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萨丕尔沃夫假说

萨丕尔沃夫假说
萨丕尔沃夫假说是一种关于恒星演化的假说。

根据这个假说，恒星在宇宙的演化过程中经历了几个不同的阶段，将其称为“萨丕尔沃夫序列”。

萨丕尔沃夫序列包括以下几个阶段：
1. 主序阶段：恒星的主要阶段是在主序带中度过的。

在这个阶段，恒星的核聚变反应以氢为燃料进行，通过核融合反应将氢转化为氦。

2. 超巨星阶段：当恒星的核心的氢燃料耗尽时，恒星会膨胀成超巨星。

它的外部层会呈现出巨大的体积，并释放出大量的能量。

在这个阶段，恒星的亮度和温度会显著增加。

3. 红巨星阶段：在某些情况下，当恒星核心的氦耗尽时，恒星会进一步膨胀成红巨星。

它的外部层会更大，亮度更高。

4. 氦闪阶段：当恒星进入红巨星阶段时，核心内的氦开始进行核融合反应，这会导致恒星外层的聚变增加，恒星会经历一个“氦闪”阶段。

在这个阶段，恒星外层的存储能量会突然释放，导致恒星的亮度和温度显著增加。

根据萨丕尔沃夫假说，恒星的演化过程受恒星的质量影响。

质量较大的恒星会有更短的寿命，并在更早的阶段进入超巨星和红巨星阶段。

质量较小的恒星会经历更长时间的主序阶段，直到最终变成白矮星或中子星。

萨丕尔沃夫假说在恒星演化理论中起着重要作用，对研究恒星的形成、演化和结构提供了基本框架。

然而，具体的恒星演化仍然是一个复杂的研究领域，需要进一步的观测和理论研究来完善我们对恒星演化的了解。

陕西省宝鸡市2025届高三上学期一模试题语文含答案

陕西省宝鸡市2025年高考模拟检测试题(一)语文试卷（答案在最后）注意事项：1.答卷前，考生务必将自己的姓名、准考证号填写在答题卡上。

2.回答选择题时，选出每小题答案后，用铅笔把答题卡上对应题目的答案标号涂黑。

如需改动，用橡皮擦干净后，再选涂其他答案标号。

回答非选择题时，将答案写在答题卡上。

写在本试卷上无效。

3.考试结束后，将本试卷和答题卡一并交回。

一、现代文阅读(35分)(一)现代文阅读I(本题共5小题，19分)阅读下面的文字，完成1~5题。

材料一：人类自20世纪50年代开始向海洋中排放塑料垃圾，现在每年的排放量已1000万吨左右。

大部分进入海洋的塑料的密度低于海水，这就意味着如果它们能在海洋中存在几年以上，我们应该会发现数千万吨的塑料漂浮在海面上。

这比我们实际看到的要多几个数量级——那么，塑料都到哪里去了呢?“塑料的失踪不像暗物质,而是另一种失踪,”荷兰乌得勒支大学(Utrecht University)的海洋学家和气候科学家埃里克·范·赛伯(Erik van Sebille)说，“没有人知道海底有多少，海滩上有多少，有多少被动物摄入，又有多少已经被细菌降解。

这就是我们面临的谜题。

”无论我们在哪里发现塑料污染，都能看到它对野生动物的影响，要量化这些影响的广泛程度——如果我们想要减轻这些影响，量化是必不可少的一步——我们需要更好地了解海洋塑料是如何分布的，以及被丢弃后的几年或几十年里是如何漂流的。

如果我们想在清理海洋垃圾方面有所成就的话，这一点也是必要的，因为只收集最显眼的1%塑料垃圾可能会没有什么效果。

(摘编自马立克·史蒂芬斯《消失的海洋塑料：它们都去哪儿了?》祝叶华译)材料二：2004年5月7日英国普利茅斯大学的理查德·汤普森教授在《科学》杂志上发表《消失在海洋：塑料垃圾都去哪里了?》，文中首次提出微塑料的概念，最新定义微塑料是指直径小于5毫米的塑料碎片。

钱德拉塞卡(Subrahmanyan.

钱德拉塞卡(Subrahmanyan Chandrasekhar, 1910-1995)因对恒星结构和演化过程的理论研究，特别是对白矮星结构和变化的精确预言，福勒(William A. Fowler, 1911-1995)因创立化学元素起源的核合成理论，共同分享了1983年度诺贝尔物理学奖。

钱德拉塞卡（左图）的主要贡献是发展了白矮星的理论。

白矮星的特性是在1915年由美国天文学家亚当斯 (W. S. Adams) 发现的。

1925年，英国物理学家 R. H. 福勒 (R. H. Fowler) 提出物质简并假说来解释白矮星的巨大密度。

按照这个假说，电子和离子(即电离的原子核)在极大的压力下组成高度密集的物质。

1926年，爱丁顿 (A. S. Eddington)建议，氢转变为氦可能是恒星能量的来源。

这就为恒星演化理论奠定了基础。

在白矮星的研究中，钱德拉塞卡找到了决定恒星生命的基本参数。

他借助于相对论和量子力学，具体地说，是利用简并电子气体的物态方程，为白矮星的演化过程建立了合理的模型，并做出了如下预测：(1)白矮星的质量越大，其半径越小; (2)白矮星的质量不会大于太阳质量的 1.44倍(这个值后来被称为钱德拉塞卡极限)；(3)质量更大的恒星必须通过某些形式的质量转化，也许要经过大爆炸，才能最后归宿为白矮星。

钱德拉塞卡的理论成功地解释了恒星的晚期演化，因此对宇宙学做出了重大贡献。

20世纪30年代末，贝特分别与克里奇菲尔德 (C. Critchfield) 和魏茨塞克 (C. F. Von Weizsacker) 各自独立地提出了太阳和恒星的能源主要来自它们内部的氢通过p – p 链或以12C为催化剂的碳氮（CN）循环燃烧转化为氦。

在考虑了恒星的各种模型之后，贝特指出：p– p链和CN循环中的一系列核反应足以提供恒星的辐射能量，从而帮助天文学家弄清了令人困惑的恒星能源问题。

因此，贝特荣获了1967年度诺贝尔物理学奖。

大胆探索:利用磁白矮星作为宇宙航行的中转站

大胆探索：利用磁白矮星作为宇宙航行的中转站韩厚康;韩厚健【期刊名称】《太空探索》【年(卷),期】2011(000)005【总页数】4页(P52-55)【作者】韩厚康;韩厚健【作者单位】【正文语种】中文太阳系以外的“恒星－行星”系统（艺术想像图。

恒星为红矮星）1995年10月6日，瑞士天文学家首次确认太阳系外有颗行星（其主星“飞马座51”离地球51光年）。

以后随着高解像度光谱学技术的进步，到2009年被发现的系外行星已超过400颗。

2011年2月20日，美国航宇局项目主管威廉·波鲁克奇宣布：上天不到两年的开普勒空间望远镜，在普查银河系局部区域时，新发现了1235颗潜在的行星，其中54颗位于既不冷又不热的“可居带”里（专家们由此推测，银河系“可居带”中的行星可能有5亿多颗）。

此消息使“宇宙航行”、“寻找地外生命，接触地外文明”再度成为科技界和媒体大众的热门话题。

尽管笔者深知飞往“系外行星”困难重重，但受此消息的鼓舞，决定把思考了几年的“利用磁白矮星作为宇宙航行的中转站”写出来，与有兴趣的朋友们讨论交流。

从钱学森“设想”说起1963年2月，中国航天事业奠基人钱学森的专著《星际航行概论》出版。

在书中，钱老精辟地分析了“恒星世界的宇宙航行”，并提出了重要的“设想”，其三个要点如下：1.不能靠化学燃料动力实现宇宙航行。

从太阳系飞到其它恒星，路途极其遥远。

钱老认为：“如果用现在我们能达到的每秒几十千米的速度，就是到最近的几颗恒星附近去也要几万年，这种旅行是不可设想的”。

2. 实现宇宙航行必须大大地增加航行速度，而且必需付出相当大的能量减速。

钱老认为：“要突破太阳系的范围，进入大宇宙，必须大大地增加航行速度，直到接近光的速度”。

为此钱老设想了一个飞向半人马座α星的技术方案：“设想宇宙飞船最高速度为0.80倍光速，而喷气速度为光速的0.60倍。

用二级火箭（每一级的质量比6.24）,一级用来加速火箭到0.80倍光速，一级作刹车之用。

2024湘豫名校联考11月份高三语文试题含答案

湘豫名校联考11月份高三一轮复习诊断考试语文试题阅读下面的文字，完成1～5题。

有一个深奥的问题——宇宙从何而来、如何产生？这个问题催生出宇宙大爆炸理论。

20世纪20年代，俄国科学家亚历山大·弗里德曼和比利时宇宙学家乔治·勒梅特通过求解爱因斯坦引力场方程，发现宇宙是膨胀的，但是当时这样的观念没有被科学界所接受，就连引力场方程的创造者爱因斯坦也极力反对。

