十个μas量级的误差。
1 基本坐标系及其之间的关系
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假设O为地球质量中心，令单位向量$\boldsymbol{I}_3= \overrightarrow{{OI}_3}$和$\boldsymbol{H}_3=\overrightarrow{{OH}_3}$分别指向参考历元的北平天极方向和北黄极方向(图 1)。通过$O$点，且与$\boldsymbol{I}_3$垂直的平面称为参考历元平赤道面，与$\boldsymbol{H}_3$垂直的平面称为参考历元黄道面。令：
$\left\{\begin{array}{l}\boldsymbol{I}_3 \cdot \boldsymbol{H}_3=\cos \varepsilon_0 \\\boldsymbol{I}_3 \times \boldsymbol{H}_3=\sin \varepsilon_0 \overrightarrow{O \mathit{Υ}_0}\end{array}\right.$
式中，ε0称为参考历元平黄赤交角。单位向量$\overrightarrow{O \mathit{Υ}}_0$指向参考历元的平春分点方向。在赋予以上符号物理意义后，可建立以下2个参考坐标系:
1) 参考历元黄道坐标系(ECRS)：坐标原点选择在地球质量中心O点，第3轴指向$\boldsymbol{H}_3$，第1轴指向$\boldsymbol{H}_1 \equiv \overrightarrow{O \mathit{Υ}_0}$，第2轴由$\boldsymbol{H}_2=\boldsymbol{H}_3 \times \boldsymbol{H}_1$确定。单位向量$\boldsymbol{H}_1 、\boldsymbol{H}_2$和$\boldsymbol{H}_3$称为参考历元黄道坐标系的单位坐标基，满足正交性和准惯性条件。
2) 参考历元平赤道坐标系(MERS)：坐标原点选择在地球质量中心O点，第3轴指向$\boldsymbol{I}_3$，第1轴指向$\boldsymbol{I}_1 \equiv \overrightarrow{O \mathit{Υ}_0}$，第2轴由$\boldsymbol{I}_2=\boldsymbol{I}_3 \times \boldsymbol{I}_1$确定。单位向量$\boldsymbol{I}_1 、\boldsymbol{I}_2$和$\boldsymbol{I}_3$称为参考历元平赤道坐标系的单位坐标基，满足正交性和准惯性条件。
由图 1可知，ECRS和MERS之间的关系式为[13]：
$\left\{\begin{array}{l}{[\mathrm{MERS}]=\boldsymbol{R}_1\left(-\boldsymbol{\varepsilon}_0\right)[\mathrm{ECRS}]} \\{[\mathrm{ECRS}]=\boldsymbol{R}_1\left(+\boldsymbol{\varepsilon}_0\right)[\mathrm{MERS}]}\end{array}\right.$
式中，R1(+ε0)表示绕第1轴逆时针旋转ε0角的旋转矩阵。式(2)可理解为空间一点在2个坐标系中的坐标分量变换，也可理解为2个参考系标架之间的变换。
IAU2006岁差模型(P03岁差模型)表述的是天球参考轴相对ECRS第3轴和MERS第3轴的运动，而本文的目的是给出天球参考轴相对地球质心时空参考系(GCRS)的运动。GCRS是在相对论框架下建立的, 其坐标原点选择在O点；第1轴$\boldsymbol{G}_1 \rightarrow \overrightarrow{O \mathit{Υ}_0}$；第3轴$\boldsymbol{G}_3 \rightarrow \boldsymbol{I}_3$；第2轴由$\boldsymbol{G}_2=\boldsymbol{G}_3 \times \boldsymbol{G}_1$确定。单位向量$\boldsymbol{G}_1 、\boldsymbol{G}_2$和$\boldsymbol{G}_3$称为GCRS的单位坐标基，也满足正交性和准惯性条件。参考系MERS和GCRS之间的关系为[13]：
$\left\{\begin{array}{l}{[\mathrm{MERS}]=\boldsymbol{R}_1\left(-\eta_0\right) \boldsymbol{R}_2\left(+\xi_0\right) \boldsymbol{R}_3} \\\left(+\mathrm{d} \alpha_0\right)[\mathrm{GCRS}]=\boldsymbol{B}[\mathrm{GCRS}] \\{[\mathrm{GCRS}]=\boldsymbol{R}_3\left(-\mathrm{d} \alpha_0\right) \boldsymbol{R}_2\left(-\xi_0\right) \boldsymbol{R}_1} \\\left(+\eta_0\right)[\mathrm{MERS}]=\boldsymbol{B}^{-1}[\mathrm{MERS}]\end{array}\right.$
