Highways in the Sky

Abstract

Calculation of orbits of members of the solar system started early but was hampered by the idea of a static Earth. In modern times, Copernicus first suggested that the Earth actually moves round the Sun. Galileo’s discovery of the Jovian satellites played an important role in general acceptance of the heliocentric system; it later also allowed the first measurement of the velocity of light. Once the heliocentric system was accepted, progress was swift. Kepler postulated elliptical orbits which led to Newton’s law of universal gravitation. Measurement of the size of the solar system followed immediately. The discovery of the planet Neptune from calculation of the orbit of Uranus was a triumph of Newton’s law. Later, deviation of the orbit of Mercury from the prediction of Newton’s law paved the way to the General Theory of Relativity. Observation of motion of binary stars allowed us to measure their masses contributing significantly to understanding them. The same approach also led tho the discovery of exoplanets. Finally, the motion of the stars in galaxies has been instrumental in deducing the property of the black holes at their centres, at the same time it has also thrown up a puzzle that still remains to be solved.

4.1 Introduction

From the very beginning, humankind has looked up in the sky and has seen the Sun, the Moon, the stars and the planets travel across the celestial dome. Even before the birth of the first hominid, birds in the sky and animals in the sea and on the land have followed the stars in their migratory paths. The stars rise in the east and set in the west. It was obvious to our distant ancestors that the universe rotates around our abode. Slowly, it was understood that it is the Earth which rotates around itself while orbiting the Sun. The solar system is a part of a huge star system, the Milky Way galaxy which is majestically rotating around its axis with a time period of more than two hundred million years. These various regular motions can surely be termed as the rhythms of the universe. Study of these rhythms has extended our knowledge about the universe and has contributed significantly to our knowledge of physical science.

4.2 The Solar System

The Earth is a member of the solar system. As we have seen in Chap. 3 the motion of the Sun and the Moon controlled the calendars created by different societies. Thus, the first attempts to trace the path of astronomical bodies naturally were concentrated on the members of the solar system.

4.2.1 From Aristotle to Copernicus

It was inevitable that the first theory of the universe would envisage a static Earth with everything rotating around it. There were a few dissident voices, but they remained muted in the face of the apparent absurdity of a moving Earth. Aristarchus (c. 310–c. 230 BCE) in Greece suggested a universe with the Sun at its Centre. Pythagoras (c. 570–c. 495 BCE) postulated a secret fire hidden from us about which everything is rotating. However, their ideas remained philosophical; there was no evidence to support their ideas.

The first comprehensive attempt to understand the universe was made by Aristotle (384 BCE–322 BCE). He believed that the Earth was at the centre of the universe. There are a number of spheres that encircle the Earth. Nearest is the sphere of the Moon. Then comes the spheres for Mercury, Venus, Sun, Mars, Jupiter, and Saturn. The furthest sphere was the sphere of the distant stars. These spheres rotate around the Earth with different speeds. They were made of ‘aether’, a heavenly staff which is not available on the Earth. The path of the heavenly bodies was circular because circles and spheres are perfect and heavenly bodies are also perfect.

Ptolemy (c. 100–170 CE) built on Aristotle’s model but made a number of changes. Aristotle’s model failed to explain the apparent retrograde motion of the outer planets. Excellent observations using naked eye also showed up departures form the circular motions of planets. Ptolemy used epicycles, i.e. circles with centres moving in circular orbits to explain the observations. He also placed the Earth away from the centre of the planetary orbits, though it was still at rest. Figure 4.1 shows how epicycles can be used to explain retrograde motion of the outer planets.

Fig. 4.1

Ptolemy used epicycles to explain retrograde motion of planets

  Arab scientists in the middle ages accepted the model of Aristotle and Ptolemy, whose writings later returned to Europe via translations by Islamic scientists and occupied the central place in the theory of the universe.

The Polish polymath and catholic canon Nicolaus Copernicus (1473–1543) was the first to suggest a radically different picture of the universe. Although he circulated a manuscript ‘Commentariolus’ (‘Little Commentary’), containing a heliocentric hypothesis, among his friends before 1514, he refused to publish his calculations fearing scorn, if not hostile reception. He went on working on his book ‘De revolutionibus orbium coelestium’ (On the Revolutions of the Heavenly Spheres) and finally agreed to its publication after being encouraged by some of his friends and students. It was finally published in 1543; legend has it that he received a copy in his death bed. This publication marks the decisive break with the Aristotelean tradition.

Figure 4.2 shows how heliocentric model can explain observation of retrograde motion of outer planets in the picture T and M show different positions of Earth and Mars, respectively. Earth moves faster than Mars and overtakes it. Against the backdrop of distant stars, Mars appears to move forward from positions from 1 to 3. As Earth overtakes Mars, it appears to go backwards from 3 to 5 in the sky and then forward to 6 again.

Fig. 4.2

Credit MLWatts/Wikipedia, Public domain

Explanation of apparent retrograde motion of Mars.

  Copernicus also used circular orbits, thus he also had to take recourse to epicycles, though less in number than Ptolemy. His book was also very technical, almost inaccessible to any but very advanced mathematics experts of the day. The importance of the Copernican hypothesis was lost on the learned also because of a preface put in the book by a Lutheran priest, Andreas Osiander, who oversaw the printing. Osiander described the hypothesis as a mere calculational tool, not necessarily true. Copernicus never learned of the preface. Thus, though there was immediate opposition from various influential members of the Church, the book was put on the list of banned books only in 1616. Among the later enthusiastic supporters of Copernicus, Giordano Bruno (1548–17 February 1600) was burnt at the stake for heresy. Two of the greatest scientists of the seventeenth century, Galileo (1564–1642) and Kepler (1571–1630), were responsible for the general acceptance of the Copernican worldview.

4.2.2 Galileo and the Jovian Satellites

Galileo’s strong public support for the heliocentric system incurred the ire of the Catholic church. In a celebrated trial, he was made to recant and was put under house arrest for the rest of his life. His experiments with falling bodies, projectiles and motion on inclined planes strongly contradicted Aristotle’s idea of motion and led to the development of the concept of inertia by Newton later. His discovery of the four Jovian satellites, mentioned in Chap. 3, showed that Earth is not unique in possessing a moon. A major argument against a moving Earth was that the Moon would fall behind; Galileo’s observations proved otherwise. Galileo also was the first to attempt to measure the speed of light. He failed, but his discovery of the Jovian satellites later was instrumental in the first successful measurement.