这样的僵局直到1929年天才科学家埃德温·哈勃通过天文观测发现确实如此，人们才开始接受宇宙一直在膨胀的事实。

既然如此，回溯到很久以前，宇宙被限制在一个极其狭小的空间内。

换句话说，宇宙起源于一次极其猛烈的大爆炸，也就是说，宇宙是“炸”出来的。

尽管弗里德曼和勒梅特一直都孕育着这一思想，但是正式撰文提出宇宙大爆炸理论的是弗里德曼的学生乔治·伽莫夫。

1948年他和同事们提出了标准的热大爆炸模型。

但即便人们接受宇宙膨胀的事实，伽莫夫的热大爆炸模型在当时也不吃香，强有力的反对者便是大名鼎鼎的英国天文学家弗雷德·霍伊尔，“大爆炸”正是他的嘲讽之词。

伽莫夫提出的热大爆炸模型认为，宇宙开始于高温高密的原初物质，温度超过几十亿度，整个宇宙是各向同性的，物质分布是均匀的。

随着宇宙膨胀，温度和密度逐渐下降，慢慢演化形成了现在的星系等天体。

他们预言大爆炸之后38万年的时候，宇宙已经冷却到电子和原子核结合形成中性原子，这时光子失去碰撞对象电子，成为背景光子（即微波背景辐射），至今依然弥漫在宇宙当中，当前整个世界浸泡在背景光子海洋当中，且背景光子的温度在今天约为几开尔文。

可以说宇宙微波背景辐射是宇宙大爆炸理论的直接证据，能否找到它，对这一理论能否立足至关重要。

幸运的是，1964年美国贝尔实验室的无线电工程师阿诺·彭齐亚斯和罗伯特·威尔逊偶然间发现了宇宙微波背景辐射，这强有力地支持了大爆炸理论。

随后，美国航天局和欧洲宇航局对宇宙微波背景辐射进行了更加精细的探测，如1989年美国发射的微波背景探测者卫星COBE探测到的背景辐射谱是完美的黑体辐射谱，这给宇宙大爆炸理论提供了更有力的证明。

接触双小行星(4179) Toutatis的形成机制研究

接触双小行星(4179) Toutatis的形成机制研究胡寿村【期刊名称】《《天文学报》》【年(卷),期】2019(060)005【总页数】3页(P114-116)【作者】胡寿村【作者单位】中国科学院紫金山天文台南京 210033【正文语种】中文接触双小行星是一类明显由两部分结构相接而构成的单小行星.地面雷达观测结果表明直径大于200 m的近地小行星中大约14%为接触双小行星,而且目前3个小天体探测任务(隼鸟号、嫦娥二号和罗塞塔号)的探测目标也都具有接触双星结构.接触双小行星是一类重要的小行星类型,对其形成机制开展研究能够为深入理解小行星的形成演化提供重要线索.(4179) Toutatis是一颗受关注较多的S型近地小行星,自从1989年发现以来地面雷达和光学望远镜就对该小行星进行了大量观测,发现其具有缓慢的非主轴自转特性,并且从反演的雷达形状模型判断其是一颗接触双小行星.嫦娥二号于2012年12月13日从距离其表面770 m处飞越了该小行星,获得了大量高精度光学图像数据,结果证实Toutatis确实是一颗明显由两部分结构相接组成的接触双小行星,并且在接触位置有一个边缘清晰的直角结构,其体积较大的部分(主瓣)有明显的延长型形状,且两部分的连接点位于主瓣的长轴上.从动力学角度来看Toutatis的形状结构处于一个较不稳定的状态.多数学者猜测Toutatis的形状是由其两个组成部分在低速碰撞下形成,但具体的形成过程,包括如何产生这种不稳定的结构仍不清楚.利用Toutatis的雷达形状模型以及嫦娥二号获得的光学探测数据,本文主要开展了以下3个部分的研究工作.首先,利用嫦娥二号探测数据并结合已有的雷达形状模型,通过轮廓匹配方法获得了Toutatis新的3维形状模型.在此基础上,基于Toutatis的形状参数,假设其母体是一颗主星和卫星均为球体的双轨旋同步双小行星,我们通过采用散体动力学数值模拟方法研究了在变化的参数空间下该双小行星“母体”近距离飞越地球的动力学过程.结果表明地球引力摄动可能导致卫星与主星发生m/s量级的低速相撞,但撞击本身不会对主星的形状产生显著影响; 而在选定合适的飞越距离后(约1.4–1.5倍地球半径),地球潮汐效应在主星和卫星相撞之前会明显改变卫星和主星的形状与自转,并且卫星和主星相撞并合后可以形成类似Toutatis形状的延长型接触双小行星,因此该机制为延长型接触双小行星的形成提供了一种合理的解释.其次,从嫦娥二号获得的Toutatis光学图像上可以发现其表面分布有222颗直径在10 m到61 m之间的碎石以及70多个撞击坑,特别是在主瓣端部有一个直径约800 m的撞击坑.在强度域假设下,基于描述高速撞击溅射物参数分布的标度律方法,我们获得了从800 m撞击坑内抛射出来的溅射物粒子的大小、抛射位置和抛射速度分布,并通过数值模拟研究了这些抛射出来的粒子回落到Toutatis小行星上的比例和位置,结果表明回落粒子的总数目和总体积均明显低于嫦娥二号给出的观测值,这说明Toutatis表面分布的碎石大部分都不来自于撞击,而有可能来源于其形成时候的“母体”.最后,小行星附近的引力场环境可能对小行星的形成演化机制提供重要约束.为了研究从800 m撞击坑内抛射出的溅射物粒子回落到Toutatis表面的位置分布情况,需要对溅射物粒子在小行星引力场影响下的轨道进行外推计算,此时若采用多面体法计算引力场会显著增加计算成本.为了克服这一问题,我们提出应用3维空间中的Chebyshev多项式拟合方法来计算不规则形状小行星附近的引力场.该工作比较了4种引力加速度拟合方式,在充分运用小行星附近引力场变化规律的基础上,提出了一种先沿着地平坐标系方向分解再拟合的处理方式,并且提出采用自适应八叉树算法来减小小行星表面附近的引力场拟合误差.以Toutatis为例的数值算例表明该方法能显著提高引力场计算效率,在精度要求不是特别高的情况下可用该方法做轨道积分.Contact binary asteroids are a kind of single asteroids with bifurcated (or bi-lobed)configurations.Ground-based radar observations have shown that about 14% of near-Earth asteroids larger than 200 meters in diameter may be contact binaries.Specially,the targets of the three minor planet missions,Hayabusa,C hang’e-2,and Rosetta,have contact binary configurations.Contact binaries are an important type of asteroids for completely understanding the formation and evolution of asteroids in the solar system.(4179) Toutatis is a prominent S-type near-Earth asteroid,and has been observed by many ground-based radar and optical campaigns since it was discovered in 1989.These observations reveal that the asteroid is rotating very slowly around a non-principal axis,and its 3D shape model constructed by radar data indicat es that it is a contact binary.Chang’e-2 spacecraft flew by Toutatis on 13 December 2012,with a nearest distance of approximately 770 meters away from its surface.A series of high-resolution optical images taken during the flyby confirm that Toutatis is a distinct contact binary composed of two components.A sharplyperpendicular silhouette is observed near the connection area,the big lobe has an obviously elongated shape,and the connection locates at the long axis of the big lobe.From a dynamical point of view,the configuration of Toutatis is in an unstable state.Many researchers suggest that the Toutatis’ configuration may come from a lowspeed impact between two components,but the detailed formation process that how such an unstable state become realistic is still not well understood.By employing the radarderived shape model and the optical images obtained by Chang’e-2,the research presented in this dissertation mainly includes the following three parts.First,a new 3D shape model of Toutatis is derived by matching the silhouette between the Chang’e-2’s optical images and the existing radar model.With that,we assume the precursor of Toutatis is a doubly synchronous binary asteroid composed of two spherical bodies.The dynamical scenario that the binary precursor closely encounters the Earthis investigated by applying the granular dynamical simulations in a wide parameter space.The results show that the gravitational perturbation of Earth may lead to a m/s-level low-speed impact between the primary and secondary,while the impact has a negligible influence on the shape of the primary.But if we choose an appropriate flyby distance (about 1.4–1.5 Earth radii),the Earth’s tide may significantly alter the shape and spin of the components prior to their mutual impact,and a Toutatis-like contact binary asteroid may be reconstructed after the impact coalescence.This mechanism gives a new reasonable interpretation for the formation ofToutatis-like elongated contact binary asteroids.Second,the optical images obtained by Chang’e-2 show that 222 boulders from 10 m to 61 m across,and as well as over 70 craters,are identified from the optical images obtained by Chang’e-2,especially a depression with a diameter of 800 meters locates in the endpoint of the big lobe.We use a scaling-law method to obtain the size,position,and velocity distribution of ejecta particles from the depression under strength regime.Numerical simulations of the particles’ orbits are conducted,and the re-impact portions and their position distributions are obtained.The results show that the total number and the volume of reimpact ejecta particles are obviously lower than the results identified by the observations of Chang’e-2,which means that most of the boulders on the surface of Toutatis did not come from the crater,but may originate during formation process of the parent body.Finally,we discuss the constraint of the gravitational field nearby an asteroid on its formation and evolution.In order to study the distribution of re-impact ejecta coming from the depression on Toutatis,we need to compute the orbits of these ejecta under the gravitational acceleration of Toutatis.However,the computation cost will be high if we use the polyhedral method to calculate the gravity.To eliminate this problem,we present a new method to calculate the gravity near an irregularly-shaped asteroid by adopting the 3D Chebyshev polynomial interpolation.We have compared four different gravity interpolation methods,and the best one is recommended for the efficiency,in which the gravitational acceleration willbe decomposed along the local horizontal coordinate system,and then interpolated separately.An error-adaptive octree division scheme is also introduced to reduce the interpolation error near the surface of asteroid.We take Toutatis as an example to show that the new method may greatly improve the computation efficiency of near-surface gravitational acceleration,and it may be well used to perform the orbit propagation in case that the precision is not rigorous.。

