強(qiáng)不規(guī)則天體引力場中的動力學(xué)研究進(jìn)展

姜宇,寶音賀西

(1.西安衛(wèi)星測控中心宇航動力學(xué)國家重點(diǎn)實(shí)驗(yàn)室,西安710043; 2.清華大學(xué)航天航空學(xué)院,北京100084)

強(qiáng)不規(guī)則天體引力場中的動力學(xué)研究進(jìn)展

姜宇1,2,寶音賀西2

(1.西安衛(wèi)星測控中心宇航動力學(xué)國家重點(diǎn)實(shí)驗(yàn)室,西安710043; 2.清華大學(xué)航天航空學(xué)院,北京100084)

小行星探測與彗星探測是深空探測的重要方面。一般來說,小行星和彗星因質(zhì)量都不足以使得萬有引力克服應(yīng)力達(dá)到流體靜力學(xué)平衡,而具有強(qiáng)不規(guī)則的外形。研究強(qiáng)不規(guī)則天體引力場中的動力學(xué)行為及其內(nèi)在機(jī)制,是探測器被不規(guī)則天體捕獲并對其形成近距離探測軌道的基礎(chǔ)。從引力場模型和動力學(xué)行為兩個方面綜述了強(qiáng)不規(guī)則天體引力場中動力學(xué)的研究進(jìn)展,在引力場模型的研究方面介紹了強(qiáng)不規(guī)則天體引力場建模的球諧函數(shù)攝動展開模型、簡單特殊體模型及多面體模型的研究現(xiàn)狀,在動力學(xué)機(jī)制的研究方面介紹了強(qiáng)不規(guī)則天體引力場中的周期軌道和擬周期軌道、平衡點(diǎn)、流形、分岔與共振以及混沌運(yùn)動的研究現(xiàn)狀,指出了這些方面研究的重點(diǎn)與難點(diǎn)。分析了強(qiáng)不規(guī)則體引力場中動力學(xué)的研究趨勢。

強(qiáng)不規(guī)則天體;小行星探測;彗星探測;宇航動力學(xué);動力學(xué)規(guī)律

0 引 言

自古以來,人類從未放棄太空探索的夢想。從神話傳說中的嫦娥奔月、夸父逐日,到敦煌壁畫飛天;從帝堯陶唐氏的“夢攀天而上”,到名相伊尹乘坐宇宙飛船的夢想——“伊摯將應(yīng)湯命,夢乘船過日月之傍”;從屈原的《天問》,到李白的“俱懷逸興壯思飛,欲上青天覽(攬)明月”,無不寄托著人們對太空探索的不倦追求。目前,人類的太空探索早已不局限于使用繞地衛(wèi)星對地球表面及地球附近的探索,而是走向了行星際空間,包括對大行星及其衛(wèi)星的環(huán)繞[1-2]、觀測[3]、著陸[4]、化學(xué)成分分析[57]等,對小行星的飛越[8]、成像[810]、不規(guī)則外形與引力場建模[11-12]、著陸采樣[1314]、碎石堆結(jié)構(gòu)分析[15-21]等,對彗核的結(jié)構(gòu)與分裂分析[22-28]、對彗發(fā)形態(tài)與物質(zhì)成分的分析[2931]等。

太陽系的天體除了八大行星、矮行星以及大行星的大衛(wèi)星之外,絕大多數(shù)天體都是強(qiáng)不規(guī)則形的[3233]。其中矮行星又叫侏儒行星,是環(huán)繞太陽運(yùn)行的天體中大小介于大行星和小行星之間的、質(zhì)量大到行星上物質(zhì)之間的萬有引力足以克服應(yīng)力而達(dá)到流體靜力學(xué)平衡、而質(zhì)量又沒有大到能清除該行星軌道上的小天體的那些天體[34]。小行星與矮行星沒有明確的大小界限,國際天文聯(lián)合會曾經(jīng)擬建議以800 km直徑為小行星與矮行星的界限,但未形成最終決議,目前小行星與矮行星之間仍無統(tǒng)一的被廣泛接受的界限。小行星[3536]可以小至幾十米,大至數(shù)百千米。比小行星更小的稱為流星體[37-39],流星體和小行星之間亦無明確的界限。英國皇家天文學(xué)會曾將100μm至10 m之間的天體以及50 m以下的近地天體定義為流星體,比流星體大的則為小行星,比小行星大的則為矮行星。然而也有例外,最新發(fā)現(xiàn)的小天體2014 HL129的直徑只有約7.6 m,也被稱為小行星[40]。一般直徑在100 m以上的小行星能提供足夠的引力使得探測器可以環(huán)繞該小行星飛行從而成為其衛(wèi)星,例如阿波羅雙小行星2003 SS84的直徑僅有約120 m,其衛(wèi)星S/2004(2003 SS84)1的直徑約為60 m[4142],直徑如此之小的小行星也能提供足夠的引力形成雙小行星系統(tǒng),這是迄今為止人類發(fā)現(xiàn)的最小的雙小行星系統(tǒng)。