式中，ξ0和η0为天极补偿，dα0为分点改正，统称为框架偏差。
因为框架偏差是微小量，框架偏差的正弦值就等于框架偏差本身，其余弦值则等于1。因此，式(3)中的框架偏差矩阵可写为：
$\boldsymbol{B}=\left(\begin{array}{ccc}1 & +\mathrm{d} \alpha_0 & -\xi_0 \\-\mathrm{d} \alpha_0 & 1 & -\eta_0 \\+\xi_0 & +\eta_0 & 1\end{array}\right)$
且B-1=BT。
2 相对MERS的IAU2006岁差模型
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北天极区域附近岁差见图 2，其中参考历元北黄极指向为$\boldsymbol{H}_3=\overrightarrow{O H_3}$，其与天球表面的交点为$\boldsymbol{H}_3$；参考历元北平天极方向为$\boldsymbol{I}_3=\overrightarrow{{OI}_3}$，其与天球表面的交点为$\boldsymbol{I}_3$。观测历元北平天极方向为$\boldsymbol{P}_3=\overrightarrow{O P_3}$，其与天球表面交点为$P_3$；观测历元北黄极为$Q_3=\overrightarrow{O Q_3}$，其与天球表面交点为$Q_3$。
需要注意的是，图 2中的直线应该理解为单位球面上的大圆弧，图 2仅适用于短期(参考历元前后几十年或百年内)岁差模型，不适用于长期(几百年或者几千年)岁差模型。在图 2中2点之间的大圆弧，或2个大圆弧之间的夹角，统称为岁差角。岁差角xA的表达式为[5]：
$\begin{gathered}x_{\mathrm{A}}=x_{\mathrm{A} 0}+x_{\mathrm{A} 1} t+x_{\mathrm{A} 2} t^2+x_{\mathrm{A} 3} t^3+ \\x_{\mathrm{A} 4} t^4+x_{\mathrm{A} 5} t^5\end{gathered}$
式中，t是从参考历元起算的儒略世纪数。式(5)称为岁差模型，国际上曾经主要采用IAU1976岁差模型[14]和IAU2000岁差模型[15]，目前采用的是P03岁差模型, 或称为IAU2006岁差模型[6]，见表 1。
需要注意的是，与参考文献[5]相比，本文符号变化如下：φJ2000→φA，γJ2000→γA，ψJ2000→τA，其余符号与参考文献[5]保持相同。与其他岁差角相比较，P03岁差模型中的岁差角ΠA的0阶项少了2位有效数字，其余少了1位有效数字(表 1中第6个岁差角)。另外，P03岁差模型也没有给出岁差角ΛA的多项式展开，本文给出了计算岁差角ΛA的2个表达式:
$\begin{gathered}\cos \varLambda_{\mathrm{A}}=\cos \left(\varPi_{\mathrm{A}}+\psi_{\mathrm{A}}\right) \cos \chi_{\mathrm{A}}+ \\\sin \left(\varPi_{\mathrm{A}}+\psi_{\mathrm{A}}\right) \sin \chi_{\mathrm{A}} \cos \omega_{\mathrm{A}}\end{gathered}$
$\begin{gathered}\sin \varLambda_{\mathrm{A}}=\sin \pi_{\mathrm{A}} \sin \chi_{\mathrm{A}} \sin \omega_{\mathrm{A}}+\cos \pi_{\mathrm{A}} \cos \chi_{\mathrm{A}} \sin \left(\varPi_{\mathrm{A}}+\right. \\\left.\psi_{\mathrm{A}}\right)-\cos \pi_{\mathrm{A}} \sin \chi_{\mathrm{A}} \cos \left(\varPi_{\mathrm{A}}+\psi_{\mathrm{A}}\right) \cos \omega_{\mathrm{A}}\end{gathered}$
计算发现，式(6)和式(7)计算出的ΛA结果完全一致。
岁差参数PA和QA的表达式为[5, 13]：
$\left\{\begin{array}{l}P_{\mathrm{A}}=\sin \pi_{\mathrm{A}} \sin \varPi_{\mathrm{A}} \\Q_{\mathrm{A}}=\sin \pi_{\mathrm{A}} \cos \varPi_{\mathrm{A}}\end{array}\right.$
称为黄道岁差基本量。另外，岁差参数PA的表达式为[5, 13]：
$P_{\mathrm{A}}=\psi_{\mathrm{A}}-\chi_A \cos \omega_{\mathrm{A}} 。$
称为黄经总岁差。本文给出2个更严格的表达式：
$\left\{\begin{array}{l}P_{\mathrm{A}}=\psi_{\mathrm{A}}-\sin \chi_{\mathrm{A}} \cos \omega_{\mathrm{A}} \\\sin P_{\mathrm{A}}=\cos \chi_{\mathrm{A}} \sin \psi_{\mathrm{A}}-\sin \chi_{\mathrm{A}} \cos \psi_{\mathrm{A}} \cos \omega_{\mathrm{A}}\end{array}\right.$