黑洞的准正模式(quasinormal modes)

Quasi-Normal Modes of Stars and Black HolesKostas D.KokkotasDepartment of Physics,Aristotle University of Thessaloniki,Thessaloniki54006,Greece.kokkotas@astro.auth.grhttp://www.astro.auth.gr/˜kokkotasandBernd G.SchmidtMax Planck Institute for Gravitational Physics,Albert Einstein Institute,D-14476Golm,Germany.bernd@aei-potsdam.mpg.dePublished16September1999/Articles/Volume2/1999-2kokkotasLiving Reviews in RelativityPublished by the Max Planck Institute for Gravitational PhysicsAlbert Einstein Institute,GermanyAbstractPerturbations of stars and black holes have been one of the main topics of relativistic astrophysics for the last few decades.They are of partic-ular importance today,because of their relevance to gravitational waveastronomy.In this review we present the theory of quasi-normal modes ofcompact objects from both the mathematical and astrophysical points ofview.The discussion includes perturbations of black holes(Schwarzschild,Reissner-Nordstr¨o m,Kerr and Kerr-Newman)and relativistic stars(non-rotating and slowly-rotating).The properties of the various families ofquasi-normal modes are described,and numerical techniques for calculat-ing quasi-normal modes reviewed.The successes,as well as the limits,of perturbation theory are presented,and its role in the emerging era ofnumerical relativity and supercomputers is discussed.c 1999Max-Planck-Gesellschaft and the authors.Further information on copyright is given at /Info/Copyright/.For permission to reproduce the article please contact livrev@aei-potsdam.mpg.de.Article AmendmentsOn author request a Living Reviews article can be amended to include errata and small additions to ensure that the most accurate and up-to-date infor-mation possible is provided.For detailed documentation of amendments, please go to the article’s online version at/Articles/Volume2/1999-2kokkotas/. Owing to the fact that a Living Reviews article can evolve over time,we recommend to cite the article as follows:Kokkotas,K.D.,and Schmidt,B.G.,“Quasi-Normal Modes of Stars and Black Holes”,Living Rev.Relativity,2,(1999),2.[Online Article]:cited on<date>, /Articles/Volume2/1999-2kokkotas/. The date in’cited on<date>’then uniquely identiﬁes the version of the article you are referring to.3Quasi-Normal Modes of Stars and Black HolesContents1Introduction4 2Normal Modes–Quasi-Normal Modes–Resonances7 3Quasi-Normal Modes of Black Holes123.1Schwarzschild Black Holes (12)3.2Kerr Black Holes (17)3.3Stability and Completeness of Quasi-Normal Modes (20)4Quasi-Normal Modes of Relativistic Stars234.1Stellar Pulsations:The Theoretical Minimum (23)4.2Mode Analysis (26)4.2.1Families of Fluid Modes (26)4.2.2Families of Spacetime or w-Modes (30)4.3Stability (31)5Excitation and Detection of QNMs325.1Studies of Black Hole QNM Excitation (33)5.2Studies of Stellar QNM Excitation (34)5.3Detection of the QNM Ringing (37)5.4Parameter Estimation (39)6Numerical Techniques426.1Black Holes (42)6.1.1Evolving the Time Dependent Wave Equation (42)6.1.2Integration of the Time Independent Wave Equation (43)6.1.3WKB Methods (44)6.1.4The Method of Continued Fractions (44)6.2Relativistic Stars (45)7Where Are We Going?487.1Synergism Between Perturbation Theory and Numerical Relativity487.2Second Order Perturbations (48)7.3Mode Calculations (49)7.4The Detectors (49)8Acknowledgments50 9Appendix:Schr¨o dinger Equation Versus Wave Equation51Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt41IntroductionHelioseismology and asteroseismology are well known terms in classical astro-physics.From the beginning of the century the variability of Cepheids has been used for the accurate measurement of cosmic distances,while the variability of a number of stellar objects(RR Lyrae,Mira)has been associated with stel-lar oscillations.Observations of solar oscillations(with thousands of nonradial modes)have also revealed a wealth of information about the internal structure of the Sun[204].Practically every stellar object oscillates radially or nonradi-ally,and although there is great diﬃculty in observing such oscillations there are already results for various types of stars(O,B,...).All these types of pulsations of normal main sequence stars can be studied via Newtonian theory and they are of no importance for the forthcoming era of gravitational wave astronomy.The gravitational waves emitted by these stars are extremely weak and have very low frequencies(cf.for a discussion of the sun[70],and an im-portant new measurement of the sun’s quadrupole moment and its application in the measurement of the anomalous precession of Mercury’s perihelion[163]). This is not the case when we consider very compact stellar objects i.e.neutron stars and black holes.Their oscillations,produced mainly during the formation phase,can be strong enough to be detected by the gravitational wave detectors (LIGO,VIRGO,GEO600,SPHERE)which are under construction.In the framework of general relativity(GR)quasi-normal modes(QNM) arise,as perturbations(electromagnetic or gravitational)of stellar or black hole spacetimes.Due to the emission of gravitational waves there are no normal mode oscillations but instead the frequencies become“quasi-normal”(complex), with the real part representing the actual frequency of the oscillation and the imaginary part representing the damping.In this review we shall discuss the oscillations of neutron stars and black holes.The natural way to study these oscillations is by considering the linearized Einstein equations.Nevertheless,there has been recent work on nonlinear black hole perturbations[101,102,103,104,100]while,as yet nothing is known for nonlinear stellar oscillations in general relativity.The study of black hole perturbations was initiated by the pioneering work of Regge and Wheeler[173]in the late50s and was continued by Zerilli[212]. The perturbations of relativistic stars in GR wereﬁrst studied in the late60s by Kip Thorne and his collaborators[202,198,199,200].The initial aim of Regge and Wheeler was to study the stability of a black hole to small perturbations and they did not try to connect these perturbations to astrophysics.In con-trast,for the case of relativistic stars,Thorne’s aim was to extend the known properties of Newtonian oscillation theory to general relativity,and to estimate the frequencies and the energy radiated as gravitational waves.QNMs wereﬁrst pointed out by Vishveshwara[207]in calculations of the scattering of gravitational waves by a Schwarzschild black hole,while Press[164] coined the term quasi-normal frequencies.QNM oscillations have been found in perturbation calculations of particles falling into Schwarzschild[73]and Kerr black holes[76,80]and in the collapse of a star to form a black hole[66,67,68]. Living Reviews in Relativity(1999-2)5Quasi-Normal Modes of Stars and Black Holes Numerical investigations of the fully nonlinear equations of general relativity have provided results which agree with the results of perturbation calculations;in particular numerical studies of the head-on collision of two black holes [30,29](cf.Figure 1)and gravitational collapse to a Kerr hole [191].Recently,Price,Pullin and collaborators [170,31,101,28]have pushed forward the agreement between full nonlinear numerical results and results from perturbation theory for the collision of two black holes.This proves the power of the perturbation approach even in highly nonlinear problems while at the same time indicating its limits.In the concluding remarks of their pioneering paper on nonradial oscillations of neutron stars Thorne and Campollataro [202]described it as “just a modest introduction to a story which promises to be long,complicated and fascinating ”.The story has undoubtedly proved to be intriguing,and many authors have contributed to our present understanding of the pulsations of both black holes and neutron stars.Thirty years after these prophetic words by Thorne and Campollataro hundreds of papers have been written in an attempt to understand the stability,the characteristic frequencies and the mechanisms of excitation of these oscillations.Their relevance to the emission of gravitational waves was always the basic underlying reason of each study.An account of all this work will be attempted in the next sections hoping that the interested reader will ﬁnd this review useful both as a guide to the literature and as an inspiration for future work on the open problems of the ﬁeld.020406080100Time (M ADM )-0.3-0.2-0.10.00.10.20.3(l =2) Z e r i l l i F u n c t i o n Numerical solutionQNM fit Figure 1:QNM ringing after the head-on collision of two unequal mass black holes [29].The continuous line corresponds to the full nonlinear numerical calculation while the dotted line is a ﬁt to the fundamental and ﬁrst overtone QNM.In the next section we attempt to give a mathematical deﬁnition of QNMs.Living Reviews in Relativity (1999-2)K.D.Kokkotas and B.G.Schmidt6 The third and fourth section will be devoted to the study of the black hole and stellar QNMs.In theﬁfth section we discuss the excitation and observation of QNMs andﬁnally in the sixth section we will mention the more signiﬁcant numerical techniques used in the study of QNMs.Living Reviews in Relativity(1999-2)7Quasi-Normal Modes of Stars and Black Holes 2Normal Modes–Quasi-Normal Modes–Res-onancesBefore discussing quasi-normal modes it is useful to remember what normal modes are!Compact classical linear oscillating systems such asﬁnite strings,mem-branes,or cavitiesﬁlled with electromagnetic radiation have preferred time harmonic states of motion(ωis real):χn(t,x)=e iωn tχn(x),n=1,2,3...,(1) if dissipation is neglected.