小行星951 Gaspra[43]是人類第一次造訪過的小行星,“伽利略號”探測器于1991年飛越該小行星,最近距離1 600 km。小行星951 Gaspra的尺寸為18.2×10.5×8.9 km[44],具有強(qiáng)不規(guī)則的外形[45],其軌道半長軸為2.6 AU,逃逸速度為6 m/s,一個Gaspra天為7.042 h。人類第一次發(fā)現(xiàn)的雙小行星系統(tǒng)是Ida-Dactyl系統(tǒng)[46]。1993年“伽利略號”探測器[4748]飛越了小行星243 Ida,最近距離2 400 km。小行星243 Ida[47-51]的軌道半長軸為2.991 AU,尺寸為53.6×24.0×15.2 km,其小月亮Dactyl的尺寸為1.6×1.4×1.2 km,可見該雙小行星系統(tǒng)是由兩個強(qiáng)不規(guī)則天體組成,兩個小行星的距離為90 km,一個Ida天為4.63 h,小月亮Dactyl的公轉(zhuǎn)周期約20 h[47,49,52]。除了對小行星等太陽系強(qiáng)不規(guī)則小天體進(jìn)行飛越觀測之外,還要進(jìn)行表面登陸及采樣返回[14,53]。其中,“隼鳥號”是人類第一個對小行星進(jìn)行采樣返回的探測器[8,13-15]?！傲_塞塔號”是人類第一個對彗星核進(jìn)行登陸的探測器,已于2014年11月登陸彗星67P/Churyumov-Gerasimenko[54-56]。

1 引力場模型研究現(xiàn)狀

人們對天體引力場建模方法的研究,經(jīng)歷了從簡單到復(fù)雜、從低維到高維的研究階段[57-69]。經(jīng)典的引力場建模方法是對球諧函數(shù)疊加攝動項(xiàng)并進(jìn)行級數(shù)展開,一個均質(zhì)球的引力場等效于一個點(diǎn)質(zhì)量產(chǎn)生的引力場,點(diǎn)是測度為零的零維對象,因此球諧函數(shù)攝動展開方法是在一個測度為零的對象上進(jìn)行攝動處理[57-59]。對于地球、火星等近球形天體,由于級數(shù)收斂較快,球諧函數(shù)攝動展開方法的效果良好[57,70]。而對于小行星951 Gaspra[43-44]、216 Kleopatra[7175]與1620 Geographos[7678]等強(qiáng)不規(guī)則天體[7995]來說,在小行星附近的一些區(qū)域上,級數(shù)發(fā)散,在另外一些區(qū)域上,級數(shù)收斂速度非常慢[96-109]。因此,探索能有效解決這一難題的引力場建模方法成為太陽系小天體幾何形狀與物理特性研究領(lǐng)域的重要內(nèi)容[68-69,96-99]。此后,質(zhì)點(diǎn)群模型[110]與簡單特殊體模型[60-67]被用來嘗試克服這一難題。其中,質(zhì)點(diǎn)群模型對強(qiáng)不規(guī)則天體的建模精確程度遠(yuǎn)好于球諧函數(shù)攝動展開模型,但質(zhì)點(diǎn)群模型計(jì)算量較大,從測度論的意義上講,質(zhì)點(diǎn)群模型的有限個質(zhì)點(diǎn)的測度仍然為零,也是零維的物體[110]。此后由Werner于1994年建立的小行星多面體模型法[68]不僅能克服處理球諧函數(shù)攝動展開模型在特殊區(qū)域發(fā)散的問題,還能仿真出強(qiáng)不規(guī)則天體的幾何外形與質(zhì)量瘤[68-69,111],目前已經(jīng)能采用成千上萬個點(diǎn)與面來對強(qiáng)不規(guī)則天體進(jìn)行幾何與物理建模[112-127]。此外多面體是三維的物體,其測度不為零[112,123,126]。特別是對于雙小行星系統(tǒng),多面體模型法是目前唯一能同時(shí)解決收斂與強(qiáng)不規(guī)則幾何外形仿真的方法[128-130]。

1.1 球諧函數(shù)攝動展開模型

將探測器看作一個質(zhì)點(diǎn),考慮該質(zhì)點(diǎn)在小行星等強(qiáng)不規(guī)則天體引力場中的運(yùn)動。如果質(zhì)點(diǎn)距離強(qiáng)不規(guī)則天體足夠遠(yuǎn),則該天體的引力場可近似看作一個均值圓球產(chǎn)生的引力場,或者說近似看作一個質(zhì)點(diǎn)產(chǎn)生的引力場,因?yàn)榫祱A球的引力場等效于一個質(zhì)點(diǎn)產(chǎn)生的引力場,此時(shí),需要考慮太陽引力對質(zhì)點(diǎn)運(yùn)動的影響。倘若該天體的外形接近球形,例如地球、火星等,則經(jīng)典的Legendre級數(shù)方法可以用來近似其引力場[57-59];并且天體的外形越接近球形,則級數(shù)的收斂速度越快[131-132]。Hu和Scheeres (2002,2004)[133-134]采用Legendre級數(shù)展開的2階項(xiàng)來近似勻速自旋天體的引力場,并分析了考慮2階項(xiàng)時(shí)的軌道運(yùn)動的穩(wěn)定區(qū)域。如果天體的形狀是不規(guī)則的,則天體附近Legendre級數(shù)難以收斂[101,131-132],在一些點(diǎn)或區(qū)域上,Legendre級數(shù)還會發(fā)散[61,63-64]。