其中，式(10)的第2式适合长期岁差研究。
3 2组岁差角之间的关系
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如果将图 2中的参考历元北平天极方向$\boldsymbol{I}_3= \overrightarrow{O I_3}$变为GCRS的第3轴$\boldsymbol{G}_3=\overrightarrow{O G_3}$，其与天球表面的交点为G3，其余向量保持不变，那么图 2中有关岁差角需要进行如下改变：参考历元平黄赤交角ε0变为φ0，其他岁差角由xA变为xG。可证明它们之间的关系式为：
$\left\{\begin{array}{l}\varphi_0=\varepsilon_0-\eta_0 \\\psi_{\mathrm{G}}=\psi_{\mathrm{A}}+\frac{1}{\sin \varepsilon_0} \xi_0 \\\omega_{\mathrm{G}}=\omega_{\mathrm{A}} \\\theta_{\mathrm{G}}=\theta_{\mathrm{A}}+\xi_0 \cos \zeta_{\mathrm{A}}-\eta_0 \sin \zeta_{\mathrm{A}} \\\zeta_{\mathrm{G}}=\zeta_{\mathrm{A}}+\frac{\cos \varepsilon_0}{\sin \varepsilon_0} \xi_0\end{array}\right.$
$\left\{\begin{array}{l}\pi_{\mathrm{G}}=\pi_{\mathrm{A}} \\\varPi_{\mathrm{G}}=\varPi_{\mathrm{A}}-\frac{1}{\sin \varepsilon_0} \xi_0 \\\varphi_{\mathrm{G}}=\varphi_{\mathrm{A}}+\xi_0 \sin \gamma_{\mathrm{A}}-\eta_0 \cos \gamma_{\mathrm{A}} \\\gamma_{\mathrm{G}}=\gamma_{\mathrm{A}}-\frac{\cos \varepsilon_0}{\sin \varepsilon_0} \xi_0+\frac{\cos \varphi_{\mathrm{A}}}{\sin \varphi_{\mathrm{A}}}\left(\xi_0 \cos \gamma_{\mathrm{A}}+\right. \\\left.\eta_0 \sin \gamma_{\mathrm{A}}\right) \\\varepsilon_{\mathrm{G}}=\varepsilon_{\mathrm{A}} \\\chi_{\mathrm{G}}=\chi_{\mathrm{A}} \\\tau_{\mathrm{G}}=\tau_{\mathrm{A}}+\frac{1}{\sin \varphi_{\mathrm{A}}}\left(\xi_0 \cos \gamma_{\mathrm{A}}+\eta_0 \sin \gamma_{\mathrm{A}}\right)\end{array}\right.$
$\left\{\begin{array}{l}z_{\mathrm{G}}=z_{\mathrm{A}}+\left(\xi_0 \sin \zeta_{\mathrm{A}}+\eta_0 \cos \zeta_{\mathrm{A}}\right) \sin \theta_{\mathrm{A}} \\\varLambda_{\mathrm{G}}=\varLambda_{\mathrm{A}}\end{array}\right.$
其中，式(11)~(13)是忽略框架偏差二次项的结果。
下面给出岁差角φ0、ψG和ΠG的证明过程，其余岁差角xG的证明过程基本类似。
3.1 相对GCRS的平黄赤交角φ0
根据式(2)和式(3)可得：
$\begin{gathered}\boldsymbol{G}_3=-\boldsymbol{\xi}_0 \boldsymbol{H}_1+\left(\sin \varepsilon_0-\eta_0 \cos \varepsilon_0\right) \boldsymbol{H}_2+ \\\left(\cos \varepsilon_0+\eta_0 \sin \varepsilon_0\right) \boldsymbol{H}_3\end{gathered}$
岁差角φ0的定义为：
$\left\{\begin{array}{l}\cos \varphi_0 \equiv \boldsymbol{G}_3 \cdot \boldsymbol{H}_3=\cos \varepsilon_0+\eta_0 \sin \varepsilon_0 \\\sin \varphi_0 \overrightarrow{O \mathit{Υ}_{\mathrm{G}}} \equiv \boldsymbol{G}_3 \times \boldsymbol{H}_3=\left(\sin \varepsilon_0-\right. \\\left.\eta_0 \cos \varepsilon_0\right) \boldsymbol{H}_1+\xi_0 \boldsymbol{H}_2\end{array}\right.$
