Einstein ring: Weighing a star with light
EEinstein ring: Weighing a star with light
Jorge Pinochet , Michael Van Sint Jan Facultad de Educaci´onUniversidad Alberto Hurtado, Erasmo Escala 1825, Santiago, Chile. [email protected] Woodtech S.A.El Golf 150, Las Condes, Santiago, Chile
January 3, 2018
Abstract
In 1936, Albert Einstein wrote a brief article where he suggested the possibility that amassive object acted as a lens, amplifying the brightness of a star. As time went by,this phenomenon –known as gravitational lensing– has become a powerful research toolin astrophysics. The simplest and symmetrical expression of a gravitational lens is knownas Einstein ring. This model has recently allowed the measurement of the mass of a star,the white dwarf Stein 2051 B. The purpose of this work is to show an accessible andup-to-date introduction to the effect of gravitational lensing, focused on the Einstein ringand the measurement of the mass of Stein 2051 B. The intended audience of this articleare non-graduate students of physics and similar fields of study, and requires only basicknowledge of classical physics, modern physics, algebra and trigonometry.
Keywords : General relativity, gravitational lens, Einstein ring, star’s mass measurement,undergraduate physics students.
In 1915, Albert Einstein presented a new theory of gravity, known as general relativity, whichperfects and widens Isaacs Newton’s law of universal gravitation [1]. Using this theory, Ein-stein predicted that the trajectory of a beam of light that goes through the gravitational fieldof a massive body will deviate in proportion to the mass of the body. It was this predictionthat allowed the first empirical confirmation of general relativity, and that made Einstein aninternational celebrity. This confirmation constitutes one of the greatest milestones of the20th century. It was carries out in 1919 by an English expedition through the observation ofa total solar eclipse that occurred on Prince island, off the coast of Africa [2].Two decades after, in 1936, Einstein wrote a short article where he extended his predictionof 1916. In this article, he suggested that by deviating light, a massive body can act as alens, greatly increasing a star’s brightness [3]. Although he expressed his skepticism aboutthe possibility to directly observe this phenomenon, the great physicist underestimated theprogress of astronomy in the next decades. Indeed, the gravitational lens effect, as is nowknown, has become a powerful tool in astronomical investigation, and has allowed the studyof such diverse objects as extrasolar planets, dark matter and quasars. The last astronomical1 a r X i v : . [ phy s i c s . pop - ph ] D ec ilestone based on the gravitational lens effect occurred only months ago and consisted in thefirst accurate measurement of the mass of a near star [4]. We are talking about Stein 2051 B,a white dwarf some 18 light-years away from us, that is part of a binary system along witha red dwarf (Stein 2051 A).As is the case with many of Einstein’s great ideas, the gravitational lens effect has become asubject in the leading edge of modern physics and astronomy, that is powering a wide researchprogram with unforeseen reach. Because gravitational light deviation is a prediction of generalrelativity -a complex theory that is usually studied at the PhD level in physics or astronomy-the analysis of this effect and its important applications in astronomy are normally left out ofundergraduate courses of physics and similar careers. This fact can be verified by reviewingthe most widely known modern physics textbooks, and noting that the subject is absent or,at best, treated qualitatively and superficially [5-9]. The goal of this article is to presentan accessible and updated introduction to the gravitational lens effect. However, given thecomplexity and broadness of the subject, the article focuses in a particular phenomenon thatallowed the measurement of the mass of the white dwarf Stein 2051 B. This phenomenon isknown as Einstein ring, and is the simplest and symmetrical expression of the gravitationallens effect. To achieve the abovementioned goal, the article starts by introducing some basicnotions of general relativity, presenting a heuristic derivation of Einstein’s equation for thelight deviation angle, based on the principle of equivalence. Then we present a simplifiedanalysis of Einstein’s ring and explain how it is used to determine the mass of a star. For thiswe examine the specific case of Stein 2051 B, using the original data taken by the team thatmeasured the star’s mass. Finally, we briefly discuss the importance of the gravitational lenseffect in astronomy and the meaning and implications of using Einstein ring to measure themass of Stein 2051 B.To make the article accessible to the broadest possible audience, we have tried to keep thetechnical terms to a minimum, and we have put special emphasis in mathematical simplicity.In fact, the article only assumes general knowledge of classical and modern physics and basicknowledge of algebra, geometry and trigonometry. From