We have already seen that Galileo proposed a method for the determination of longitude by observing eclipses of the Jovian satellites. In 1675, the observation of these eclipses helped Ole Roemer to measure the velocity of light with reasonable accuracy. He observed that when the Earth is moving towards Jupiter, the time interval between successive eclipses decreases. On the other hand, it increases when Earth moves away from Jupiter. He correctly ascribed it to the finite velocity of light.

Fig. 4.3

Figure accompanying Roemer’s announcement of measurement of the velocity of light

  Figure 4.3 is from a paper by Roemer. Here B is Jupiter, C and D denote the position of a satellite just before and after the eclipse, A is the Sun, and F, G, H, L, K are the different positions of Earth in its orbit around the Sun. When Earth travels from F to G, the interval between successive eclipses decreases because light requires progressively less time to cover the distance from the satellite to the Earth. Exactly the opposite happens when Earth moves from point L to point K. Assuming spherical orbit for the Earth, we denote the radius of the orbit by D. The time period of the satellite is t. If in the time interval $T_1$ when the Earth is moving from H to E, that is from perigee to apogee position with respect to Jupiter, we observe that the satellite completes n revolutions, we may write

$$\begin{aligned} T_1 =nt+D/c, \end{aligned}$$

(4.1)

where c is the velocity of light in free space. On the other hand, when moving from E to H, the time to observe n revolutions is $T_2$, we have

$$\begin{aligned} T_2=nt-D/c. \end{aligned}$$

(4.2)

Subtracting, we get

$$\begin{aligned} T_1-T_2=2D/c \end{aligned}$$

so that we get the velocity of light as

$$\begin{aligned} c=\frac{2D}{T_1-T_2}. \end{aligned}$$

(4.3)

Roemer found that, for the satellite Io, $T_1-T_2=22$ min, that is it takes 22 min for the light to cross the Earth’s orbits. The diameter of the orbit was not very accurately known at that time. Later measurements of the orbit led to a value $c=2,20,000$ km s$^{-1}$, a respectable first measurement of a tremendously large quantity.

4.2.3 Kepler and Newton

  Besides the geocentric and the heliocentric models, there was an alternate model of the universe. Martinius Capella, in the fifth century, suggested a model in which the two inner planets, Mercury and Venus rotate around the Sun, which, in turn, orbit the Earth along with the Moon, the outer planets and the stars. Copernicus praised his ideas in his book. In India, Nilakntha Somayaaji proposed a similar model in the fifteenth century, where the planets rotate around the Sun, which in turn orbits the Earth. In the sixteenth century, Tycho Brahe (1546–1601) proposed a similar model. In Fig. 4.4, a picture of Tycho’s universe from a book has been shown. Tycho was the pre-eminent astronomer of his day. Working in his Uraniborg observatory, he accumulated data on the motion of the planets unmatched in quality. In the last period of his life, Johannes Kepler was his assistant. After the untimely death of Tycho, Kepler inherited Tycho’s observations. 

Fig. 4.4

Tycho Brahe’s model of the universe from Hevelius’ Selenographia, (1647). In it, the Sun, Moon, and sphere of stars orbit the Earth, while the five known planets orbit the Sun

Kepler was a Copernican, he tried to explain the observed movements of the planets within a heliocentric model. He finally recognised that the orbits of the planets are not circles but ellipses. In 1604, he formulated two laws of planetary motion, though he published them in 1609. The third law of planetary motion was published ten years later in 1619. The three laws are:

• 1. Planets move in elliptical orbits with the Sun at one of the two foci.

• 2. A line segment joining a planet and the Sun sweeps out equal areas in equal intervals of time.

• 3. The square of a planet’s orbital period is proportional to the cube of the length of the semi-major axis of its orbit.

The laws can be clearly understood in terms of Fig. 4.5 where the various terms have been indicated.

A deduction of Kepler’s third law for circular orbits starting from Newton’s law of universal gravitation may be found in Sect. 4.3.2. A visualisation of the inner solar system¹ shown in Fig. 4.6 clearly shows the eccentricity of the orbits.

Fig. 4.5

Motion of planets in Kepler’s law

Fig. 4.6

Credit NASA’s Scientific Visualization Studio

Visualisation of the inner solar system. The eccentricity of the orbits is visible.

  Kepler was followed by Christian Huygens (1629–1685) who obtained an expression for centripetal acceleration assuming that the force of the Sun is pulling the planets in a curved path. The need of the hour was to discover the form of the force that keeps the planets in orbit around the Sun. Of course, the concept of force was not fully developed, it would take the genius of Newton (1643–1727) to establish the definition of force as we understand today. Nevertheless, Kepler speculated that the Sun emits some corporeal body responsible for the force. He speculated that the force would vary inversely with distance. Interestingly, the same argument led Kepler to correctly surmise that the light intensity would vary as inverse square power of the distance. The French astronomer Ismaël Bullialdus criticised Kepler and advocated that the force should vary as an inverse square of the distance. Robert Hooke and Edmund Halley also held that the force should vary as the inverse square of distance. However, this argument obviously stood on a flimsy ground. Newton brilliantly used Kepler’s laws to establish the inverse-square law. He correctly understood that the motion of the Moon around the Earth and a body falling freely under gravity are acted upon by the same force. He proved that the planets will follow an elliptical path under such a law. Newtonian gravity is  quite simple: the gravitational force between two point-like objects is proportional to their masses and is inversely proportional to the square of the distance between them. It acts along the line joining the two centres (such forces are called central forces; electrostatic force is also central in nature). Mathematically, it looks like

$$\begin{aligned} \boldsymbol{F} = G \, \frac{M_1M_2}{r^2}\, \boldsymbol{\hat{r}}\,, \end{aligned}$$

(4.4)

where $M_1$ and $M_2$ are the masses of the two bodies and r is the distance between them. The central nature is evident from $\boldsymbol{\hat{r}}$, the unit vector along the joining line. G is a constant, called Newton’s gravitational constant.

4.2.3.1 Precession of the Equinoxes and the Shape of the Earth

Hipparchus, arguably the greatest astronomer of antiquity, first proposed the precision of the Earth’s axis of rotation around an axis perpendicular to the plane of its orbit around the Sun. He arrived at this conclusion by studying the star positions noted by his predecessors which he measured to be different. His estimation of the rotation was close to the presently accepted value. Newton was the first to explain this movement as the effect of the gravitation of the Moon and the Sun of the equatorial bulge of the Earth. His theory predicted the Earth to have an oblate shape, that is flattened along the rotation axis, as the centripetal force due to rotation that opposes the gravitational attraction is maximum at the equator and vanishes at the two poles of the Earth. 