1.2 簡單特殊體模型

在強(qiáng)不規(guī)則天體附近動力學(xué)行為研究的早期,人們通過研究簡單特殊體附近的動力學(xué)行為來幫助理解一般強(qiáng)不規(guī)則天體引力場中的可能運(yùn)動狀態(tài)。這些簡單特殊體包括細(xì)直棒[63,66,98100,103,131-132]、圓環(huán)[60,64,106]、圓餅[62,65,102]、三角盤與正方形盤[101]、立方體[67,104-105,107]、啞鈴體[108]等。

細(xì)直棒是一維的物體,具有良好的對稱性,僅用一維坐標(biāo)就可以表示細(xì)直棒上的點(diǎn),然而研究表明,其附近的動力學(xué)行為異常復(fù)雜,棒體之外有4個平衡點(diǎn),大范圍軌道和平衡點(diǎn)周圍的局部軌道均存在共振與混沌的現(xiàn)象[66,99,103]。Elipe和Riaguas (2003)[63]研究了旋轉(zhuǎn)細(xì)直棒所產(chǎn)生的對數(shù)形式引力場中的有效勢與平衡點(diǎn)。Elipe和Lara(2003)[99]將小行星433 Eros的物理模型簡化為細(xì)直棒,并考慮細(xì)直棒產(chǎn)生的引力場中的平衡點(diǎn)、周期軌道族以及分岔行為。Lindner等(2010)[103]發(fā)現(xiàn)細(xì)直棒附近存在穩(wěn)定的同步軌道、一般的混沌軌道、不穩(wěn)定周期軌道、旋轉(zhuǎn)穩(wěn)定軌道等。Najid等(2011)[66]通過計(jì)算Poincaré截面展示了細(xì)直棒附近的復(fù)雜動力學(xué)行為。三角盤、正方形盤、圓環(huán)以及圓餅都是二維的物體,也具有良好的對稱性,用兩維坐標(biāo)系可以表示該物體上的點(diǎn)。通過繪制Poincaré截面知,三角盤和正方形盤引力場中都存在周期軌道,其Poincaré截面上存在不動點(diǎn)和孤島[101]。

1.3 多面體模型

通常的引力多體問題,考慮多個質(zhì)點(diǎn)在相互之間的引力作用下的運(yùn)動,屬于有限個點(diǎn)質(zhì)量產(chǎn)生的引力場,其測度仍然為零;若將有限個點(diǎn)增加到可數(shù)無窮個點(diǎn),乃至連通的不可數(shù)無窮個點(diǎn)形成的點(diǎn)集,研究其引力場中的運(yùn)動,就要考慮一般的非零測集的三維空間的強(qiáng)不規(guī)則天體引力場中的動力學(xué)機(jī)制。Werner(1994)[68]給出了使用均質(zhì)多面體描述不規(guī)則小天體幾何外形與引力場物理模型的方法,并將該方法應(yīng)用到模擬火衛(wèi)一的幾何外形與引力場物理模型中,使用了146個頂點(diǎn)和288個三角形面。Scheeres等(1996)[112]使用小行星4769 Castalia的由雷達(dá)觀測數(shù)據(jù)生成的多面體物理模型,研究了小行星附近的質(zhì)點(diǎn)運(yùn)動,給出并討論了質(zhì)點(diǎn)運(yùn)動的Jacobi積分與零速度面的表達(dá)形式,并計(jì)算了若干個周期軌道族。Werner和Scheeres(1997)[69]進(jìn)一步詳細(xì)推導(dǎo)了常密度多面體外部引力的解析表達(dá),包括引力勢、引力、引力梯度矩陣等,并將其應(yīng)用到小行星4769 Castalia的幾何外形與引力物理模型的建立之中,給出了應(yīng)用多面體處理小行星4769 Castalia質(zhì)量瘤的方法。

將多面體模型同球諧函數(shù)模型和簡單特殊體模型相比,球諧函數(shù)建模[57-59,131-134]方法是在一個點(diǎn)質(zhì)量的基礎(chǔ)上疊加攝動來近似天體,點(diǎn)質(zhì)量的測度為零,不能描述天體的不規(guī)則性,且球諧函數(shù)方法不能解決一些區(qū)域級數(shù)發(fā)散的問題[61,63-64]。簡單特殊體模型[60-67,98-108,131-132]只能用來作為對不規(guī)則天體引力場中動力學(xué)行為的初步探索和理解使用,不存在任何一個小行星是一維的細(xì)直棒、平面盤或者立方體等簡單特殊體形狀[63-64,68-69]。因此,自從Werner給出了使用多面體描述不規(guī)則小行星幾何外形與引力場物理模型的方法[68]之后,多面體模型已經(jīng)在小天體附近動力學(xué)的研究中成為最為先進(jìn)的方法[69,109-127]。此后采用多面體模型對小天體進(jìn)行建模并研究其引力場中的動力學(xué)行為的包括小行星4179 Toutatis[113]、433 Eros[114]、216 Kleopatra[120-125]、2002 AT 4[118]、1989 ML[118]、1620 Geographos[123-124],以及小行星4769 Castalia[123]和6489 Golevka[123]。