this perspective, we hope that thearticle can be rewarding for non-graduate students of physics and for non-specialists. Thearticle may be particularly useful for teachers giving courses of astronomy or modern physicsat under-graduate level, and who are looking for updated material that examines the latestadvances in physics and astronomy. The so-called equivalency principle, first enounced by Einstein in 1911, and considered by himas the foundation of his theory of general relativity, states that: a homogeneous gravitationalfield is completely equivalent to a uniformly accelerated frame of reference. An alternativeway to formulate this principle is as follows: experiments made in a frame of reference uni-formly accelerating wit acceleration (cid:126)a with respect to an inertial frame, produces the exactsame experimental results as an inertial frame of reference in a uniform gravitational field − (cid:126)a [8, 9]. A White dwarf is a cold and stable star that has exhausted its nuclear fuel and that maintains its state ofequilibrium because of the electrons repelling each other by the principle of exclusion. g , far from any gravitational influence. Onthe right, the spaceship is standing on the surface of a planet with a surface acceleration ofgravity − g . In both cases the scale shows mg .A simple example allows us to better understand the meaning of the equivalency principle.Imagine a spaceship moving with constant acceleration g , in a region of space away from anygravitational influence (see figure 1). Inside the ship an astronaut of mass m is standing overa weighing scale. What value does the instrument show? As the only force acting upon theastronaut is the normal N exerted by the scale in the same direction as g , from Newton’ssecond law we know that the value shown is N = mg . On the other hand, if we consider thesame spaceship standing still on the surface of a planet where the gravitational accelerationis − g , the scale will also show the value mg .From the equivalency principle, Einstein predicted that a beam of light in a gravitationalfield will deviate from its trajectory [10]. To see how one could arrive at this conclusion, let’simagine that in the same spaceship from the previous example, a photon enters through awindow located in the left wall, traveling a distance x before arriving at the opposite wall. Fig-ure 2 shows two possible perspectives of this photon (drawn as a red dot) in four equidistantinstants in time, while the spaceship travels upward with constant acceleration of magnitude g .The external observer’s point of view is shown on the left image. This observer is in the samereference system from which the photon is emitted. To this observer, the photon follows astraight path. The point of view of an observer located inside the spaceship is shown on theright. The effect of the spaceship’s acceleration, as seen from inside, is a deviation of thephoton’s path, which enters the ship through the upper left side and arrives at the lower rightside. The photon travels a horizontal distance x , and deviates a vertical distance y . Thinkingin terms of a beam of photons, i.e. a ray of light, the conclusion is the same. What is theangle of deviation of the photon in the ship’s frame of reference? To answer this question, weonly need some considerations from kinematics.3igure 2: On the left, the trajectory of the photon is shown from outside the accelerated frameof reference, in four different instants of time. On the right, the trajectory of the photon isshown inside the accelerated frame of reference.The vertical distance that the photon travels is: y = 12 gt (1)On the other hand, the time it takes the photon to fall this distance is given by: t = xc (2)where c = 3 . × m · s − is the speed of light in the vacuum. Introducing this result intoequation (1): y = x g c (3)Due to the huge value of the speed of light, in general this deviation angle will be very small.Thus, if we call this angle α , from equation (3) we can express it in radians ( rad ) as: α ≈ tan α = yx = xg c (4)In virtue of the equivalency principle, we conclude that what happens to the ray of light inthe frame of reference with constant acceleration g , will also occur in a uniform gravitationalfield g . That is to say, a ray of light inside a constant gravitational field g , will experience adeviation from its straight trajectory. This deviation is bigger as g increases.We can use the equivalency principle to approximate the deviation angle of a ray of light ina gravitational field of magnitude g . Let us consider a spherically symmetrical mass distribu-tion, such a star. We will call this object lens because it deviates light rays that pass closeto it. Behind the lens there is another star, which we call source. The source emits lightthat is deviated by the lens. Figure 3 illustrates this situation where the minimum distance4igure 3: A ray of light (continuous red line) emitted from a source, is deviated by an angle α when passing close to a lens of mass M . The minimum distance between the light ray and thelens is b . The source