式中，$\overrightarrow{O \mathit{Υ}_G}$为单位向量。另外，根据式(15)的第2式可得：
$\sin \varphi_0=\sin \varepsilon_0-\eta_0 \cos \varepsilon_0$
再根据式(15)的第1式和式(16)，可得式(11)的第1式。
3.2 相对GCRS的赤道岁差角ψG
基于式(15)的第2式，并考虑到式(16)，可得：
$\overrightarrow{O \mathit{Υ}_G}=\boldsymbol{H}_1+\xi_0 \frac{1}{\sin \varepsilon_0} \boldsymbol{H}_2$
此处略去了框架偏差的二次项。
根据图 2，球面角ψA定义为：
$\left\{\begin{array}{l}-\overrightarrow{O \mathit{Υ}_1} \cdot-\overrightarrow{O \mathit{Υ}_0}=\cos \psi_{\mathrm{A}} \\-\overrightarrow{O \mathit{Υ}_1} \times-\overrightarrow{O \mathit{Υ}_0}=\sin \psi_{\mathrm{A}} \boldsymbol{H}_3\end{array}\right.$
由于$\overrightarrow{O \mathit{Υ}_0}=\boldsymbol{H}_1$，根据式(18)可得：
$\overrightarrow{O \mathit{Υ}_1}=\cos \psi_{\mathrm{A}} \boldsymbol{H}_1-\sin \psi_{\mathrm{A}} \boldsymbol{H}_2$
类似式(18)，定义岁差角ψG为：
$\left\{\begin{array}{l}-\overrightarrow{O \mathit{Υ}_1} \cdot-\overrightarrow{O \mathit{Υ}_{\mathrm{G}}}=\cos \psi_{\mathrm{G}} \\-\overrightarrow{O \mathit{Υ}_1} \times-\overrightarrow{O \mathit{Υ}_{\mathrm{G}}}=\sin \psi_{\mathrm{G}} \boldsymbol{H}_3\end{array}\right.$
根据式(17)、式(19)和式(20)可得:
$\left\{\begin{array}{l}\sin \psi_{\mathrm{G}}=\sin \psi_{\mathrm{A}}+\xi_0 \frac{\cos \psi_{\mathrm{A}}}{\sin \varepsilon_0} \\\cos \psi_{\mathrm{G}}=\cos \psi_{\mathrm{A}}-\xi_0 \frac{\sin \psi_{\mathrm{A}}}{\sin \varepsilon_0}\end{array}\right.$
由于ψG-ψA是微小量，因此利用式(21)可得式(11)的第2式。
3.3 相对GCRS的黄道岁差角ΠG
根据图 2所示，球面角ΠA定义为：
$\left\{\begin{array}{l}\overrightarrow{O \mathit{Υ}_0} \cdot \overrightarrow{O P}=\cos \varPi_{\mathrm{A}} \\\overrightarrow{O \mathit{Υ}_0} \times \overrightarrow{O P}=\sin \varPi_{\mathrm{A}} \boldsymbol{H}_3\end{array}\right.$
式中，$\overrightarrow{O H_3} \times \overrightarrow{O Q_3}=\sin \pi_{\mathrm{A}} \overrightarrow{O P}$，由于$\overrightarrow{O \mathit{Υ}_0}=\boldsymbol{H}_1$，根据式(22)可得：
$\overrightarrow{O P}=\cos \varPi_{\mathrm{A}} \hat{H}_1+\sin \varPi_{\mathrm{A}} \hat{H}_2$
类似于式(22)，定义：
$\left\{\begin{array}{l}\overrightarrow{O \mathit{Υ}_{\mathrm{G}}} \cdot \overrightarrow{O P}=\cos \varPi_{\mathrm{G}} \\\overrightarrow{O \mathit{Υ}_{\mathrm{G}}} \times \overrightarrow{O P}=\sin \varPi_{\mathrm{G}} \boldsymbol{H}_3\end{array}\right.$