A rival theory of Rene Descartes predicted the Earth to be prolate, that is elongated along the axis. The French academy, in the decade following the death of Newton, decided to experimentally determine the shape. An oblate shape would mean that the length of the arc corresponding to one degree of latitude would be larger at the pole and smaller at the equator. An expedition was sent to Lapland to measure the length near the pole. It returned with 1 year and reported that the length of the arc was longer than that measured at the latitude of Paris. Another expedition to the present-day Ecuador and Peru to measure the length near the equator suffered a lot of hardship and the first report reached Paris only after 10 years. Their result also vindicated Newton. Thus the observation of the periodic motion of the Moon and the planets was instrumental in formulating and establishing the law of universal gravitation.

4.2.4 The Transit of Venus

Even after the discovery of Kepler’s laws and Newton’s theory of gravitation, the distance of the Sun and other planets was not known to any reasonable accuracy. That is because the gravitational constant ‘G’ could not be directly measured in the laboratory before 1798 when Henry Cavendish used a torsional balance to obtain a value of $G= 6.74\times 10^{-11}\textrm{m}^3\textrm{kg}^{-1}\textrm{s}^{-2}$. This is very close to the presently accepted value, $G=6.67430\pm 0.00015$ m$^3$/kg/s$^2$, mentioned in Chap. 2. Earlier estimates, one by Newton himself, depended on the assumption of a value for an average density of the Earth. The requirement was to measure the distance of at least one of the planets or the Sun; the rest of the distances could be easily obtained using Kepler’s law.   The first measurement was actually done in 1671 even before Newton published the law of gravitation. Jean Picard, Jean-Dominique Cassini and Jean Richer measured the distance of Mars by measuring it from different positions on Earth and using trigonometry. However, their measurement had certain problems and yielded a value of the distance of the Sun which we now know to be off by nearly 7%.

Fig. 4.7

Credit ESO

Measurement of distance using transit (Not to scale).

James Gregory first suggested that the transit of Venus can be used to measure the scale of the solar system. The idea is presented in Fig. 4.7. During transit of Venus, it comes between the Earth and the Sun. It appears as a small black disk in front of the Sun. From different points on the Earth, Venus will appear at different positions with respect to the disk of the Sun. In the figure, from points A and B on the Earth, Venus will appear at points C and D, respectively. The distance between the points A and B can be easily found out from the location of the points. The ratio VC/VA is known from Kepler’s third law. Thus, if the angle at V is measured, we can find out the distances VE and SE, i.e. thus distance to Venus and the Sun, respectively.

Edmund Halley had observed the transit of Mercury on 7 November 1677. He strongly supported the idea of observation of the transit of Venus. as Venus being closer to Earth will offer a better opportunity for accurate measurement. He died at the age of 85 in 1742. He knew that he would not live to see the next transit which would occur on 6 June 1761 CE. Still, he devised plans for the observation. It is difficult to measure the location of any point on the bright nearly featureless disk of the Sun. Halley suggested an alternate method of measuring transit time from different locations. The idea can be understood from Fig. 4.8. Both the observers, A and B, will measure the velocity of Venus to be same, but due to their different positions, the transit time would be different. Thus, measuring the transit time, it is possible to measure the angle at V in Fig. 4.7.

Fig. 4.8

Measurement of distance using transit path

Transit was observed from Great Britain and France, as well as their colonies and many other places in 1761 CE. In India, observations were made from Kolkata and Chattagram in Bengal and Chennai and Tarangamwadi in Madras provinces. There were many misfortunes, adventures and tragedies. Finally, results from 122 different observations were collected. However, the final outcome proved disappointing because the observations had very large errors. When Venus came very close to the rim of the Sun, its image suddenly elongated. It became circular after it had travelled a considerable distance in front of the solar disk. This phenomenon, called the black drop effect, was completely unexpected and ruined the measurements.

However, scientists did not give up. The next transit was to be on 3–4 June in 1769 CE. The noted telescope maker James Short devised a way of eliminating the black drop effect. Finally, again after tragedies and mishaps, the results were collected to yield a Sun-Earth distance of 151.7 million kilometres, about 1.4% larger than the presently measured value of 149.6 million kilometres.

4.2.5 The Discovery of Neptune

    Uranus was the first planet to be discovered in modern times. It was discovered by William Herschel using a telescope he made himself. However, the motion of Uranus could not be completely explained by Newton’s law of gravitation considering even the effects of Jupiter and Saturn besides that of the Sun. This led John Adams in England and Urbain Le Verrier to independently predict the possible existence of a new heavy planet and based on their calculations, to suggest its position. Adams’ prediction came earlier, and he informed the Astronomer Royal George Airy about it. Contrary to a popular myth, Airy paid sufficient attention to Adams, but Adams himself was reticent and tended to change his predictions.

It was Le Verrier who sent letters to various observatories suggesting they look for the new planet. Airy then instructed James Challis, director of the Cambridge Observatory, to search the sky following the predictions of Adams and Le Verrier. Johann Galle and his assistant Heinrich d’Arrest in Berlin observatory also wasted no time once they got Le Verrier’s letter and discovered the new planet on the first night of observation on 23 September 1846 within one degree of the position suggested by Le Verrier. Once the discovery was announced, Challis realised that he had actually observed the planet but could not identify it as his star map was out of date. Airy later wrote about the British search for the planet [1]. A more recent account, including some hitherto unpublished material, can be found in Ref. [2].  Actually, Neptune was observed earlier by Galileo, but it is believed that he considered it a fixed star. Recently, it has been shown from Galileo’s notebook that he was aware of the motion of the newly observed body [3]. However, even if had considered it as a planet, he did not follow up his observations.

4.2.6 Comets, Asteroids, and Dwarf Planets

Comets are balls made of rock, ice, frozen gases and dust that orbit the Sun. They travel in highly eccentric orbits so that at aphelion they are far away from the Sun, while, at perihelion, they may come closer to the Sun than even Mercury. When they come closer to the Sun, they warm up emitting gases and vapours which form a large head and a very long tail.

Comets have been known from antiquity. Because of their distinctive appearance, they were taken to be omens of calamities or upheavals. Frequently, mention of comets was added to portend an impending catastrophe. In India, for example, Rajtarangini, a political history of Kashmir from 2000 BCE to 1600 CE, was written by five historians. Srivara, the third of them, mentions a few important astronomical events which have been studied by Shylaja [4]. The appearance of a comet and two eclipses occurring within a fortnight have been narrated as heralding the death of Sultan Zain-Ul-Abidin who died in 1470. No known sighting of comet or pair of eclipses matches the year 1470 CE. It is thought that Srivara had added them to signify the passing of a great ruler. It is also possible that the Srivara had seen Halley’s comet  which appeared in 1456 and has added its description to foreshadow the death of the Sultan.