is observed to be in a (apparent) position different from its real position.The region where the effects of the gravitational field generated by the lens are significant hasa diameter 2 b .between the light ray and the lens is b , and the angle of deviation is α . In the figure, this angleis extremely magnified, because in reality it is very small, on the order of micro-seconds ofarc. Due to this deviation of the light, a distant observer detects the source in an (apparent)position different from its real position.Let M be the mass of the lens, and let us suppose that ∼ b defines the diameter of the regionwhere the effects of the gravitational field over the light beam are significant. If the lens isa common star (which excludes objects such as black holes or neutron stars), g is weak, andwe can use the law of universal gravitation. According to this law, the gravitational field’smagnitude of a gravitational lens at a distance b from the ray of light is: g = GMb (5)where G = 6 . × − N · m · kg − is the universal gravitational constant. If we assumeroughly that g is given by equation (5) throughout the path of the light beam in the regionof length ∼ b (uniform gravitational field simplification), we can replace this value of g inequation (4), taking x = 2 b . Under these conditions we get: α ≈ GMc b (6)Although this result is a very crude estimation, it only differs by a factor of 4 from the exactvalue found by Einstein in 1916 [1]: α = 4 GMc b (7)5et’s imagine that the lens is our own Sun, which mass is 1 . × kg , and let us assumethat the beam of light passes very close to the Sun, such that b is on the order of the solarradius: 6 . × m . Because 2 πrad = 360 ◦ , and each degree contains 3600 (cid:48)(cid:48) (seconds), then1 rad = (360 × / π ) (cid:48)(cid:48) , and therefore: α = 4(6 . × − N · m · kg − )(1 . × kg )(3 × m · s − )(6 . × m ) (cid:18) × π (cid:19) (cid:48)(cid:48) = 1 . (cid:48)(cid:48) (8)This is the result predicted by Einstein in 1916 for the deviation of a ray of light in the vicinityof the Sun, and allowed the first observational confirmation of the theory of general relativity[1, 2]. As we shall see in the next section, equation (7) is the basis for the concept of Einsteinring. The gravitational lens effect is produced when the light originating from distant objects isdeviated or curved around massive bodies, producing a phenomenon with significant similar-ities to what happens when light passes through a lens. The deviation of light near massiveobjects studied in the preceding section, is a particular case of the gravitational lens effect,where only a shift in the apparent position of a distant star is appreciated. In a more generalsituation, the effect can generate multiple images, increase in brightness and distortions inthe object’s shape.The Einstein ring is the simplest and most symmetrical effect of gravitational lens, generatedwhen the source, the lens and the observer are perfectly aligned. As its name suggests, theEinstein ring corresponds to the situation where the observer detects the light deviated bythe lens as if it were a circumference or ring. Figure 4 illustrates this situation, where theEinstein ring is represented by a dashed yellow line circumference, centered on the source andbeing looked from the side. It can be observed that the rays of light that travel directly tothe observer (continuous red lines) subtend an angle θ E , known as Einstein angle. Which isthe characteristic magnitude of the Einstein ring. It also can be appreciated the deviationangle α calculated in the previous section. Again, this angle is very accentuated in the figure,because in practice, α and θ E are on the order of microseconds of arc.Figure 5 shows a version of figure 4 which is more useful to calculate θ E . Here are shownthe distance between the source and the lens, d LS , the distance between the lens and theobserver, d L , the distance between the source and the observer, d S , and the angle β betweenthe undisturbed ray of light and the straight line between the source and the lens. To calculate θ E , let us note that by the exterior angle theorem we must have that: α = θ E + β (9)If M is the mass of the lens, combining equations (7) and (9) we get: α = 4 GMc b = θ E + β (10)Because α y θ E are so small, when expressing them in radians we can use the approximations:6igure 4: A source, a lens and an observer are perfectly aligned, which creates an Einsteinring. The drawing simplifies the phenomenon by showing only two of the light-rays that formthe ring (continuous red lines), that subtend an Einstein’s angle θ E . θ E ≈ tan θ E = bd L , β ≈ tan β = bd LS (11)Introducing this expressions into equation (10):4 GMc b = bd L + bd LS (12)Solving for b: b = (cid:18) GMc d L d LS d L + d LS (cid:19) / = (cid:18) GMc d L d LS d S (cid:19) / (13)To express this result in rad , we divide member for member by the distance between theobserver and the lens, obtaining Einstein angle: θ E = bd L = (cid:18) GMc d LS d L d S (cid:19) / (14)For a more technical examination of this equation and the effect of gravitational