利用式(17)和式(23)，可将式(24)表示为：
$\left\{\begin{array}{l}\cos \varPi_{\mathrm{G}}=\cos \varPi_{\mathrm{A}}+\xi_0 \frac{\sin \varPi_{\mathrm{A}}}{\sin \varepsilon_0} \\\sin \varPi_{\mathrm{G}}=\sin \varPi_{\mathrm{A}}-\xi_0 \frac{\cos \varPi_{\mathrm{A}}}{\sin \varepsilon_0}\end{array}\right.$
考虑到ΠG-ΠA是微小量，根据式(25)可得式(12)的第2式。
4 相对GCRS的IAU2006岁差模型
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式(3)和式(4)中框架偏差的数值为[13]：
$\left\{\begin{array}{l}\xi_0=-0.016\;617\;0^{\prime \prime} \pm 0.000\;010\;0^{\prime \prime} \\\eta_0=-0.006\;819\;2^{\prime \prime} \pm 0.000\;010\;0^{\prime \prime} \\\text{d} \alpha_0=-0.014\;60^{\prime \prime} \pm 0.000\;50^{\prime \prime}\end{array}\right.$
由于
$\begin{aligned}\left(\xi_0\right)^2= & \left(-0.016\;617\;0^{\prime \prime} \frac{2 \pi}{360 \times 3\;600}\right)^2 \times \\& \frac{360 \times 3\;600}{2 \pi} \approx 1.34^{\prime \prime} \times 10^{-9}\end{aligned}$
因此，在涉及框架偏差的运算中只保留其一阶项。
结合表 1给出的数值和式(26)，利用式(11)~ (13)可得表 2。
需要注意的是，与参考文献[5]相比，本文符号变化如下：φGCRS→φG，γGCRS→γG，ψGCRS→τG。参考文献[5]只给出了以上3个岁差角的多项式展开，其余岁差角未给出。另外，γG的0阶项的数值为[5]：
$\gamma_{\mathrm{G} 0}=-0.052\;928^{\prime \prime}$
根据式(12)的第4式和表 1中的数据，只能得出γG0=0，见表 2第8行。
5 北平天极和北黄极运动的初始方向
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岁差角ψA和ωA称为赤道岁差基本量，PA和QA称为黄道岁差基本量[5]。除参考历元平黄赤交角ε0外，根据赤道和黄道岁差基本量，可导出表 1中其他岁差角的表达式。下面利用ψA和ωA及φA和γA，分别计算北平天极和北黄极运动的初始方向，由此检验岁差角多项式展开系数的准确性。
5.1 北平天极运动的初始方向
观测历元北平天极的表达式为：
$\begin{gathered}\boldsymbol{P}_3=\sin \omega_{\mathrm{A}} \sin \psi_{\mathrm{A}} \boldsymbol{H}_1+\sin \omega_{\mathrm{A}} \cos \psi_{\mathrm{A}} \boldsymbol{H}_2+ \\\cos \omega_{\mathrm{A}} \boldsymbol{H}_3\end{gathered}$
其运动速度可表示为：
$\begin{gathered}\boldsymbol{v}=\frac{\mathrm{d} \boldsymbol{P}_3}{\mathrm{~d} t}=\left[\cos \omega_{\mathrm{A}} \sin \psi_{\mathrm{A}} \boldsymbol{H}_1+\cos \omega_{\mathrm{A}} \cos \psi_{\mathrm{A}} \boldsymbol{H}_2-\right. \\\left.\sin \omega_{\mathrm{A}} \boldsymbol{H}_3\right] \frac{\mathrm{d} \omega_{\mathrm{A}}}{\mathrm{~d} t}+\left[\cos \psi_{\mathrm{A}} \boldsymbol{H}_1-\sin \psi_{\mathrm{A}} \boldsymbol{H}_2\right] \\\sin \omega_{\mathrm{A}} \frac{\mathrm{~d} \psi_{\mathrm{A}}}{\mathrm{~d} t}\end{gathered}$
考虑到式(5)和表 1给出的多项式系数，在t=0时有：
$\begin{gathered}\left.\boldsymbol{v}\right|_{t=0}=\omega_{\mathrm{Al}}\left(\cos \varepsilon_0 \boldsymbol{H}_2-\sin \varepsilon_0 \boldsymbol{H}_3\right)+ \\\psi_{\mathrm{A} 1} \sin \varepsilon_0 \boldsymbol{H}_1\end{gathered}$