Aristotle was possibly the first to attempt a scientific explanation of comets. As he did not believe that any change can occur in the heavens beyond the Moon, he postulated comets as atmospheric phenomena. This held sway for nearly two thousand years before Tycho Brahe showed that a comet that appeared in 1577 was further than the Moon. Tycho also opposed Aristotle’s idea of planets being held by celestial spheres made of hard and impervious matter as he found the comet to easily pass through these hypothetical spheres.

Because of their long periods and the fact that a comet on its return usually appears at a different place in the sky, their periodicity was not understood for a long time. Edmund Halley studied the literature on comets and concluded that the comets that appeared in 1531 CE, 1607 CE, and 1682 CE have nearly the same orbital elements. For this, he used the newly discovered Newton’s law of gravity and considered the influence of the Sun, Jupiter, and Saturn. He predicted that all the appearances were of the same comet that have a period of 76 years and will return again to the inner solar system in 1758. It duly returned but, by that time, Halley was dead. This was one of the first tests of the predictive power of Newton’s law. The comet was named later in his honour.

In ancient China, astronomers recorded the appearance of comets and their descriptions and positions in the sky. Confirmed records go back to 613 BCE. These have been useful to the later day astronomers. For example, efforts to calculate the past orbit of Halley’s comet could not be extended back before 837 CE because of a perturbation in the orbit due to its close approach to the Earth. Chinese records have been used to constrain the calculations [5]. The first confirmed sighting dates back to 239 BCE. Comets can be divided into two classes based on their periods. Short-period comets have periods less than two hundred years. Their paths usually lie close to the ecliptic plane and are thought to originate from the Kuiper belt.² $^{,}$³ Long-period comets  have longer periods extending to millions of years but are gravitationally bound to the Sun. Their possible origin is the Oort cloud.⁴ Some comets also travel in hyperbolic orbits; after one visit to the inner solar system, they escape the gravity of the Sun. It is generally believed that comets brought water to Earth after it was formed. Comets have been studied in detail from the ground as well by a number of space missions as they contain the furthest part of the solar system and are likely to contain pre-solar material. Halley’s comet was visited by five spacecrafts during its last visit in 1985–1986. In 2013, NASA’s Deep Impact mission sent a special impactor into the path of the comet Tempel 1 to study inner materials and learn about the composition and structure of a comet. Rosetta spacecraft of European Space Agency (ESA) visited the comet 67P/Churyumov-Gerasimenko and landed a lander Philae in 2014. Rosetta was later impacted by the comet.

Asteroids are minor planets that orbit the Sun within the inner solar system. Their size varies from a few metres to nearly one thousand kilometres in the case of the largest asteroid Ceres, which is nowadays classified as a dwarf planet. More than a million asteroids have been discovered.

Ceres was the first asteroid to be discovered. The history of its discovery is interesting. In the late eighteenth century, a numerical relationship was found between the distances of planets. It was called Titius-Bode law following the astronomers whose work suggested the relationship. Though now the law is mostly discredited, it was widely accepted at that time. It suggested that there should be a planet between Mars and Jupiter at a distance of 2.8 AU. Searches were initiated to find the new planet which found a few asteroids, but the first discovery was purely accidental. Giuseppe Piazzi, a Catholic priest at the Academy of Palermo, Sicily, discovered Ceres while looking to locate a particular star. However, the position of Ceres shifted towards the glare of the Sun before other astronomers could confirm his discovery. It was to be again visible a few months but the observations of Piazzi were too few for an accurate calculation of the orbit at that time for a meaningful search. The famous mathematician Carl Friedrich Gauss, then at a young age of 24, developed a completely new method which involved no assumption about the form and character of unperturbed orbit except that it had to be a conic section. His predictions were instrumental in finding Ceres later [6]. Still, it posed a problem that is showed no disk but was a point in the telescopic view. With the discovery of other asteroids, it was understood that they are small bodies. Initially, it was thought that these were fragments from the rupture of a planet. Now we understand that these to be pieces which failed to form a planet being disturbed by the massive gravity of Jupiter. Based on their orbits, asteroids can be divided into several groups. The main asteroid belt lies between Mars and Jupiter. Trojan asteroids are captured by the gravity of the Sun and one of the massive planets and exist at the two Lagrange points 60$\circ$ ahead and behind the planet. Near-Earth asteroids (NEA) come close to Earth. Asteroids that actually cross the orbit of the Earth are called Earth-crossers. Orbits of some of the main belt asteroids sometimes change by impact or under the gravity of Jupiter to come towards the inner solar system making them NEAs.

The NEAs are of concern to humanity. In the life of the Earth, there have been several collisions with such objects with catastrophic consequences. It is almost certain that such a collision 65 million years ago decimated the dinosaurs and the era of mammals began. Hence, the orbits of these asteroids are carefully monitored. These are small bodies and shine brightly in the infrared. NASA’s Wide-field Infrared Survey Explorer (WISE) mission was rechristened Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE) and brought out of hibernation. It orbits the Earth and sweeps the sky looking for asteroids and comets. As of August 2023, NEOWISE has made measurements of 1,499 near-Earth objects and 272 comets. In 2022, NASA’s Double Asteroid Redirection Test (DART) mission had the spacecraft impacting with the asteroid Diomorphos which is actually one component of a double asteroid. It was observed that the orbit of Diomorphos about its partner Didymos changed due to impact. NASA has approved the plans for the launch of a Near-Earth Observer mission in 2028 which will detect the path of NEAs in the infrared.

The composition of asteroids is also important to understand the origin of the solar system as well as provide clue to the beginning of life. Some asteroids have undergone little change since the formation of the solar system and can provide us with information about its composition before the birth of the Sun. Organic molecules necessary for life have been found in meteorite and comet samples. To study this, NASA’s OSIRIS-REx mission travelled to the NEA Bennu and brought back samples to Earth in 2023. Japan’s two Hayabusa missions have already returned samples from the asteroids Itokawa and Ryugu. 