(We assumeχto be some complex valuedﬁeld.) There is generally an inﬁnite collection of such periodic solutions,and the“gen-eral solution”can be expressed as a superposition,χ(t,x)=∞n=1a n e iωn tχn(x),(2)of such normal modes.The simplest example is a string of length L which isﬁxed at its ends.All such systems can be described by systems of partial diﬀerential equations of the type(χmay be a vector)∂χ∂t=Aχ,(3)where A is a linear operator acting only on the spatial variables.Because of theﬁniteness of the system the time evolution is only determined if some boundary conditions are prescribed.The search for solutions periodic in time leads to a boundary value problem in the spatial variables.In simple cases it is of the Sturm-Liouville type.The treatment of such boundary value problems for diﬀerential equations played an important role in the development of Hilbert space techniques.A Hilbert space is chosen such that the diﬀerential operator becomes sym-metric.Due to the boundary conditions dictated by the physical problem,A becomes a self-adjoint operator on the appropriate Hilbert space and has a pure point spectrum.The eigenfunctions and eigenvalues determine the periodic solutions(1).The deﬁnition of self-adjointness is rather subtle from a physicist’s point of view since fairly complicated“domain issues”play an essential role.(See[43] where a mathematical exposition for physicists is given.)The wave equation modeling theﬁnite string has solutions of various degrees of diﬀerentiability. To describe all“realistic situations”,clearly C∞functions should be suﬃcient. Sometimes it may,however,also be convenient to consider more general solu-tions.From the mathematical point of view the collection of all smooth functions is not a natural setting to study the wave equation because sequences of solutionsLiving Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt8 exist which converge to non-smooth solutions.To establish such powerful state-ments like(2)one has to study the equation on certain subsets of the Hilbert space of square integrable functions.For“nice”equations it usually happens that the eigenfunctions are in fact analytic.They can then be used to gen-erate,for example,all smooth solutions by a pointwise converging series(2). The key point is that we need some mathematical sophistication to obtain the “completeness property”of the eigenfunctions.This picture of“normal modes”changes when we consider“open systems”which can lose energy to inﬁnity.The simplest case are waves on an inﬁnite string.The general solution of this problem isχ(t,x)=A(t−x)+B(t+x)(4) with“arbitrary”functions A and B.Which solutions should we study?Since we have all solutions,this is not a serious question.In more general cases, however,in which the general solution is not known,we have to select a certain class of solutions which we consider as relevant for the physical problem.Let us consider for the following discussion,as an example,a wave equation with a potential on the real line,∂2∂t2χ+ −∂2∂x2+V(x)χ=0.(5)Cauchy dataχ(0,x),∂tχ(0,x)which have two derivatives determine a unique twice diﬀerentiable solution.No boundary condition is needed at inﬁnity to determine the time evolution of the data!This can be established by fairly simple PDE theory[116].There exist solutions for which the support of theﬁelds are spatially compact, or–the other extreme–solutions with inﬁnite total energy for which theﬁelds grow at spatial inﬁnity in a quite arbitrary way!From the point of view of physics smooth solutions with spatially compact support should be the relevant class–who cares what happens near inﬁnity! Again it turns out that mathematically it is more convenient to study all solu-tions ofﬁnite total energy.Then the relevant operator is again self-adjoint,but now its spectrum is purely“continuous”.There are no eigenfunctions which are square integrable.Only“improper eigenfunctions”like plane waves exist.This expresses the fact that weﬁnd a solution of the form(1)for any realωand by forming appropriate superpositions one can construct solutions which are “almost eigenfunctions”.(In the case V(x)≡0these are wave packets formed from plane waves.)These solutions are the analogs of normal modes for inﬁnite systems.Let us now turn to the discussion of“quasi-normal modes”which are concep-tually diﬀerent to normal modes.To deﬁne quasi-normal modes let us consider the wave equation(5)for potentials with V≥0which vanish for|x|>x0.Then in this case all solutions determined by data of compact support are bounded: |χ(t,x)|<C.We can use Laplace transformation techniques to represent such Living Reviews in Relativity(1999-2)9Quasi-Normal Modes of Stars and Black Holes solutions.The Laplace transformˆχ(s,x)(s>0real)of a solutionχ(t,x)isˆχ(s,x)= ∞0e−stχ(t,x)dt,(6) and satisﬁes the ordinary diﬀerential equations2ˆχ−ˆχ +Vˆχ=+sχ(0,x)+∂tχ(0,x),(7) wheres2ˆχ−ˆχ +Vˆχ=0(8) is the homogeneous equation.The boundedness ofχimplies thatˆχis analytic for positive,real s,and has an analytic continuation onto the complex half plane Re(s)>0.Which solutionˆχof this inhomogeneous equation gives the unique solution in spacetime determined by the data?There is no arbitrariness;only one of the Green functions for the inhomogeneous equation is correct!All Green functions can be constructed by the following well known method. Choose any two linearly independent solutions of the homogeneous equation f−(s,x)and f+(s,x),and deﬁneG(s,x,x )=1W(s)f−(s,x )f+(s,x)(x <x),f−(s,x)f+(s,x )(x >x),(9)where W(s)is the Wronskian of f−and f+.If we denote the inhomogeneity of(7)by j,a solution of(7)isˆχ(s,x)= ∞−∞G(s,x,x )j(s,x )dx .(10) We still have to select a unique pair of solutions f−,f+.Here the information that the solution in spacetime is bounded can be used.The deﬁnition of the Laplace transform implies thatˆχis bounded as a function of x.Because the potential V vanishes for|x|>x0,the solutions of the homogeneous equation(8) for|x|>x0aref=e±sx.(11) The following pair of solutionsf+=e−sx for x>x0,f−=e+sx for x<−x0,(12) which is linearly independent for Re(s)>0,gives the unique Green function which deﬁnes a bounded solution for j of compact support.Note that for Re(s)>0the solution f+is exponentially decaying for large x and f−is expo-nentially decaying for small x.For small x however,f+will be a linear com-bination a(s)e−sx+b(s)e sx which will in general grow exponentially.Similar behavior is found for f−.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt 10Quasi-Normal mode frequencies s n can be deﬁned as those complex numbers for whichf +(s n ,x )=c (s n )f −(s n ,x ),(13)that is the two functions become linearly dependent,the Wronskian vanishes and the Green function is singular!The corresponding solutions f +(s n ,x )are called quasi eigenfunctions.Are there such numbers s n ?From the boundedness of the solution in space-time we know that the unique Green function must exist for Re (s )>0.Hence f +,f −are linearly independent for those values of s .However,as solutions f +,f −of the homogeneous equation (8)they have a unique continuation to the complex s plane.In [35]it is shown that for positive potentials with compact support there is always a countable number of zeros of the Wronskian with Re (s )<0.What is the mathematical and physical signiﬁcance of the quasi-normal fre-quencies s n and the corresponding quasi-normal functions f +?First of all we should note that because of Re (s )<0the function f +grows exponentially for small and large x !The corresponding spacetime solution e s n t f +(s n ,x )is therefore not a physically relevant solution,unlike the normal modes.If one studies the inverse Laplace transformation and expresses χas a com-plex line integral (a >0),χ(t,x )=12πi +∞−∞e (a +is )t ˆχ(a +is,x )ds,(14)one can deform the path of the complex integration and show that the late time behavior of solutions can be approximated in ﬁnite parts of the space by a ﬁnite sum of the form χ(t,x )∼N n =1a n e (αn +iβn )t f +(s n ,x ).(15)Here we assume that Re (s n +1)<Re (s n )<0,s n =αn +iβn .The approxi-mation ∼means that if we choose x 0,x 1, and t 0then there exists a constant C (t 0,x 0,x 1, )such that χ(t,x )−N n =1a n e (αn +iβn )t f +(s n ,x ) ≤Ce (−|αN +1|+ )t (16)holds for t >t 0,x 0<x <x 1, >0with C (t 0,x 0,x 1, )independent of t .The constants a n depend only on the data [35]!This implies in particular that all solutions deﬁned by data of compact support decay exponentially in time on spatially bounded regions.The generic leading order decay is determined by the quasi-normal mode frequency with the largest real part s 1,i.e.slowest damping.On ﬁnite intervals and for late times the solution is approximated by a ﬁnite sum of quasi eigenfunctions (15).It is presently unclear whether one can strengthen (16)to a statement like (2),a pointwise expansion of the late time solution in terms of quasi-normal Living Reviews in Relativity (1999-2)11Quasi-Normal Modes of Stars and Black Holes modes.For one particular potential(P¨o schl-Teller)this has been shown by Beyer[42].Let us now consider the case where the potential is positive for all x,but decays near inﬁnity as happens for example for the wave equation on the static Schwarzschild spacetime.Data of compact support determine again solutions which are bounded[117].Hence we can proceed as before.Theﬁrst new point concerns the deﬁnitions of f±.It can be shown that the homogeneous equation(8)has for each real positive s a unique solution f+(s,x)such that lim x→∞(e sx f+(s,x))=1holds and correspondingly for f−.These functions are uniquely determined,deﬁne the correct Green function and have analytic continuations onto the complex half plane Re(s)>0.It is however quite complicated to get a good representation of these func-tions.If the point at inﬁnity is not a regular singular point,we do not even get converging series expansions for f±.