其运动方向可表示为：
$\begin{gathered}\hat{\boldsymbol{v}}_0=\left.\hat{\boldsymbol{v}}\right|_{t=0}=\frac{1}{\sqrt{\left(\omega_{\mathrm{A} 1}\right)^2+\left(\psi_{\mathrm{A} 1} \sin \varepsilon_0\right)^2}} \times \\{\left[\omega_{\mathrm{A} 1}\left(\cos \varepsilon_0 \boldsymbol{H}_2-\sin \varepsilon_0 \boldsymbol{H}_3\right)+\right.} \\\left.\psi_{\mathrm{A} 1} \sin \varepsilon_0 \boldsymbol{H}_1\right]\end{gathered}$
在MERS中，式(32)可表示为：
$\begin{gathered}\hat{\boldsymbol{v}}_0=\left.\hat{\boldsymbol{v}}\right|_{t=0}=\frac{1}{\sqrt{\left(\omega_{\mathrm{A} 1}\right)^2+\left(\psi_{\mathrm{A} 1} \sin \varepsilon_0\right)^2}} \times \\{\left[\omega_{\mathrm{A} 1} I_2+\psi_{\mathrm{A} 1} \sin \varepsilon_0 I_1\right]}\end{gathered}$
根据图 1所示，大圆弧$\overparen{H_3 I_3}$在I3点的切线方向为-I2，于是有：
$\begin{gathered}\cos \left(90^{\circ}-\zeta_{\mathrm{A} 0}\right)=-\boldsymbol{I}_2 \cdot \hat{\boldsymbol{v}}_0= \\\frac{-\omega_{\mathrm{A} 1}}{\sqrt{\left(\omega_{\mathrm{A} 1}\right)^2+\left(\psi_{\mathrm{A} 1} \sin \varepsilon_0\right)^2}}\end{gathered}$
将表 1中的有关数值代入上式可得：
$\zeta_{\mathrm{A} 0}=+2.650\;516\;556^{\prime \prime}$
这一数值与表 1中的数值ζA0=+2.650 545″有28 μas的差异。
5.2 北黄极运动的初始方向
观测历元北黄极可表示为：
$\begin{gathered}Q_3=\sin \varphi_{\mathrm{A}} \sin \gamma_{\mathrm{A}} \boldsymbol{I}_1-\sin \varphi_{\mathrm{A}} \cos \gamma_{\mathrm{A}} \boldsymbol{I}_2+ \\\cos \varphi_{\mathrm{A}} \boldsymbol{I}_3\end{gathered}$
其运动速度可表示为：
$\begin{gathered}\boldsymbol{v}=\frac{\mathrm{d} \boldsymbol{Q}_3}{\mathrm{~d} t}=\left[\cos \varphi_{\mathrm{A}} \sin \gamma_{\mathrm{A}} \boldsymbol{I}_1-\cos \varphi_{\mathrm{A}} \cos \gamma_{\mathrm{A}} \boldsymbol{I}_2-\right. \\\left.\sin \varphi_{\mathrm{A}} \boldsymbol{I}_3\right] \frac{\mathrm{d} \varphi_{\mathrm{A}}}{\mathrm{~d} t}+\sin \varphi_{\mathrm{A}}\left[\cos \gamma_{\mathrm{A}} \boldsymbol{I}_1+\sin \gamma_{\mathrm{A}} \boldsymbol{I}_2\right] \frac{\mathrm{d} \gamma_{\mathrm{A}}}{\mathrm{~d} t}\end{gathered}$
考虑到式(5)和表 1给出的多项式系数，当时间t=0时有：
$\begin{gathered}\left.\boldsymbol{v}\right|_{t=0}=\gamma_{\mathrm{A} 1} \sin \varepsilon_0 \boldsymbol{I}_1-\varphi_{\mathrm{A} 1} \cos \varepsilon_0 \boldsymbol{I}_2- \\\varphi_{\mathrm{A} 1} \sin \varepsilon_0 \boldsymbol{I}_3\end{gathered}$