lens, reference[11] is a publication that can be consulted. Let us note that θ E increases as M grows, asis to be expected, because an increase in mass implies a stronger gravitational field and amore pronounced deviation of the light rays close to the lens. From equation (5), taking d LS = d S – d L in equation (14) and solving for M , we get: M = c G d L d S θ E ( d S − d L ) = c G d L θ E (1 − d L /d S ) (15)7igure 5: Reduced version of the figure 4, where we show Einstein angle, θ E , the distancebetween the source and the lens, d LS , the distance between the lens and the observer, d L , thedistance between the source and the observer, d S , and the angle β between the undisturbedray of light and the straight line between the source and the lens.If we assume that d L (cid:28) d S , which is the situation in which we are interested here, it is possibleto neglect the term d L /d S , such that: M = c G d L θ E (16)Obtaining the mass of a star from Einstein ring is a problem of considerable technical dif-ficulties, among other reasons, because the probability of finding an astronomical system ina ring configuration is very low. However, it is possible to overcome this difficulty if a moreprobable scenario is considered, in which the source and the lens are slightly misaligned. Inthese conditions, an asymmetrical version of the Einstein ring is produced, which despite this,allows to obtain a good estimation of θ E and the other parameters that define the ring. Fromthese estimations, it’s possible to use equations (15) and (16) to calculate M with very goodaccuracy. Leaving aside the complex technical details, this was the procedure to determinethe mass of Stein 2051 B, that was done by a team of astronomers lead by Kailash Sabu fromthe Space Telescope Science Institute [4]. Using the Hubble Space Telescope, Sahu and histeam tracked the white dwarf Stein 2051 B (lens) for two years, while it crossed in front ofanother star in the background (source). From the data acquired, the value obtained for themass of Stein 2051 B was of 0 . ± . M (cid:12) , where M (cid:12) = 1 . × kg is the mass of the sun.Let us reproduce the calculation done by Sahu’s team, but leaving aside the technical issuesand omitting the calculation of the errors. As reported in their article [4], the measuredvalues are: d L = 5 . parsec ( pc ), d S = 2 × pc (note that d L /d S ∼ − (cid:28) θ E = 31 . milliarcsecond ( mas ). To obtain the mass of the lens in kilograms ( kg ) weuse that 1 pc = 3 . × m , and that 1 mas = (10 − ) (cid:48)(cid:48) , where 1 (cid:48)(cid:48) = (2 π/ × rad .Introducing the corresponding values in equation (16): M = (3 × m · s − ) (17 . × m ) (cid:18) . π . × rad (cid:19) . × − N · m · kg − ) = 1 . × kg (17)Using units of solar masses, it is found that M = 0 . M (cid:12) , which matches the value calculatedby Sahu and his team. 8 Final comments
The gravitational lens effect constitutes a powerful astronomical research tool, which over thelast decades has allowed the study of diverse phenomena. Depending on the goals, this typeof research may focus on the source or on the gravitational lens. In the first case the studiesrely on the fact that the lens can significantly intensify the light from the source, which hasenabled the detection of very faint objects, such as far away galaxies and extrasolar plan-ets with masses similar to Earth’s mass. In the second case, when studying the lens itself,the deviation of the light makes it possible to estimate the mass of the lens, which has al-lowed the study of big distributions of mass, such as galaxy clusters or the elusive dark matter.Therefore, the usage of the gravitational lens effect to study the universe is not new, and ispart of a wide research program developed long ago in different universities and astronomi-cal centers throughout the world. However, what makes the work of Sahu and his team sorelevant, is that they introduced a new and powerful procedure to calculate the mass of com-paratively small objects that cannot be easily measured by other means (the gravitational lenseffect applied to small objects, such as planets or stars, is known as microlensing). Further-more, given the symmetry of the Einstein ring, the procedure can achieve higher accuraciesthan traditional methods.Indeed, previous measurements of the mass of Stein 2051 B, based on its orbit around itscompanion star (recall that this white dwarf is part of a binary system) have underestimatedits mass. This triggered a long controversy that appeared to put into question the validity oftheoretical models of white dwarfs, based on the relation between mass and radius of thesestars. The precise measurement of the mass of Stain 2051 B using Einstein ring, allowed tosettle this controversy, supporting the validity of theoretical models. Hence, the achievementof Sahu and his team has had echoes that extend far beyond the measurement of the massof a star. As this is a pioneer work, it is to be expected that in the following years new andimportant astronomical discoveries will be made based on Einstein ring.5 References