In 2006, International Astronomical Union named the a group of small bodies that orbit the Sun, and are not satellites of any planet, as dwarf planets. Pluto is the prime example, it was earlier classified as a planet. Ceres which was earlier called an asteroid is now recognised as a dwarf planet. Pluto was discovered in a scenario reminiscent of the discovery of Neptune; the orbit of Neptune could not be explained using Newton’s law led to the idea of another planet. Though Pluto was thought to be the reason for the erratic behaviour of Neptune, later it was understood the supposed deviations were not a result of the attraction of Pluto, which, in any case, was not massive enough, but due to inaccuracy of measuring the orbital elements of Neptune. Pluto has a satellite, Charon. Following a method described in the next section, the mass of Pluto was found to be only about 5% of Mercury. It is a Kuiper belt object. With the discovery of other Kuiper belt objects of similar mass, it was decided to call them dwarf planets. The number of bodies generally accepted as dwarf planets now stands at eight; except Ceres, all are Kuiper belt objects. 

4.2.7 Precession of the Perihelion of Mercury and General Relativity

 The law of gravitation, shown earlier in Eq. 4.4 is quite simple but once coupled with Newton’s laws of motion, has far-reaching consequences. One can show that under gravity, the orbit of any planet has to be an ellipse. It can also be a parabola or a hyperbola, but objects moving along those curves will never come back to the same point; there may be comets which follow such trajectories. 

An ellipse has two axes: the major axis and the minor axis. A circle is a very special case where both the axes have the same length, and then we call it a diameter. There are two special points on the major axis, called the foci, and for the planetary motion, the Sun sits in one of the foci.⁵

There are, however, other bodies in the solar system. They also tug the orbiting planets, like Mars and Venus providing the strongest tug for Earth. The Sun is also not a perfect sphere; the oblateness results in a nonzero torque on the orbiting planet. In Newtonian theory, these two effects (the first one is much more dominant than the second one), make the elliptic orbits non-stationary. What we mean is this. The line joining the perihelion and aphelion points is the major axis of the ellipse. The ellipse slowly rotates in space, which means the positions of the perihelion and the aphelion are not fixed. When the planet comes back after a full revolution, the perihelion has moved slightly; this is called the precession of the perihelion. How far does it move?

The closer a planet is to the Sun, the higher the precession rate. For Mercury, it is $574.10\pm 0.65$ arcsec. per century, so this is indeed a tiny effect even for Mercury. Newtonian theory gives about 532 arcsec. per century, so there is a shortfall of about 42 arcsec. This is extremely tiny but was measured even in the nineteenth century. The co-discoverer of Neptune, Urbain Le Verrier, even proposed the existence of a new planet, so close to the Sun that it is invisible to us, drowned in the solar glare, but provides the extra tug to match the theory with experiment.

The whole thing was solved by Einstein in his modern theory of gravity, known as General Relativity (GR). He showed that one important effect was neglected in the Newtonian theory: the curvature of the space-time geometry near a body as massive as the Sun. The effect of the curvature is most prominent for the perihelion of Mercury; that is the closest a planet ever comes. Einstein calculated and found that the curvature causes an extra precession of about 43 arcsec. per century. Bingo!

4.3 Binary Stars

With the invention of superior telescopes, orbits were observed beyond our solar system also. The first such observation was related to binary stars. The motion of any star is affected by all bodies of the universe. When the attraction between two stars is such that their relative motion can be considered only on the basis of their mutual attraction to a very large approximation, we call them binary stars. These stars usually form a long-term system. The number of binary stars in the universe is very large. According to a simple estimate, out of every one hundred stars in our neighbourhood, more than 60 are members of binary systems. A large number of stars also occur in groups of three or more; however, we do not consider them in this discussion.

It was William Herschel who actually started the study of binary stars. As we have seen, the main argument, from a scientific point of view, against the heliocentric system was the absence of parallax, i.e. change in the relative positions of the stars for a moving Earth Galileo assumed that stars are so far off that the technology of that age could not measure their parallax. In fact, the first measurement of parallax was possible only in 1839 by Friedrich Bessel. In the seventeenth century, Kepler suggested a clever way of testing the heliocentric theory. He argued that two stars which appear to be very close in the sky are, except in very rare cases, must be actually far away from each other. The fact that they appear to be close is a coincidence. Thus, as Earth changes its position, the relative motion of the stars would be different as the star further away would appear to move slowly. Although determining the absolute position of a star to a high accuracy is difficult, it may be possible to observe the change in the relative position of two stars which appear to be close to each other.

In the eighteenth century, William Hershel took up Kepler’s suggestions and decided to study the positions of binary stars. Improvement in telescopes has already led to the identification of a number of binary stars. In fact, Johann Lambert in 1761 and John Mitchell in 1767 forwarded the idea that the binary stars are actually physically close to each other as, otherwise, the number of binaries was too improbable if one assumes that stars are randomly distributed. Herschel primarily did not believe in the idea of physical proximity; however, he was forced to change his opinion by the number of binaries he could actually observe. His list in four instalments, the last being posthumously published, and prepared with the help of his sister, Caroline, contained a total of two thousand five hundred binary stars.

Herschel did observe relative changes in position of the two stars in binaries; however, it was clear that it was due to the actual relative motion of the stars as they interact via gravitation. In fact, Edmund Halley had earlier observed that stellar positions have actually changed from Hipparchus’ observations.

4.3.1 Classes of Binary Stars

Binary stars can be classified into different categories based on the method of detection. Visual binaries are those systems where the angular separation between the two companions is large enough that they can be distinguished through telescopic observations. These are to be distinguished from optical or perspective pairs, i.e. stars which are actually widely separated but occur in nearly the same line of sight. Relative motion in perspective pairs is rectilinear. Orbital motion of visual binaries may be difficult to distinguish in long-period binaries because of their large separation and hence slow movement. Possibly the most famous example of a visual binary is the Sirius system where the star that is visible through naked eye, Sirius A, has a white dwarf companion, Sirius B.

Binary systems, where the stars are not optically resolved, can still be identified by various means. In astrometric binaries, the presence of the invisible companion may be inferred from the variable proper motion of the visible star. Sirius was the first astrometric binary, later observations detected its companion. In photometric binaries, a regular change in the luminosity is caused by the motions of its components. Usually, these are eclipsing binaries, the brightness changes due to total or partial eclipse of one component by another is schematically shown in Fig. 4.9. An actual light curve for the binary AH Virginis drawn from the data obtained by HIPPARCOS satellite, folded with period⁶ is also shown in the bottom panel of the figure.

Fig. 4.9

Credit NASA. (Bottom) Light curve of the eclipsing binary AH Virginis folded with period from HIPPARCOS data

(Top) Schematic view of light curve for eclipsing binary.