(This is particularly serious for values of s with negative real part because we expect exponential growth in x).The next new feature is that the analyticity properties of f±in the complex s plane depend on the decay of the potential.To obtain information about analytic continuation,even use of analyticity properties of the potential in x is made!Branch cuts may occur.Nevertheless in a lot of cases an inﬁnite number of quasi-normal mode frequencies exists.The fact that the potential never vanishes may,however,destroy the expo-nential decay in time of the solutions and therefore the essential properties of the quasi-normal modes.This probably happens if the potential decays slower than exponentially.There is,however,the following way out:Suppose you want to study a solution determined by data of compact support from t=0to some largeﬁnite time t=T.Up to this time the solution is–because of domain of dependence properties–completely independent of the potential for suﬃciently large x.Hence we may see an exponential decay of the form(15)in a time range t1<t<T.This is the behavior seen in numerical calculations.The situation is similar in the case ofα-decay in quantum mechanics.A comparison of quasi-normal modes of wave equations and resonances in quantum theory can be found in the appendix,see section9.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt123Quasi-Normal Modes of Black HolesOne of the most interesting aspects of gravitational wave detection will be the connection with the existence of black holes[201].Although there are presently several indirect ways of identifying black holes in the universe,gravitational waves emitted by an oscillating black hole will carry a uniqueﬁngerprint which would lead to the direct identiﬁcation of their existence.As we mentioned earlier,gravitational radiation from black hole oscillations exhibits certain characteristic frequencies which are independent of the pro-cesses giving rise to these oscillations.These“quasi-normal”frequencies are directly connected to the parameters of the black hole(mass,charge and angu-lar momentum)and for stellar mass black holes are expected to be inside the bandwidth of the constructed gravitational wave detectors.The perturbations of a Schwarzschild black hole reduce to a simple wave equation which has been studied extensively.The wave equation for the case of a Reissner-Nordstr¨o m black hole is more or less similar to the Schwarzschild case,but for Kerr one has to solve a system of coupled wave equations(one for the radial part and one for the angular part).For this reason the Kerr case has been studied less thoroughly.Finally,in the case of Kerr-Newman black holes we face the problem that the perturbations cannot be separated in their angular and radial parts and thus apart from special cases[124]the problem has not been studied at all.3.1Schwarzschild Black HolesThe study of perturbations of Schwarzschild black holes assumes a small per-turbation hµνon a static spherically symmetric background metricds2=g0µνdxµdxν=−e v(r)dt2+eλ(r)dr2+r2 dθ2+sin2θdφ2 ,(17) with the perturbed metric having the formgµν=g0µν+hµν,(18) which leads to a variation of the Einstein equations i.e.δGµν=4πδTµν.(19) By assuming a decomposition into tensor spherical harmonics for each hµνof the formχ(t,r,θ,φ)= mχ m(r,t)r Y m(θ,φ),(20)the perturbation problem is reduced to a single wave equation,for the func-tionχ m(r,t)(which is a combination of the various components of hµν).It should be pointed out that equation(20)is an expansion for scalar quantities only.From the10independent components of the hµνonly h tt,h tr,and h rr transform as scalars under rotations.The h tθ,h tφ,h rθ,and h rφtransform asLiving Reviews in Relativity(1999-2)13Quasi-Normal Modes of Stars and Black Holes components of two-vectors under rotations and can be expanded in a series of vector spherical harmonics while the components hθθ,hθφ,and hφφtransform as components of a2×2tensor and can be expanded in a series of tensor spher-ical harmonics(see[202,212,152]for details).There are two classes of vector spherical harmonics(polar and axial)which are build out of combinations of the Levi-Civita volume form and the gradient operator acting on the scalar spherical harmonics.The diﬀerence between the two families is their parity. Under the parity operatorπa spherical harmonic with index transforms as (−1) ,the polar class of perturbations transform under parity in the same way, as(−1) ,and the axial perturbations as(−1) +11.Finally,since we are dealing with spherically symmetric spacetimes the solution will be independent of m, thus this subscript can be omitted.The radial component of a perturbation outside the event horizon satisﬁes the following wave equation,∂2∂t χ + −∂2∂r∗+V (r)χ =0,(21)where r∗is the“tortoise”radial coordinate deﬁned byr∗=r+2M log(r/2M−1),(22) and M is the mass of the black hole.For“axial”perturbationsV (r)= 1−2M r ( +1)r+2σMr(23)is the eﬀective potential or(as it is known in the literature)Regge-Wheeler potential[173],which is a single potential barrier with a peak around r=3M, which is the location of the unstable photon orbit.The form(23)is true even if we consider scalar or electromagnetic testﬁelds as perturbations.The parameter σtakes the values1for scalar perturbations,0for electromagnetic perturbations, and−3for gravitational perturbations and can be expressed asσ=1−s2,where s=0,1,2is the spin of the perturbingﬁeld.For“polar”perturbations the eﬀective potential was derived by Zerilli[212]and has the form V (r)= 1−2M r 2n2(n+1)r3+6n2Mr2+18nM2r+18M3r3(nr+3M)2,(24)1In the literature the polar perturbations are also called even-parity because they are characterized by their behavior under parity operations as discussed earlier,and in the same way the axial perturbations are called odd-parity.We will stick to the polar/axial terminology since there is a confusion with the deﬁnition of the parity operation,the reason is that to most people,the words“even”and“odd”imply that a mode transforms underπas(−1)2n or(−1)2n+1respectively(for n some integer).However only the polar modes with even have even parity and only axial modes with even have odd parity.If is odd,then polar modes have odd parity and axial modes have even parity.Another terminology is to call the polar perturbations spheroidal and the axial ones toroidal.This deﬁnition is coming from the study of stellar pulsations in Newtonian theory and represents the type ofﬂuid motions that each type of perturbation induces.Since we are dealing both with stars and black holes we will stick to the polar/axial terminology.Living Reviews in Relativity(1999-2)K.D.Kokkotas and B.G.Schmidt14where2n=( −1)( +2).(25) Chandrasekhar[54]has shown that one can transform the equation(21)for “axial”modes to the corresponding one for“polar”modes via a transforma-tion involving diﬀerential operations.It can also be shown that both forms are connected to the Bardeen-Press[38]perturbation equation derived via the Newman-Penrose formalism.The potential V (r∗)decays exponentially near the horizon,r∗→−∞,and as r−2∗for r∗→+∞.From the form of equation(21)it is evident that the study of black hole perturbations will follow the footsteps of the theory outlined in section2.Kay and Wald[117]have shown that solutions with data of compact sup-port are bounded.Hence we know that the time independent Green function G(s,r∗,r ∗)is analytic for Re(s)>0.The essential diﬃculty is now to obtain the solutions f±(cf.equation(10))of the equations2ˆχ−ˆχ +Vˆχ=0,(26) (prime denotes diﬀerentiation with respect to r∗)which satisfy for real,positives:f+∼e−sr∗for r∗→∞,f−∼e+r∗x for r∗→−∞.(27) To determine the quasi-normal modes we need the analytic continuations of these functions.As the horizon(r∗→∞)is a regular singular point of(26),a representation of f−(r∗,s)as a converging series exists.For M=12it reads:f−(r,s)=(r−1)s∞n=0a n(s)(r−1)n.(28)The series converges for all complex s and|r−1|<1[162].(The analytic extension of f−is investigated in[115].)The result is that f−has an extension to the complex s plane with poles only at negative real integers.The representation of f+is more complicated:Because inﬁnity is a singular point no power series expansion like(28)exists.A representation coming from the iteration of the deﬁning integral equation is given by Jensen and Candelas[115],see also[159]. It turns out that the continuation of f+has a branch cut Re(s)≤0due to the decay r−2for large r[115].The most extensive mathematical investigation of quasi-normal modes of the Schwarzschild solution is contained in the paper by Bachelot and Motet-Bachelot[35].Here the existence of an inﬁnite number of quasi-normal modes is demonstrated.Truncating the potential(23)to make it of compact support leads to the estimate(16).The decay of solutions in time is not exponential because of the weak decay of the potential for large r.At late times,the quasi-normal oscillations are swamped by the radiative tail[166,167].This tail radiation is of interest in its Living Reviews in Relativity(1999-2)。