2 動力學(xué)機(jī)制研究現(xiàn)狀

截至2014年4月16日,已發(fā)現(xiàn)的太陽系中的小行星共有1 111 087個[135],能使用受攝開普勒軌道理論或平面圓型限制性三體問題理論研究探測器在其引力場中運(yùn)動的天體只有幾十個,這些天體以外形非常接近圓球的太陽和大行星及大行星的大衛(wèi)星為主[96-99,120-124]。太陽系的絕大多數(shù)天體都具有復(fù)雜的強(qiáng)不規(guī)則外形,經(jīng)典的軌道理論適用于描述近球形天體所產(chǎn)生的引力場中的探測器運(yùn)動,而不適用于這些強(qiáng)不規(guī)則天體[9499]。此外,受攝開普勒軌道理論依賴于球諧函數(shù)的引力場模型[57-59]。能否發(fā)展新的理論研究,適用于任何引力場模型,包括球諧函數(shù)模型、其他特殊函數(shù)模型、特殊外形模型、質(zhì)點(diǎn)群模型、多面體模型等,則需研究能描述強(qiáng)不規(guī)則天體引力場中探測器運(yùn)動的理論,適用于所有旋轉(zhuǎn)強(qiáng)不規(guī)則天體引力場中的探測器運(yùn)動,且新的理論研究不依賴于任何具體的引力場模型,而是廣泛適用于任何引力場模型。

作為共性基礎(chǔ)理論研究,開展強(qiáng)不規(guī)則引力場中動力學(xué)的研究,也有助于星系動力學(xué)[60,62,102,136-145]、天體力學(xué)[96-108,146149]、宇航動力學(xué)[109-124]以及行星地質(zhì)動力學(xué)[11-13,150-151]、天體物理散體動力學(xué)[13,21,152-153]、天體物理流體動力學(xué)[154-155]等學(xué)科研究的進(jìn)步。一般來說,小行星和彗核大都具有強(qiáng)不規(guī)則的外形[8-9,12,17]。目前已經(jīng)被作為研究對象來分析其附近動力學(xué)行為的太陽系的強(qiáng)不規(guī)則小天體包括:火衛(wèi)一[68]、小行星4 Vesta[119]、216 Kleopatra[120-125]、433 Eros[114-115,118,126]、1580 Betulia[118]、1620 Geographos[123]、4179 Toutatis[113,118]、4769 Castalia[69,109,112,123]、6489 Golevka[123]、25143 Itokawa[116-117]以及彗星67/ P CG[118]。

不規(guī)則天體附近的運(yùn)動穩(wěn)定性需要考慮Jacobi積分與零速度面,其中零速度面將空間區(qū)域分為質(zhì)點(diǎn)運(yùn)動的禁區(qū)與可行區(qū)域[112,118,120-121,123-124]。平衡點(diǎn)的穩(wěn)定性完全決定了平衡點(diǎn)附近質(zhì)點(diǎn)的運(yùn)動穩(wěn)定性[99,123]。存在一個度量,在該度量下質(zhì)點(diǎn)相對強(qiáng)不規(guī)則體的軌道是流形在等能量超曲面上賦予了該度量的測地線,反之,流形在等能量超曲面上賦予了該度量的測地線是質(zhì)點(diǎn)相對強(qiáng)不規(guī)則體的軌道[123]。下面從平衡點(diǎn)的存在性、個數(shù)與穩(wěn)定性,周期軌道族的存在性,流形與子空間結(jié)構(gòu)、分岔、共振、混沌等幾個方面來介紹國際上相關(guān)領(lǐng)域的研究進(jìn)展。