     Particularly important are spectroscopic binaries  which may be identified from the study of their relative motions through spectroscopy. Unless we happen to be along the normal to the plane of orbit of the binary system, the motion of the stars would be such that generally one has a component of velocity away from us, while the other will have a velocity component toward us. Of course, with time, the star that is going away from us will move towards us while the other will do just the opposite. Because of the relative velocity of the individual stars with respect to us, the light form the stars will be Doppler shifted. The change in velocity is superposed on the proper motion of the centre of mass of the system and can be detected from the change in wavelength in the stellar spectra. The spectra of the star whose motion is away from us will be relatively red shifted, while that of its companion will be blue shifted. If it is possible to observe the spectra of both the components, it is called a double-line spectroscopic binary. The velocity plot of such a binary is shown in Fig. 4.10. If one of the companions is not bright enough, the spectra of one component are observed. This class of systems are called single-line spectroscopic binaries. A component of a single-line spectroscopic binary may be a planet, a situation that will be discussed in the next section. More details about binary stars may be found in standard textbooks [e.g. [7, 8]].

Fig. 4.10

Velocity plot for double line spectroscopic binary

 

 

4.3.2 Measurement of Masses

 

Masses of component stars can be measured in visual binaries from the orbital elements by observing their motions. The absolute size of the orbits can be determined if the distance of the stars is known form the angular measurements. Observation provides the projection of the elliptical orbits on the plane of the sky. However, the actual orbit can be calculated by determining the orientation of the orbit keeping in mind the fact that each component star occupies a focal position of the orbit of the other and comparing it with the projected position.

It is possible to deduce Kepler’s law in the case of circular orbits and find expressions for the masses of the components rather easily. Referring to Fig. 4.11, one can write, from the definition of the centre of mass of the system (O),

Fig. 4.11

Binary stars in circular orbits

$$\begin{aligned} \frac{a_1}{a_2}=\frac{m_2}{m_1}, \end{aligned}$$

(4.5)

where $a_1$ and $a_2$ are the radii of the orbits of the two components, and $m_1$ and $m_2$ their masses. The radius of the relative orbit is given by

$$\begin{aligned} a=a_1+a_2. \end{aligned}$$

(4.6)

Combining the above equations, we get

$$\begin{aligned} a_1=\frac{m_2a}{m_1+m_2}. \end{aligned}$$

(4.7)

The period of rotation for each body around their common centre of mass is given by P. For a circular orbit, the velocity is a constant given by

$$\begin{aligned} v_1=\frac{2\pi a_1}{P} \end{aligned}$$

(4.8)

    Now, the centripetal force on the first body may be rewritten, using the above two equations, as

$$\begin{aligned} F_1=\frac{m_1v_1^2}{a_1}=\frac{4\pi ^2m_1a_1}{P^2}=\frac{4\pi ^2 m_1m_2a}{P^2(m_1+m_2)}. \end{aligned}$$

(4.9)

The force between the two bodies is given by Newton’s law,

$$\begin{aligned} F_1=F_2=\frac{Gm_1m_2}{a^2}. \end{aligned}$$

(4.10)

Comparing the expressions for $F_1$, we get the Kepler’s third law as

$$\begin{aligned} P^2=\frac{4\pi ^2}{G(m_1+m_2)}a^3. \end{aligned}$$

(4.11)

Although we have deduced Eq. 4.11 for circular orbits, this is also true for elliptical orbits, though the mathematics is a bit more involved. For the Sun-Earth system, the mass of the Earth ($m_2$) is negligible compared to that of the Sun ($m_1=M_\odot$). For this system, Eq. 4.11 then reduces to

$$\begin{aligned} P^2=\frac{4\pi ^2}{GM_\odot }a^3. \end{aligned}$$

For the Sun-Earth system, the period of rotation is one year and the semi-major axis of Earth’s orbit is 1 AU. In this system of units, assuming that the mass is expressed in terms of solar mass ($M_\odot =1.99\times 10^{30}$ kg), the factor $4\pi ^2/G$ equals one, and it is possible to rewrite Kepler’s law as

$$\begin{aligned} m_1+m_2=\frac{a^3}{P^2}. \end{aligned}$$

(4.12)

Using Eqs. 4.5 and 4.12, it is then possible to find the masses of the stars $m_1$ and $m_2$ in terms of $M_\odot$.

Sirius A and B are separated by a distance of approximately 19.8 AU and the time period of rotation is 50.1 years. Thus, the total mass is

$$\begin{aligned} m_1+m_2=\frac{19.8^3}{50.1^2}=3.09 M_\odot \end{aligned}$$

The ratio of the distances from the centre of mass is in the ratio 2:1. Thus, the masses are in the ratio 1:2. Hence, we conclude that the masses of Sirius A and B are 2.06 and 1.03 solar mass, respectively.

It is possible to get an estimate of stellar masses for spectroscopic binaries. For simplicity, we assume that the orbits of the stars are circular around their centre of mass. Let us assume that the line of sight makes an angle i with the orbital plane. Doppler shift measurement allows us to measure the component of the velocities with respect to the common centre of mass, $v_{1obs}$ and $v_{2obs}$, respectively, for the two stars, along our line of sight in a double-line spectroscopic binary. Then the true velocities are related to the observed velocities through the relations

$$\begin{aligned} v_1=v_{1obs}\sin i~~~v_2=v_{2obs}\sin i. \end{aligned}$$

(4.13)

For a circular orbit, the true orbital velocities are related to the radii of the orbits, $a_1$ and $a_2$, and the period of rotation, P through the relations

$$\begin{aligned} v_1=\frac{2\pi a_1}{P}~~~v_2=\frac{2\pi a_2}{P}. \end{aligned}$$

(4.14)

From Eq. 4.6, using these results, we obtain

$$\begin{aligned} a=a_1+a_2=\frac{P}{2\pi \sin i}(v_{1obs}+v_{2obs}) \end{aligned}$$

(4.15)

so that Sect. 4.3.2 reduces to

$$\begin{aligned} m_1+m_2=\frac{P}{8\pi ^3\sin ^3 i}(v_{1obs}+v_{2obs})^3 \end{aligned}$$

(4.16)

    It is difficult to measure i but in the case of eclipsing binaries, we can get a reasonable estimate. Since we can see the eclipse, the orbital pane of the two stars must be along the line of sight unless they are very close to each other. In such cases, $\sin i$ must be close to 1. Generally, binaries are separated by such distances that $i>75^\circ$ corresponding to $\sin ^3i> 0.89$, so that there is approximately 10% error in the determined masses.