数学-科学的王后与仆人

数学: 科学的王后和仆人Mathematics: Queen and Servant of Science北京理工大学叶其孝本文的题目是已故的美国科学院院士、著名数学家、数学史学家和科普作家Eric Temple Bell(贝尔, 1883, 02, 07 ~ 1960, 12, 21)于1951年写的一本书的书名Mathematics: Queen and Servant of Science (数学: 科学的王后和仆人). 该书主要是为大学生和非数学领域的人士写的, 介绍纯粹和应用数学的各个方面, 更着重在说明数学科学的极端重要性.The Mathematical Association of America, 1996, 463 pages实际上这是他1931年写的The Queen of the Sciences (科学的王后)和1937年写的The Handmaiden of the Sciences (科学的女仆)这两本通俗数学论著的合一修订扩大版.Eric Temple Bell Alexander Graham Bell (1847 ~ 1922) 按常识的理解, 女王是优美、高雅、无懈可击、至尊至贵的, 在科学中只有纯粹数学才具有这样的特点, 简洁明了的数学定理一经证明就是永恒的真理, 极其优美而且无懈可击;另一方面, 科学和工程的各个分支都在不同程度上大量应用数学, 这时数学科学就是仆人, 这些仆人是否强有力, 用起来是否得心应手是雇佣这些仆人的主人最为关心的事. 事实上, servant这个字本身就有“供人们利用之物, 有用的服务工具”的意思. 毫无疑问, 我们的目的不是为数学争一个好的名分, 而是想说明数学是怎样通过数学建模来解决各种实际问题的; 数学(数学建模)的极端重要性, 以及探讨正确认识和理解数学科学的作用对于发展我国科学技术、经济以及教育, 从而争取在21世纪把我国真正建设成为屹立于世界民族之林的强国，乃至个人事业发展的至关重要性. 当然, 我们也希望说明王后和仆人集于一身并不矛盾. 历史上, 很多特别受人尊敬的科学家, 不仅仅是由于他们的科学成就, 更因为他们的科学成就能够服务于人类.数学是科学的王后, 算术是数学的王后. 她常常放下架子为天文学和其他科学效劳, 但是在所有情况下, 第一位的是她(数学)应尽的责任. (高斯)Mathematics is the Queen of the Sciences, and Arithmetic the Queen of Mathematics. She often condescends to render service to astronomy and other natural sciences, but under all circumstance the first place is her due.— Carl Friedrich Gauss (卡尔·弗里德里希·高斯, 1777, 4, 30 ~ 1855, 2, 23)From: Bell, Eric T., Mathematics: Queen and Servant of Science, MAA, 1951, p.1;Men of Mathematics, Simon and Schuster, New York, 1937, p. xv.***************************************************自古以来，数学的发展始终与科学技术的发展紧密相连，反之亦然. 首先, 我们来看一下导致我们现在这个飞速发展的信息社会的19、20世纪几乎所有重大科学理论的发展和完善过程中数学(数学建模)所起到的不可勿缺的作用.数学研究的成果往往是重大科学发明的催生素(仅就19、20世纪而言, 流体力学、电磁理论、相对论、量子力学、计算机、信息论、控制论、现代经济学、万维网和互联网搜索引擎、生物学、CT、甚至社会政治学领域等). 但是20世纪上半世纪, 数学虽然也直接为工程技术提供一些工具, 但基本方式是间接的: 先促进其他科学的发展, 再由这些科学提供工程原理和设计的基础. 数学是幕后的无名英雄.现在, 数学无处不在, 数学和工程技术之间,在更广阔的范围内和更深刻的程度上, 直接地相互作用着, 极大地推动了科学和工程科学的发展, 也极大地推动了技术的发展. 数学不仅是幕后的无名英雄, 很多方面开始走向“前台”. 但是对数学的极端重要性迄今尚未有共识, 取得共识对加强一个国家的竞争力来说是至关重要的.硬能力―一位美国朋友谈及对未来中国人的看法: 20年后, 中国年轻人会丢了中国人现在的硬能力, 他们崇拜各种明星, 不愿献身科学, 不再以学术研究为荣, 聪明拔尖的学生都去学金融、法律等赚钱的专业; 而美国人因为认识到其硬能力(例如数学)不行, 进行教育改革, 20年后, 不但保持了其软实力即非专业能力的优势, 而且在硬能力上赶上中国人.‖“正在丢失的硬实力”, 鲁鸣, 《青年文摘》2011年第5期动向：美国很多州新办STEM高中, 一些大学开始开设STEM课程等.STEM = Science + Technology + Engineering + Mathematics2012年2月7日公布的美国总统科技顾问委员会给总统的报告，参与超越：培养额外的100万具有科学、技术、工程和数学学位的大学生(Engage to Excel: Producing One Million Additional College Graduates with Degrees in Science, Technology, Engineering, and Mathematics)The Mathematical Sciences in 2025, the National Academies Press, 2013人们使用的数学科学思想、概念和方法的范围在不断扩大的同时，数学科学的用途也在不断扩展. 21世纪的大部分科学与工程将建立在数学科学的基础上.This major expansion in the uses of the mathematical sciences has been paralleled by a broadening in the range of mathematical science ideas and techniques being used. Much of twenty-first century science and engineering is going to be built on a mathematical science foundation, and that foundation must continue to evolve and expand.数学科学是日常生活的几乎每个方面的组成部分.互联网搜索、医疗成像、电脑动画、数值天气预报和其他计算机模拟、所有类型的数字通信、商业和军事中的优化问题以及金融风险的分析——普通公民都从支撑这些应用功能的数学科学的各种进展中获益，这样的例子不胜枚举.The mathematical sciences are part of almost every aspect of everyday life. Internet search, medical imaging, computer animation, numerical weather predictions and othercomputer simulations, digital communications of all types, optimization in business and the military, analyses of financial risks —average citizens all benefit from the mathematical science advances that underpin these capabilities, and the list goes on and on.调查发现:数学科学研究工作正日益成为生物学、医学、社会科学、商业、先进设计、气候、金融、先进材料等许多研究领域不可或缺的重要组成部分. 这种研究工作涉及最广泛意义下数学、统计学和计算综合，以及这些领域与潜在应用领域的相互作用. 所有这些活动对于经济增长、国家竞争力和国家安全都是至关重要的，而且这种事实应该对作为整体的数学科学的资助性质和资助规模产生影响. 数学科学的教育也应该反映数学科学领域的新的状况.Finding: Mathematical sciences work is becoming an increasingly integral and essential component of a growing array of areas of investigation in biology, medicine, social sciences, business, advanced design, climate, finance, advanced materials, and many more. This work involves the integration of mathematics, statistics, and computation in the broadest sense and the interplay of these areas withareas of potential application. All of these activities are crucial to economic growth, national competitiveness, and national security, and this fact should inform both the nature and scale of funding for the mathematical sciences as a whole. Education in the mathematical sciences should also reflect this new stature of the field.****************************************************************为了以下讲述的方便, 我们先来了解一下什么是数学建模.数学模型(Mathematical Model)是用数学符号对一类实际问题或实际发生的现象的(近似的)描述.数学建模(Mathematical Modeling)则是获得该模型并对之求解、验证并得到结论的全过程.数学建模不仅是了解基本规律, 而且从应用的观点来看更重要的是预测和控制所建模的系统的行为的强有力的工具.数学建模是数学用来解决各种实际问题的桥梁.↑→→→→→→→→↓↑↓↑↓↓↑↓←←←←←通不过↓↓通过)定义：数学建模就是上述框图多次执行的过程数学建模的难点观察、分析实际问题, 作出合理的假设, 明确变量和参数, 形成明确的数学问题. 不仅仅是翻译的问题; 涉及的数学问题可能是复杂、困难的, 求解也许涉及深刻的数学方法. 如何作出正确的判断, 寻找合适、简洁的(解析或近似) 解法; 如何验证模型.简言之:合理假设、模型建立、模型求解、解释验证.记住这16个字, 将会终生受用.数学建模的重要作用：源头创新当然数学建模也有局限性, 不能单独包打天下, 因为实际问题是非常复杂的, 需要多学科协同解决.