2.1 周期軌道和擬周期軌道

旋轉(zhuǎn)不規(guī)則天體引力場附近存在周期軌道,這種周期軌道的幾何形狀可能非常復(fù)雜[96,101,112113,118,121,123]。這些周期軌道可以通過不同的方式進(jìn)行分類[112,121,123]。根據(jù)周期軌道的直觀幾何形狀,Scheere等[112]以小行星4769 Castalia為例,將其附近的周期軌道分為三類,包括準(zhǔn)赤道順行(quasi-equatorial direct)周期軌道、準(zhǔn)赤道逆行(quasi-equatorial retrograde)周期軌道以及非赤道(nonequatorial)周期軌道。Broucke和Elipe (2005)[64]依據(jù)軌道的對稱性與直觀幾何外形,將巨形環(huán)引力場中的周期軌道分為10種類型。周期軌道族的尋找是非常復(fù)雜的,在2013年以前,人類只找到了三質(zhì)點(diǎn)體問題的三族周期軌道,Lagrange-Euler族、Broucke-Hénon族以及Moore等于1993年發(fā)現(xiàn)的圖-8族(figure-eight family)[146-147]。?uvakov和Dmitra?inovi于2013年找到了三體問題的13族新的周期解[148],發(fā)表在物理學(xué)領(lǐng)域頂尖雜志Phys.Rev.Lett.上,Science雜志以新聞報(bào)道的方式搶先披露了這一研究進(jìn)展(Jon Cartwright[149])。如果天體從質(zhì)點(diǎn)變?yōu)橐话愕膹?qiáng)不規(guī)則體,則情況更為復(fù)雜。Yu和Baoyin(2012)[121]提出一種周期軌道分層網(wǎng)格搜索算法并以該算法為分類原則,找出小行星216 Kleopatra附近的29族周期軌道。Riaguas等(1999)[96]找到了細(xì)直棒引力場中的多組具有不同穩(wěn)定特性的周期軌道。Jiang等(2014)[123]依據(jù)平衡點(diǎn)附近流形的拓?fù)浣Y(jié)構(gòu)將平衡點(diǎn)附近的周期軌道分為不同的拓?fù)漕愋?對于有的拓?fù)淝樾蔚钠胶恻c(diǎn)的附近還存在擬周期軌道,圖1給出了小行星216 Kleopatra的一個平衡點(diǎn)附近的擬周期軌道,該軌道屬于二維環(huán)面上的擬周期軌道。

通過周期軌道的延拓[121,123]可以由1條周期軌道得到相同拓?fù)涮匦缘臒o窮多條周期軌道。如果將可以延拓得到的周期軌道族中的所有周期軌道捏成1條周期軌道,則開普勒二體問題僅有1條周期軌道,限制性三體問題的單個平衡點(diǎn)附近最多有3條周期軌道,日地系統(tǒng)5個平衡點(diǎn)附近共有12條局部周期軌道,而在小行星6489 Golevka附近的局部周期軌道共有10條[123]。

2.2 平衡點(diǎn)

針對簡單特殊體引力場中平衡點(diǎn)的存在性、個數(shù)與穩(wěn)定性,有若干頗有意思的研究成果。Elipe和Riaguas(2003)[63]找到了旋轉(zhuǎn)對數(shù)函數(shù)引力場及旋轉(zhuǎn)有限長細(xì)直棒引力場中的4個外部平衡點(diǎn),并且討論了這4個平衡點(diǎn)的穩(wěn)定性。Scheeres等(2004)[116-117]發(fā)現(xiàn)了小行星25143 Itokawa引力場中的4個外部平衡點(diǎn)并給出了它們在小行星本體坐標(biāo)系的坐標(biāo)位置。Mondelo等(2010)[119]發(fā)現(xiàn)了小行星4 Vesta引力場中的4個外部平衡點(diǎn),并且給出了它們的坐標(biāo)及穩(wěn)定性;他們的研究表明其中2個平衡點(diǎn)是穩(wěn)定的,另外2個是不穩(wěn)定的。Yu和Baoyin(2012)[120]發(fā)現(xiàn)了小行星216 Kleopatra引力場中的4個外部平衡點(diǎn)并給出了這些平衡點(diǎn)的位置坐標(biāo)、特征值與線性穩(wěn)定性,發(fā)現(xiàn)這4個平衡點(diǎn)附近存在6族不同的周期軌道。Scheeres(2012)[118]發(fā)現(xiàn)了小行星1580 Betulia引力場中的6個外部平衡點(diǎn)以及彗星67P/CG引力場中的4個外部平衡點(diǎn)。

圖1 小行星216 Kleopatra的一個平衡點(diǎn)附近的擬周期軌道[123]Fig.1 A quasi-periodic orbit near an equilibrium point in the potential field of the asteroid 216 Kleopatra[123]