For a single-line spectroscopic binary, it is possible to determine a mass function from the above relation

$$\begin{aligned} \frac{m_2^3\sin ^3 i}{(m_1+m_2)^2}=\frac{P v_{1obs}^3}{8\pi ^3}. \end{aligned}$$

(4.17)

Of course, a proper determination of the mass depends on measuring the inclination i. 

4.4 Exoplanets

Exoplanets are planets outside our solar system. The first positive identification of an exoplanet was made in 1992 when Wolszczan and Frail used the 305 m Arecibo radio telescope to demonstrate that two planet size bodies are orbiting the 6.2-ms pulsar PSR1257 +12 [9]. As we will see in Chap. 7, pulsars are rotating neutron stars which emit very regular radio pulses. Wolszczan and Frail made precise measurements of the pulses to show a periodic variation consistent with two planets of masses at least 2.8 $M_\oplus$ and 3.4 $M_\oplus$, at distances of 0.47 Au and 0.36 AU, respectively. Here $M_\oplus$ refers to the mass of the Earth and AU is the mean Sun-Earth distance. The method is essentially similar to the one used for single-line spectroscopic binaries discussed earlier.

Later measurements have utilised optical spectra also to detect planets. In 1995, the first detection of a planet around a main sequence star ⁷ was made by Michel Mayor and Didier Queloz [10]. This discovery earned them a Nobel Prize in 2019. In Fig. 4.12, the time-velocity plot of Proxima Centauri, the nearest neighbour of our solar system, has been shown. The graph clearly shows a periodic variation indicating the presence of a planet around the star.

Fig. 4.12

Credit ESO/G. Anglada-Escudé

Time-velocity curve of Proxima Centauri showing the presence of a planet.

Measuring Doppler shift of the primary stars is one of two two principal methods that are used to detect exoplanets.  The other is the method of eclipsing binaries. If the Earth happens to lie along the orbital plane of the exoplanet, then periodically it will come between its primary star and us causing a drop in the light intensity. Figure 4.13 shows the light curves for the first five exoplanet discoveries by the Kepler mission. The drop in intensity also is an indication of the planet’s inclination and its distance from the primary star. The Kepler space telescope of NASA worked between 2009 and 2018 to find a large number of exoplanets in this method. Figure 4.13 shows the light curve for the first planet discoveries by it. It has now been succeeded by Transiting Exoplanet Survey Satellite (TESS) which has been operational since 2018. Spitzer infrared space telescope of NASA made the first weather map of a gas giant exoplanet. Brief descriptions of these space telescopes may be found in Appendix A. Absorption spectra can be used to find out about the atmospheric composition of the planet. Till date, more than 5,500 exoplanets have been discovered. Many Earth-sized planets have been found in habitable zone.⁸

Fig. 4.13

Credit NASA/Kepler Mission

Light curve for the first five planet discoveries by the Kepler Mission.

  Equation 4.17 can be utilised to estimate the mass of exoplanets. Since the mass of the planet ($m_2$) with respect to that of the star ($m_1$), the equation can be approximated as

$$\begin{aligned} {m_2^3 \sin ^3 i}=\frac{P v_{1obs}^3m_1^2}{8\pi ^3}. \end{aligned}$$

(4.18)

If transit is observed, we can take $\sin i\approx 1$. The mass of the primary star may be estimated from its class. This method has been used to estimate the masses of exoplanets.

4.5 Galactic Rotation

Milky Way is a spiral galaxy. Viewed from outside, they look like a flat disk, with a central bulge and spiral arms coming out of it. One of our nearest galaxies, Andromeda, is also a spiral galaxy, and there are lots and lots of such spiral galaxies in the universe. Figure 4.14 shows the image of the Andromeda galaxy in infrared taken by the Spitzer space telescope clearly indicating the spiral arms.

Fig. 4.14

Credit: NASA/JPL-Caltech/Karl D. Gordon, University of Arizona

Image of the Andromeda Galaxy taken by Spitzer in infrared, 24 mm.

The spiral arms of galaxies were created as the disk rotated about its centre. The centre is thickly populated, with most of the stars packed up in a very small region.⁹ The arms spread out like some sort of cosmic wisp. They are not devoid of stars, but the population is sparse. Our own Sun, and thus the entire solar system, lives in one such spiral arm of the Milky Way.

The spiral arms slowly rotate about the central bulge, a remnant of the original rotation from which they were created. The rotational speed can be easily measured with the standard Doppler effect.  And then comes the puzzle.

Suppose there is a star of mass m at a distance R from the central bulge, and rotating with a velocity v. It is pulled by the central bulge and the intervening medium, but it does not fall in because of the centrifugal force. The two forces balance, and the star maintains a steady orbit. But the centrifugal force is just $mv^2/R$. What is the gravitational pull?

The gravitational pull is exerted by all matter inside the radius R. If we assume a sphere of radius R centered on the galactic centre so that the star sits on its surface, all the matter inside the sphere pulls the star in—but not the matter outside the sphere. We do not know the exact matter distribution inside the sphere, but to a very good approximation, almost all the visible mass of the galaxy—made of the stars and the interstellar gas and dust—may be found in the small central bulge. If that mass is M, we can write

$$\begin{aligned} \frac{GMm}{R^2} = \frac{mv^2}{R}\,, \end{aligned}$$

(4.19)

so that the velocity v falls as $1/\sqrt{R}$; the farther a star is from the centre, the slower it moves.

Fig. 4.15

Credit: Mario de Leo/Wikipedia CC By-SA 4.0 https://commons.wikimedia.org/wiki/File:Rotation_curve_of_spiral_galaxy_Messier_33_(Triangulum).png

Rotation curve of spiral galaxy M33 (yellow and blue points with error bars), and a predicted one from distribution of the visible matter (grey line). The data and the model predictions are from Corbelli and Salucci [11].

In Fig. 4.15, the rotation curve expected on the visible line and the curve actually observed have been compared for the galaxy M33 (the 33rd object of the catalogue made by Charles Messier). The data for all the galaxies showed something completely different; v is almost flat, no matter what R is. There is no sign of that $1/\sqrt{R}$ fall as predicted by Newtonian dynamics. In 1936, astronomer Fritz Zwicky postulated that the mass distribution that we have assumed must be wrong. We were talking about the visible mass, objects that emit or absorb light (or some other form of electromagnetic radiation, like radio waves), but suppose there exists a huge lot of mass that does not interact with light. Like useless people in top-level positions, this matter does not illuminate anything but just puts its weight in it.