在图灵(A. M. Turing)的文章: The Chemical Basis of Morphogenesis (形态生成的化学基础), Philosophical Transactions of the Royal Society of London (伦敦皇家学会哲学公报), Series B (Biological Sciences),v.237(1952), 37-72.1. 一个胚胎的模型. 成形素本节将描述一个正在生长的胚胎的数学模型. 该模型是一种简化和理想化, 因此是对原问题的篡改. 希望本文论述中保留的一些特征, 就现今的知识状况而言, 是那些最重要的特征.1. A model of the embryo. MorphogensIn this section a mathematical model of the growing embryo will be described. This model will be asimplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge.想单靠数学建模本身来解决重大的生物学问题是不可能的，另一方面，想仅仅依靠实验来获得对生物学的合理、完整的理解也是极不可能的. There is no way mathematical modeling can solve major biological problems on its own. On the other hand, it ishighly unlikely that even a reasonably complete understanding could come solely from experiment.—— J. D. Murray, Why Are There No 3-Headed Monsters? Mathematical Modeling in Biology, Notices of the AMS,v. 59 (2012), no. 6, p.793.自古以来公平、公正的竞赛都是培养、选拔人才的重要手段, 科学和数学也不例外.中学生IMO (国际数学奥林匹克(International Mathematical Olympiad), 1959 ～)北美的大学生Putnbam数学竞赛(1938 ～)全国大学生数学竞赛(2010 ～)Mathematical Contest in Modeling (MCM, 1985 ～)美国大学生数学建模竞赛Interdisciplinary Contest in Modeling (ICM, 1999～)美国大学生跨学科建模竞赛China Undergraduate Mathematical Contest in Modeling (CUMCM, 1992～) 中国大学生数学建模竞赛中国大学生参加美国大学生数学建模竞赛情况中国大学生数学建模竞赛情况在以下讲述中涉及物理方面的具体的数学模型 (问题)的叙述和初步讨论可参考《物理学与偏微分方程》, 李大潜、秦铁虎编著, (上册, 1997; 下册, 2000), 高等教育出版社.Seven equations that rule your world (主宰你生活的七个方程式), by Ian Stewart, NewScientist, 13 February 2012.Fourier transformation 2ˆ()()ix f f x e dx πξξ∞--∞=⎰Wave equation 22222u u c t x ∂∂=∂∂ Ma xwell‘s equation110, , 0, H E E E H H c t c t∂∂∇⋅=∇⨯=-∇⋅=∇⨯=∂∂Schrödinger‘s equation ˆψH ψi t∂=∂Ian Stewart, In Pursuit of the Unknown:17 Equations That Changed the World (追求对未知的认识：改变世界的17个方程), Basic Books, March 13, 2012.目录(Contents)Why Equations? /viii1. The squaw on the hippopotamus ——Pythagoras‘sTheorem/12. Shortening the proceedings —— Logarithms/213. Ghosts of departed quantities —— Calculus/354. The system of the world ——Newton‘s Law ofGravity/535. Portent of the ideal world —— The Square Root ofMinus One/736. Much ado about knotting ——Euler‘s Formula forPolyhedra/837. Patterns of chance —— Normal Distribution/1078. Good vibrations —— Wave Equation/1319. Ripples and blips —— Fourier Transform/14910. The ascent of humanity —— Navier-StokesEquation/16511. Wave in the ether ——Maxwell‘s Equations/17912. Law and disorder —— Second Law ofThermodynamics /19513. One thing is absolute —— Relativity/21714. Quantum weirdness —— Schrödinger Equation/24515. Codes, communications, and computers ——Information Theory/26516. The imbalance of nature —— Chaos Theory/28317. The Midas formula —— Black-Scholes Equation/195Where Next?/317Notes/321Illustration Credits/330Index/331相对论Albert Einstein(1879, 3, 14 ～1955, 4, 18)20世纪最伟大的科学成就莫过于Einstein(爱因斯坦)的狭义和广义相对论了, 但是如果没有Minkowski (闵可夫斯基)几何、Riemann(黎曼)于1854年发明的Riemann几何, 以及Cayley(凯莱), Sylvester(西勒维斯特)和Noether(诺特)等数学家发展的不变量理论, Einstein的广义相对论和引力理论就不可能有如此完善的数学表述. Einstein自己也不止一次地说过.早在1905年, 年仅26岁的爱因斯坦就已提出了狭义相对论. 狭义相对论推倒了牛顿力学的质量守恒、能量守恒、质量能量互不相关、时空永恒不变的基本命题. 这是一场真正的科学革命.为了导出狭义相对论，爱因斯坦作出了两个假设：运动的相对性(所有匀速运动都是相对的)和光速为常数(光的运动例外, 它是绝对的). (1)狭义相对性原理,即在所有惯性系中, 物理学定律具有相同的数学表达形式;(2)光速不变原理,真空中光沿各个方向传播的速率都相等，与光源和观察者的运动状态无关.时空不是绝对独立的.由此可以导出一些推论: 相对论坐标变换式和速度变换式, 同时的相对性, 钟慢尺缩效应和质能关系式等.他的好友物理学家P.Ehrenfest指出实际上还蕴涵着第三个假设, 即这两个假设是不矛盾的. 物体运动的相对性和光速的绝对性, 两者之间的相互制约和作用乃是相对论里一切我们不熟悉的时空特征的根源.(部分参阅李新洲:《寻找自然之律--- 20世纪物理学革命》, 上海科技教育出版社, 2001.)1907 年德国数学家H. Minkowski (1864 ～1909) 提出了―Minkowski 空间‖,即把时间和空间融合在一起的四维空间1,3R. Minkowski 几何为Einstein 狭义相对论提供了合适的数学模型.“没有任何客观合理的方法能够把四维连续统分离成三维空间连续统和一维时间连续统. 因此从逻辑上讲, 在四维时空连续统(space- time continuum)中表述自然定律会更令人满意. 相对论在方法上的巨大进步正是建立在这个基础之上的, 这种进步归功于闵可夫斯基(Minkowski).”—Albert Einstein, The Meaning of Relativity, 1922, Princeton University Press. 中译本, 阿尔伯特·爱因斯坦著, 相对论的意义, (普林斯顿科学文库(Princeton Science Library) 1), 郝建纲、刘道军译, 上海科技教育出版社, 2001, p. 27.有了Minkowski 时空模型后, Einstein 又进一步研究引力场理论以建立广义相对论. 1912 年夏他已经概括出新的引力理论的基本物理原理, 但是为了实现广义相对论的目标, 还必须寻求理论的数学结构, Einstein 为此花了 3 年的时间, 最后, 在数学家M. Grossmann 的介绍下学习掌握了发展相对论引力学说所必需的数学工具—以Riemann几何和Ricci, Levi - Civita的绝对微分学, 也就是Einstein 后来所称的张量分析.“根据前面的讨论, 很显然, 如果要表达广义相对论, 就需要对不变量理论以及张量理论加以推广. 这就产生了一个问题, 即要求方程的形式必须对于任意的点变换都是协变的. 在相对论产生以前很久, 数学家们就已经建立了推广的张量演算理论. 黎曼(Riemann)首先把高斯(Gauss)的思路推广到了任意维连续统, 他很有预见性地看到了……进行这种推广的物理意义. 随后, 这个理论以张量微积分的形式得到了发展, 对此里奇(Ricci)和莱维·齐维塔(Tulio Levi-Civita, 1873～1941)做出了重要贡献. ”—阿尔伯特·爱因斯坦著, 相对论的意义, 郝建纲、刘道军译, 上海科技教育出版社, 2001, p. 57.从数学建模的角度看, 广义相对论讨论的中心问题是引力理论, 其基础是以下两个假设: 1. (等效原理)惯性力场与引力场的动力学效应是局部不可分辨的，(或说引力和非惯性系中的惯性力等效)；2. (广义相对性原理) 一切参考系都是平权的，换言之，客观的真实的物理规律应该在任意坐标变换下形式不变——广义协变性(即一切物理定律在所有参考系[无论是惯性的或非惯性的]中都具有相同的形式)。

星海求知天文学的奥秘》期末答案

1泡利不相容原理具体指的是同一体系内的任意两个()不可能有完全相同的运动状态。

（1.0分）1.0?分•A、电子•B、质子•C、中子•D、原子核我的答案：A2花样滑冰运动员在冰上旋转时,哪种动作可以获得更快的转速?()（1.0分）1.0?分•A、下蹲和直立、双臂向上双腿向下并拢•B、下蹲和单足点地,其余三肢全部横向伸展•C、张开双臂和直立、双臂向上双腿向下并拢•D、张开双臂和单足点地,其余三肢全部横向伸展我的答案：A3中国古人记载的公元185年超新星爆发的文字中,有可能反映了恒星化学组成的变化的语句是()。