Jiang等(2014)[123]建立了一般的旋轉(zhuǎn)小行星平衡點(diǎn)附近的運(yùn)動理論,包括平衡點(diǎn)附近線性化的運(yùn)動方程、特征方程,平衡點(diǎn)穩(wěn)定的一個充分條件、一個充分必要條件,以及非退化平衡點(diǎn)的拓?fù)浞诸惡妥恿餍谓Y(jié)構(gòu);此外,將旋轉(zhuǎn)不規(guī)則天體引力場中的非退化平衡點(diǎn)分為8種可能的拓?fù)漕愋?其中非退化并且非共振的平衡點(diǎn)有5種拓?fù)漕愋?而非退化并且共振的平衡點(diǎn)有3種拓?fù)漕愋汀７峭嘶⑶曳枪舱竦钠胶恻c(diǎn)的拓?fù)漕愋腿绫?所示[123]。Wang等(2014)[127]計(jì)算了23個有精確外形模型的強(qiáng)不規(guī)則天體的平衡點(diǎn)的個數(shù)及拓?fù)漕愋团c穩(wěn)定性,其中15個小行星、5個大行星的衛(wèi)星、3個彗核;發(fā)現(xiàn)這23個天體均存在內(nèi)部平衡點(diǎn),除小行星216 Kleopatra有3個內(nèi)部平衡點(diǎn)外,其余天體均只有1個內(nèi)部平衡點(diǎn);發(fā)現(xiàn)除了小行星1998 KY26沒有外部平衡點(diǎn)以外,其余天體均有外部平衡點(diǎn),其中101955 Bennu有8個外部平衡點(diǎn),其余天體均有4個外部平衡點(diǎn);發(fā)現(xiàn)小行星4 Vesta、2867 Steins、6489 Golevka、52760,大行星的大衛(wèi)星M1 Phobos、N8 Proteus、S9 Phoebe以及彗核1P/Halley和9P/ Tempel 1的每一個,均有3個平衡點(diǎn)的拓?fù)漕愋蛯儆谇樾?;此外,發(fā)現(xiàn)這23個不規(guī)則天體的外部平衡點(diǎn)若屬于情形1與情形2,則不同拓?fù)漕愋偷钠胶恻c(diǎn)間隔分布,屬于情形1的外部平衡點(diǎn)個數(shù)與屬于情形2的外部平衡點(diǎn)個數(shù)相等;同樣,若這些天體的外部平衡點(diǎn)屬于情形2與情形5,則不同拓?fù)漕愋偷钠胶恻c(diǎn)也間隔分布,屬于情形2的外部平衡點(diǎn)個數(shù)與屬于情形5的外部平衡點(diǎn)個數(shù)相等。圖2和圖3給出了小行星216 Kleopatra及彗核103P/ Hartley的有效勢、平衡點(diǎn)以及平衡點(diǎn)的拓?fù)漕愋蚚123,127]。不同拓?fù)漕愋偷耐獠科胶恻c(diǎn)間隔分布、個數(shù)相等,這一現(xiàn)象對所有的強(qiáng)不規(guī)則天體都成立,還是僅對滿足某些條件的不規(guī)則天體成立,目前尚無定論,該現(xiàn)象的內(nèi)在動力學(xué)機(jī)制需深入研究。

和平衡點(diǎn)有關(guān)的問題如平衡點(diǎn)的個數(shù)、有限性等問題遠(yuǎn)未解決,菲爾茨獎和沃爾夫獎雙獎得主Smale(1998)在文獻(xiàn)[156]中列出了21世紀(jì)的18個數(shù)學(xué)領(lǐng)域的世紀(jì)性問題,其中N-質(zhì)點(diǎn)體的系統(tǒng)中相對平衡點(diǎn)個數(shù)的有限性被列為第6個世紀(jì)性問題,如果將這N個質(zhì)點(diǎn)中的部分或全部換成強(qiáng)不規(guī)則體,則情況將更加復(fù)雜,此時(shí)不規(guī)則體的引力勢和旋轉(zhuǎn)勢相互耦合。

表1 旋轉(zhuǎn)不規(guī)則天體引力場中的非退化并且非共振的平衡點(diǎn)的5種可能的拓?fù)漕愋蚚123]Table 1 Five topological cases of the non-degenerate and nonresonant equilibrium points in the potential field of irregular celestial bodies[123]

2.3 流形

《彖》曰:“大哉乾元、萬物資始,乃統(tǒng)天。云行雨施,品物流形?！绷餍问切D(zhuǎn)強(qiáng)不規(guī)則體引力場中動力學(xué)研究的一個重要方面,其中平衡點(diǎn)附近的流形可以分為漸近穩(wěn)定流形、漸近不穩(wěn)定流形以及中心流形,這些流形分別同漸近穩(wěn)定子空間、漸近不穩(wěn)定子空間以及中心子空間在平衡點(diǎn)處相切[104,119-120,123]。Mondelo等(2010)[119]討論了小行星4 Vesta平衡點(diǎn)附近的周期軌道與對應(yīng)的流形。Liu等(2011)[104]考慮了旋轉(zhuǎn)均質(zhì)立方體引力場中平衡點(diǎn)附近的流形結(jié)構(gòu),給出了漸近穩(wěn)定流形與漸近不穩(wěn)定流形在位置空間的投影,進(jìn)一步計(jì)算了不同平衡點(diǎn)之間的異宿軌道。Yu和Baoyin (2012)[120]以小行星216 Kleopatra為研究對象,計(jì)算了在單參數(shù)變化下,漸近穩(wěn)定流形、漸近不穩(wěn)定流形以及中心流形在位置空間的投影和每一個平衡點(diǎn)附近周期軌道族周期的取值區(qū)間。Jiang等(2014)[123]發(fā)現(xiàn)了一般的旋轉(zhuǎn)簡單形狀體或強(qiáng)不規(guī)則體的非退化平衡點(diǎn)的8種可能的拓?fù)漕愋蛯?yīng)的流形結(jié)構(gòu),包括線性穩(wěn)定類型1種、不穩(wěn)定兼非共振類型4種、以及共振類型3種,并將理論結(jié)果應(yīng)用到小行星216 Kleopatra、1620 Geographos、4769 Castalia和6489 Golevka中。