其初始运动方向可表示为：
$\begin{aligned}& \hat{\boldsymbol{v}}_0=\left.\hat{\boldsymbol{v}}\right|_{t=0}=\frac{1}{\sqrt{\left(\varphi_{\mathrm{A} 1}\right)^2+\left(\gamma_{\mathrm{A} 1} \sin \varepsilon_0\right)^2}} \times \\& {\left[\gamma_{\mathrm{A} 1} \sin \varepsilon_0 \boldsymbol{I}_1-\varphi_{\mathrm{A} 1} \cos \varepsilon_0 \boldsymbol{I}_2-\varphi_{\mathrm{A} 1} \sin \varepsilon_0 \boldsymbol{I}_3\right]}\end{aligned}$
在ECRS中的表达式为：
$\begin{gathered}\hat{\boldsymbol{v}}_0=\left.\hat{\boldsymbol{v}}\right|_{t=0}=\frac{1}{\sqrt{\left(\varphi_{\mathrm{Al}}\right)^2+\left(\gamma_{\mathrm{Al}} \sin \varepsilon_0\right)^2}} \\{\left[\gamma_{\mathrm{Al}} \sin \varepsilon_0 \boldsymbol{H}_1-\varphi_{\mathrm{Al}} \boldsymbol{H}_2\right]}\end{gathered}$
根据图 1所示，大圆弧$\overparen{H_3 I_3}$在H3点的切线方向为-H2，于是有：
$\begin{gathered}\cos \varPi_{\mathrm{A} 0}=-\hat{\boldsymbol{H}}_2 \cdot \hat{\boldsymbol{v}}_0= \\\frac{\varphi_{\mathrm{A} 1}}{\sqrt{\left(\varphi_{\mathrm{A} 1}\right)^2+\left(\gamma_{\mathrm{A} 1}\right)^2 \sin ^2 \varepsilon_0}}\end{gathered}$
将表 1中的有关数值代入上式可得：
$\varPi_{\mathrm{A} 0}=629\;546.793\;533^{\prime \prime}$
此数值与表 1中给出的数值ΠA0=629 546.793 600″相比有67 μas的差异。
6 讨论与结论
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岁差模型描述的是观测历元北平天极和相对参考历元北平天极的运动及观测历元北黄极相对参考历元北黄极的运动，或者说岁差模型是在MERS和ECRS中建立的。由于岁差模型已将参考系调整为国际天球参考系，这一调整涉及MERS与GCRS之间的框架偏差。本文给出了顾及框架偏差的，也就是相对GCRS和ECRS的岁差模型。
除参考历元平黄赤交角ε0外，在IAU2006岁差模型中，独立的岁差角有4个, 但在微角秒量级上，这些岁差角的数值表达式之间并不严格自洽。例如，利用赤道岁差基本量计算北平天极的初始运动方向和利用黄道岁差导出量计算北黄极的初始运动方向，都与IAU2006岁差模型有一定差异, 说明在微角秒量级上，需要重新审视不同岁差角的数值计算。
基金
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  • 国家自然科学基金(12473068)
  • 国家自然科学基金(42074002)
  • 国家自然科学基金(41931075)
文献
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2026年第46卷第2期
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文章信息
doi: 10.14075/j.jgg.2025.03.075
  • 接收时间：2025-03-10
  • 首发时间：2026-10-29
  • 出版时间：2026-02-15
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  • 收稿日期：2025-03-10
基金
国家自然科学基金(12473068)
国家自然科学基金(42074002)
国家自然科学基金(41931075)
作者信息
    1 山东理工大学建筑工程与空间信息学院, 淄博市, 255000

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孙娜, 副教授, 主要研究方向为大地测量学, E-mail:
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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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