（1.0分）1.0?分•A、客星出南门中•B、大如半筵•C、五色喜怒•D、至后年六月消我的答案：C4太阳的寿命预计还有()亿年。

（1.0分）1.0?分•A、46•B、50•C、100•D、700我的答案：B5理论上应该出现的在各个不同波段,辐射强度分布的情况,这种分布被称为( )。

（1.0分）1.0?分•A、正太分布•B、普朗克分布•C、t分布•D、泊松分布我的答案：B6黑洞二字分别指的是()。

（1.0分）1.0?分•A、这个天体是黑色的和天体存在洞穴结构•B、这个天体是黑色的和所有的物质在视界内都往中心奇点坠落•C、任何电磁波都无法逃出这个天体和天体存在洞穴结构•D、任何电磁波都无法逃出这个天体和所有的物质在视界内都往中心奇点坠落我的答案：D7以下哪位华裔科学家对暗物质探索的贡献最大?()（1.0分）1.0?分•A、杨振宁•B、李政道•C、丁肇中•D、朱棣文我的答案：C8太阳系的6重物质界限中,没有太“大块”的天体物质的是哪一重?()（1.0分）1.0?分•A、小行星带•B、柯伊伯带•C、奥尔特云•D、太阳风的最远范围我的答案：C9暗物质的特征不包括()。

（1.0分）1.0?分•A、总量比亮物质多10倍以上•B、不发出任何辐射,但存在引力•C、质量大,寿命长,作用弱•D、主体应该是已知重粒子以外的物质我的答案：A10银河在星空中的“流域”没有涵盖哪个星座?()（1.0分）1.0?分•A、天鹅座•B、天蝎座•C、南十字座•D、狮子座我的答案：B11假如冥王星上有智慧生命,则“他们”对飞掠而过的“新视野”号做出的反应可能会是()。

溴到硼酸酯

Journal of
Materials Chemistry C
Published on 20 November 2014. Downloaded on 08/12/2016 07:54:22.
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Cite this: J. Mater. Chem. C, 2015, 3, 861
However, these oligouorene functionalized oligomers may suﬀer from the unwanted long wavelength emission under long-term device operation, similar to polyuorene-based macromolecules.34–36
Received 26th September 2014 Accepted 17th November 2014 DOI: 10.1039/c4tc02173h /MaterialsC
Starburst 4,40,400-tris(carbazol-9-yl)triphenylamine-based deep-blue ﬂuorescent emitters with tunable oligophenyl length for solution-processed undoped organic lightemitting diodes†
Introduction
Since 1987, organic light-emitting diodes (OLEDs) have attracted tremendous attention due to their potential applications in at-panel displays and solid-state lightings.1–10 To achieve fullcolor displays, three primary RGB luminescent materials with excellent stability, eﬃciency and color purity are required. With respect to green and red counterparts, it seems to be a big challenge to develop blue emitters, especially deep-blue ones that have a Commission Internationale de l'Eclairage (CIE) y coordinate value of <0.10, because the intrinsic wide bandgap would inevitably result in ineﬃcient charge injection to an emitting layer (EML).11 Therefore, great eﬀorts should be paid to the design of deep-blue emitters to push forward the commercialization of full-color OLEDs.

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arXiv:astro-ph/0308233v1 13 Aug 2003DraftversionFebruary2,2008PreprinttypesetusingLATEXstyleemulateapjv.7/15/03

WHITEDWARFDONORSINULTRACOMPACTBINARIES:THESTELLARSTRUCTUREOFFINITEENTROPYOBJECTS

ChristopherJ.DeloyeDepartmentofPhysics,BroidaHall,UniversityofCalifornia,SantaBarbara,CA93106

LarsBildstenKavliInstituteforTheoreticalPhysicsandDepartmentofPhysics,KohnHall,UniversityofCalifornia,SantaBarbara,CA93106DraftversionFebruary2,2008

ABSTRACTWediscussthemass-radius(M-R)relationsforlow-mass(M<0.1M⊙)whitedwarfs(WDs)ofarbi-trarydegeneracyandevolved(He,C,O)composition.Wedosowithbothasimpleanalyticalmodelandmodelscalculatedbyintegrationofhydrostaticbalanceusingamodernequationofstatevalidforfullyionizedplasmas.TheM-RplaneisdividedintothreeregionswhereeitherCoulombphysics,degenerateelectronsoraclassicalgasdominatetheWDstructure.ForagivenMandcentraltem-perature,Tc,theM-Rrelationhastwobranchesdiﬀerentiatedbythemodel’sentropycontent.WepresenttheM-RrelationsforasequenceofconstantentropyWDsofarbitrarydegeneracyparameter-izedbyMandTcforpureHe,C,andO.Wediscusstheapplicationsofthesemodelstotherecentlydiscoveredaccretingmillisecondpulsars.Weshowtherelationshipbetweentheorbitalinclinationforthesebinariesandthedonor’scompositionandTc.InparticularweﬁndfromorbitalinclinationconstraintsthattheprobabilityXTEJ1807-294canaccommodateaHedonorisapproximately15%whileforXTEJ0929-304,itisapproximately35%.Wearguethatifthedonorsinultracompactsystemsevolveadiabatically,thereshouldbe60-160moresystemsatorbitalperiodsof40minthanatorbitalperiodsof10min,dependingonthedonor’scomposition.Tracksofourmass-radiusrelationsforHe,C,andOobjectsareavailablethroughtheelectronicversionofthispaper.Subjectheadings:binaries:close—pulsars:individual(XTEJ0929-314,XTEJ1751-305,XTEJ1807-294)—whitedwarfs—X-rays:binaries

1.INTRODUCTION

ThediscoveryofthreeX-raytransientultracom-pactaccretingmillisecondpulsars(MSPs),XTEJ1751-305(Markwardtetal.2002),XTEJ0929-314(Gallowayetal.2002),andXTEJ1807-294(Markwardtetal.2003a,b)havedemonstratedtheexistenceofbinarypulsarsystemswithlowmass,M2≈10−2M⊙,donors.Thesethreeultracompactsystems(heredeﬁnedasbinarieswithorbitalperiods,Porb<60min)areremarkablyhomogeneous,withmeasuredPorb=42.4,43.6,40.1minrespectively,wellbelowtheminimumperiodforasystemwithadonorcomposedprimarilyofhydrogen(Rappaportetal.1982).Sincethenatureofthedonorsinthesesystemstodaydependsonthepriorevolutionofthesystem,itisusefultodiscussthepotentialformationmechanismsforthesesystems.BinarysystemswithPorb<80mincanformthroughatleasttwochannels.Stablemasstrans-ferfromanevolvedmain-sequencestar(Nelsonetal.1986;Fedorova&Ergma1989;Podsiadlowskietal.2002;Nelson&Rappaport2003)oraHeburningstar(Savonijeetal.1986)ontoaneutronstar(NS)isonemechanism.Inthisscenario,themain-sequencestarisbroughtintoRochelobecontactduetoorbitalangu-larmomentumlossesfrommagneticbrakingatatimewhenthecorehasnearlycompletedHburning.Suchasystemwillevolvetoorbitalperiodscomparableto

Electronicaddress:cjdeloye@physics.ucsb.eduElectronicaddress:bildsten@kitp.ucsb.edu

theultracompactMSPsandcanreachPorb≈10min(Podsiadlowskietal.2002;Nelson&Rappaport2003).Podsiadlowskietal.(2002)andNelson&Rappaport(2003)showthattheresultingultracompactbinarieshavedonormassesM2≈0.1−0.2M⊙astheypassthroughPorb≈40minontheirwaytowardsashorterperiod.ThesemassesaresigniﬁcantlygreaterthanthosemeasuredintheultracompactMSPs(Gallowayetal.2002;Markwardtetal.2002;Bildsten2002).However,systemsevolvingthrough40minonthewayoutfromtheperiodminimumhavemassesmoreinlinewiththemea-surements(M2≈0.01M⊙)andbythistimethedonorshavebecomepartiallydegeneratewithcoretemperaturesTc∼105−106K(Nelson&Rappaport2003).ThesecondscenariothatmayformultracompactsystemsinvolvestriggeringacommonenvelopephaseduringanunstablemasstransferepisodefromthedonorontotheNS.Thecoreofthedonor,eitheraHeorC/Owhitedwarf(WD),andtheNSspiral-intoshorterorbitalperiodsuntiltheenvelopeisexpelled(Paczynski1976).Severalauthorshaveproposedbinaryevolutionscenariosinwhichthesystem,afteremergingfromthecommonenvelopephase,thensuﬀersin-spiralduetogravitationalwave(GW)emissionandeventuallyre-establishescontact(Iben&Tutukov1985;Rasioetal.2000;Dewietal.2002;Yungelsonetal.2002).DuringthislongGWin-spiral,theWDwillhavehadtimetocool,settingtheentropyofthedonorattheonsetofthesecondmasstransferphase(Bildsten2002).Tauris(1996)ﬁndsthatalargefractionoftheNS-WDbinariesthatundergoacom-