圖2 小行星216Kelopatra及彗核103P/Hartley 2的有效勢[127]Fig.2 The effective potential of the asteroid 216 Kleopatra and the comet 103P/Hartley 2[127]

圖3 小行星216Kelopatra及彗核103P/Hartley的平衡點(diǎn)及其拓?fù)漕愋蚚123,127](圓點(diǎn)表示屬于拓?fù)淝樾?對應(yīng)的平衡點(diǎn),星形表示拓?fù)淝樾?對應(yīng)的平衡點(diǎn),三角形表示拓?fù)淝樾?對應(yīng)的平衡點(diǎn))Fig.3 The topological cases of the equilibrium points for the asteroid 216 Kleopatra and the comet 103P/Hartley 2[123,127](Dot:Case 1,Star:Case 2;Triangle:Case 5)

2.4 分岔與共振

在強(qiáng)不規(guī)則體的質(zhì)量分布、幾何外形、自旋速度以及運(yùn)動質(zhì)點(diǎn)的Jacobi積分等參數(shù)變化下,運(yùn)動可能表現(xiàn)出分岔行為。Riaguas等(1999)[96]發(fā)現(xiàn)了細(xì)直棒引力場參數(shù)變化下周期軌道的分岔行為。Galán等(2002)[147]分析了三體問題圖-8解的分岔行為。此外,共振平衡點(diǎn)附近也可能會出現(xiàn)分岔[123]。

共振有多種可能的類型:Scheeres等(1996)[112]考慮使用球諧函數(shù)模型來對小行星4769 Castalia的引力場進(jìn)行建模,并分析了2階引力場解析近似引起的運(yùn)動質(zhì)點(diǎn)軌道角速度與小行星自旋角速度成整數(shù)比1∶1及2∶3引起的共振行為。Scheeres等(2000)[114]進(jìn)一步對小行星433Eros使用2階球諧函數(shù)引力場近似分析了3∶2、2∶1及1∶2共振。Yu和Baoyin(2013年)[122]從質(zhì)點(diǎn)機(jī)械能是否突變的角度來考慮,發(fā)現(xiàn)并分析了小行星附近質(zhì)點(diǎn)運(yùn)動的瞬時(shí)共振行為。此外,質(zhì)點(diǎn)軌道角速度與小行星自旋角速度成1∶1的共振軌道同小行星的平衡點(diǎn)重合[119,123]。此外,在慣性空間來看,旋轉(zhuǎn)強(qiáng)不規(guī)則天體的平衡點(diǎn)是天體旋轉(zhuǎn)角速度與質(zhì)點(diǎn)軌道角速度之比為1∶1共振的軌道,Jiang等(2014)[123]發(fā)現(xiàn)的共振平衡點(diǎn)是雙重的共振,包含天體旋轉(zhuǎn)角速度與質(zhì)點(diǎn)軌道角速度之比為1∶1的共振以及平衡點(diǎn)的純虛特征值數(shù)值之比為1∶1的共振。

2.5 混沌

混沌與分岔及共振緊密相連,若干關(guān)于簡單特殊體引力場的研究表明,其中質(zhì)點(diǎn)的運(yùn)動可能表現(xiàn)出混沌行為[99,103,105]。Elipe和Lara(2003)[99]討論了細(xì)直棒引力場中的1∶1共振引起的分岔,發(fā)現(xiàn)在參數(shù)變化下,共振將導(dǎo)致混沌的產(chǎn)生。Lindner等(2010)[103]發(fā)現(xiàn)了繞旋轉(zhuǎn)巨形細(xì)直棒公轉(zhuǎn)的質(zhì)點(diǎn)運(yùn)動的混沌行為,質(zhì)點(diǎn)的運(yùn)動包括穩(wěn)定同步軌道、一般混沌軌道以及不穩(wěn)定周期軌道和自旋穩(wěn)定軌道族。Poincaré截面是一個有效的工具,通過它不僅可以觀察周期軌道的存在性,還能有助于分析質(zhì)點(diǎn)運(yùn)動的混沌行為[64,66,101,105]。Broucke和Elipe(2005)[64]通過計(jì)算固體圓環(huán)引力場中的Poincaré截面,如圖4所示,發(fā)現(xiàn)截面上存在的顯著孤島點(diǎn),對應(yīng)著環(huán)繞著該圓環(huán)的周期軌道。Blesa(2006)[101]分別計(jì)算了平面三角盤及正方形盤引力場中的Poincaré截面,找到了若干條周期軌道。Najid等(2011)[66]計(jì)算了細(xì)直棒引力場中的若干Poincaré截面,展現(xiàn)了其中動力學(xué)行為整體結(jié)構(gòu)的一個概覽。Liu等(2011)[105]通過計(jì)算Poincaré截面找出了旋轉(zhuǎn)立方體引力場中的周期軌道。

圖4 固體圓環(huán)引力場中的Poincaré截面[64]Fig.4 Poincarésurface of sections in the potential field of a solid ring

3 結(jié)束語

預(yù)計(jì)未來強(qiáng)不規(guī)則體引力場中動力學(xué)的研究趨勢為:

1)更加注重理論研究的普適性[123-124]。此前,關(guān)于受攝二體問題軌道理論研究中,有通過J2項(xiàng)得出的太陽同步軌道和臨界軌道等,相關(guān)的理論結(jié)果依賴于球諧函數(shù)攝動展開模型。目前關(guān)于強(qiáng)不規(guī)則天體引力場的建模已有多種方法,未來動力學(xué)行為的研究預(yù)計(jì)將朝著不依賴于具體的建模方法的方向發(fā)展,而是針對一般情況,適用于任何關(guān)于強(qiáng)不規(guī)則天體引力場的建模方法,包括球諧函數(shù)展開、簡單特殊體、多面體等[123124]。

2)研究成果的一般性增強(qiáng)[123]。Kepler二體問題可以看作是限制性三體問題的特例,而使用無質(zhì)量細(xì)長桿將限制性三體問題的兩個天體連接,則可以看作是一個特殊的強(qiáng)不規(guī)則天體,因此限制性三體問題是平面對稱體引力場的特例,而平面對稱體又是一般的強(qiáng)不規(guī)則天體的特例。因此,針對一般的強(qiáng)不規(guī)則天體引力場中軌道運(yùn)動的研究結(jié)論,將適用于簡單特殊體和限制性三體等特例情形[123]。

3)從關(guān)注解析性質(zhì)和外在幾何性質(zhì)到關(guān)注內(nèi)在幾何性質(zhì)、拓?fù)湫再|(zhì)乃至代數(shù)性質(zhì)的轉(zhuǎn)變[123,127]。經(jīng)典的受攝二體問題軌道理論關(guān)注軌道根數(shù)的變化率等解析性質(zhì)以及軌道的幾何外形等外在的幾何性質(zhì)。經(jīng)典的限制性三體問題理論和受攝限制性三體問題理論關(guān)注局部性質(zhì)和解析性質(zhì)。強(qiáng)不規(guī)則天體引力場中的軌道運(yùn)動則更關(guān)注軌道的流形結(jié)構(gòu)等幾何性質(zhì)、平衡點(diǎn)及軌道的種類等拓?fù)湫再|(zhì)等[123-124,127]。

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[責(zé)任編輯:宋宏]

Research Trend of Dynamics in the Gravitational Field of Irregular Celestial Body

JIANG Yu1,2BAOYIN Hexi2
(1.State Key Laboratory of Astronautic Dynamics,Xi'an Satellite Control Center,Xi'an 710043,China; 2.School of Aerospace Engineering,Tsinghua University,Beijing 100084,China)

Both asteroid and comet exploration are important areas in the deep space exploration.The mass of an asteroid or a comet is not big enough,so its gravitational force is much smaller than the stress to satisfy the hydrostatic equilibrium,which makes the minor celestial body irregular-shaped.The research of dynamical behaviours and mechanisms in the gravitational field of irregular celestial body is the basis of minor celestial body exploration,including the catching of the explorer and the design of the orbit around the minor celestial body.This paper summarizes the research progress of the dynamics in the gravitational field of irregular celestial body through gravitational models and dynamical mechanisms.The research situation of gravitational models such as the Legendre polynomial model,the simple-shaped model and the polyhedron model are presented.In addition,the research situation of dynamical mechanisms such as the periodic orbits and quasi-periodic orbits,equilibrium points, manifolds,bifurcations and resonances,chaos,are also presented.Besides,we have analyzed key points and difficult points of these researches.Finally,the research trend of the dynamics in the gravitational field of irregular celestial body is discussed.

irregular celestial body;asteroid exploration;comet exploration;astronautic dynamics;dynamical law

V448.2

:A

:2095-7777(2014)04-0250-12

10.15982/j.issn.2095-7777.2014.04.002

姜宇(1983—),男,博士研究生,工程師,主要研究方向?yàn)椴灰?guī)則天體引力場中的拓?fù)鋭恿ο到y(tǒng)等。

E-mail:jiangyu_xian_china@163.com

寶音賀西(1972—),男,博士,博士生導(dǎo)師,教授,主要研究方向?yàn)楹教炱鬈壍览碚摰取?/p>

E-mail:baoyin@tsinghua.edu.cn

2014-07-30

2014-08-30

國家重點(diǎn)基礎(chǔ)研究發(fā)展計(jì)劃(973)計(jì)劃資助項(xiàng)目(2012CB720000);國家自然科學(xué)基金資助項(xiàng)目(11372150);宇航動力學(xué)國家重點(diǎn)實(shí)驗(yàn)室基金資助項(xiàng)目(2014ADL-DW02)