This is known as the dark matter. The galactic rotation curves were the first evidence of dark matter, albeit an indirect one, and there have been more indirect evidences since then. We now know that there are about 5 times more dark matter than visible matter; the stars and the planets make a tiny contribution to the mass budget of the universe. But we do not know as yet what exactly the dark matter is. All the direct detection experiments that look for dark matter yielded negative results so far; hopefully, we will soon have something positive. Worse, there is no model of elementary particles that accommodate a dark matter candidate. To put it simply, we know that dark matter exists, but we do not know what it is, we do not know how it can be detected. That is some puzzle. 

4.6 The Black Hole at the Centre of Our Galaxy

  The universe is full of strange, weird, and bizarre objects. They are strange not because they are rare, but because we do not see them in our immediate vicinity, and also because we do not yet fully understand their dynamics. The strangest one is arguably the black holes.

What is a black hole? It is the final stage of a supermassive dying star. A star shines by hydrogen burning. It is in a state of dynamic equilibrium, stuck between an outward radiation pressure plus the pressure of the hot gas coming out of the central region, and an inward gravitational pull. When the hydrogen runs out, the star starts to collapse under gravity. This ignites the helium that has been formed in the core, and the star again reaches another equilibrium. This process can continue till the core contains only $^{56}$Fe,¹⁰ which cannot be further ignited. Thus, no more nuclear fusion, which may resist the gravitational collapse, is possible.

The collapsing star, however, may be kept in equilibrium by a nonthermal source of pressure. One common example is the Fermi pressure or the electron degeneracy pressure that builds up because the electrons, loosely speaking, repel each other as a consequence of Fermi statistics. This happens when the electrons are so close that their wavefunctions start to overlap significantly, the same mechanism that causes the band structure in solids. A star which is held in equilibrium by the electron Fermi pressure is known as a white dwarf star. We will encounter white dwarfs in later chapters. Its typical radius is a few thousand kilometres, and its maximum mass can be about 1.4 times the solar mass. This is known as the Chandrasekhar limit, worked out in 1930 by the young Indian astrophysicist Subrahmanyan Chandrasekhar. Thus, if the electron Fermi pressure is to keep a star in balance, its maximum mass can only be about $1.4 M_{\odot }$.

A more massive star undergoes further compression—the electron Fermi pressure is not enough to resist the gravitational collapse. In such a star, protons and electrons combine to form neutrons (and neutrinos, which escape). Such stars are called neutron stars, and they are highly interesting objects. The typical radius of a neutron star is about 10 km, and it generally rotates very rapidly about its axis (which is just conservation of angular momentum; if radius goes down by a factor of $10^5$, the angular velocity increases by $10^{10}$ if no mass is lost). Further discussions about neutron stars and their discovery can be found in Sect. 7.4.  

The maximum mass of a neutron star is about 0.7 solar mass, so some mass is shed when the star collapses beyond the white dwarf stage. Modern estimates put this limit a bit higher; a star can reach a nonthermal equilibrium if its mass is not greater than something between 1.5 and 3 $M_{\odot }$, taking into account all the uncertainties in such calculations (this is known as the Tolman-Oppenheimer-Volkoff limit).  As most of the mass is not retained, one estimates the starting mass to be around 10–20 $M_\odot$. But there are a lot of stars in the sky which are more massive than this. Some of them shed a large fraction of their mass by violent explosion; still, there must be some stars which have run out of nuclear fuel, while the mass is still greater than $3M_\odot$. What happens to them?

The answer is that they are in a state of ongoing gravitational collapse, and therefore the density goes on increasing. The fate of such objects can be deduced from Einstein’s equations of General Relativity, which relate the space-time curvature to the density of matter and energy in that region. What happens for a static spherical mass was first calculated by the German astronomer and physicist Karl Schwarzschild in 1915 (he died the next year at a very young age of 42). If the mass rotates, or has electric charge, the situation becomes more complicated.

Einstein’s equations break down if the radius of the object shrinks to zero, or the density becomes infinite. This is called a singularity; there is no way to know what happens at the singularity.¹¹ But something interesting happens even before that, when the radius shrinks to a value $2GM/c^2$, where G is Newton’s gravitational constant, and c is the velocity of light. This is called the Schwarzschild radius $R_s$; it is a very small number. For Earth, $R_s$ is about 9 mm, which means you have to compress the Earth to the size of a pea to reach that radius. The Sun is more massive and has $R_s\approx 3$ km.

We know that light rays bend in a curved space-time. In particular, it is not very difficult to calculate the bending for a static spherical object. For a light ray grazing the Sun, it is 1.75 arcsecond. The geometry is much more violently curved near a body whose radius is close to $R_s$. Light rays emitted from the surface of such a collapsing star bend, but can escape the star and come to us as long as the radius is greater than $R_s$. When the radius is below $R_s$, the light rays get permanently trapped—the bending is so large that emitted rays fall back on the star, nothing comes out, and there is no way to obtain any information about the star except to observe its nearby geometry. Such a body is called a black hole, and the radius $R_s$ is known as the event horizon.¹² 

The light ray emitted at $r=R_s$ does not fall back on the star, neither does it come to us—it goes around the star like a satellite. What should one see from outside? In a very short period of time—typically $10^{-5}$ s—the collapse of a star slows down, the emitted light gets more and more red-shifted until the red shift reaches infinity, the star grows dark and reaches the black hole stage. All history of the star will be erased from us; such a black hole is characterised only by its mass M. Note that black holes, in all probability, do not constitute the dark matter that we talked about in the previous section.

This is true only for a static charge-neutral black hole. Apart from mass, a black hole can have a nonzero angular momentum; such a rotating object is known as a Kerr black hole. Rotation of black holes has been discussed in Sect. 7.5. It can also be static and charged (a Reissner-Nördstrom black hole), or rotating and charged (a Kerr-Newman black hole), but that is all for a classical black hole. To any outside observer, it is completely specified by its mass, angular momentum, and electric charge. No other information that passes through the unidirectional event horizon can ever be restored; this is cheekily referred to as a no hair theorem. 

Are there such black holes in our galaxy? We have preliminary evidence for some. If an ordinary star is near a black hole, the matter from the star gets sucked up by the black hole, and the falling matter emits X-rays, which is an indirect signal of such a black hole. We have evidence for some such X-ray signals.¹³ The black hole can act as a gravitational lens; the light coming from a galaxy behind the black hole bends around it so that we see a double image, which is called microlensing.

There is something more. All the black holes scattered here and there inside our galaxy (and in all other galaxies) have masses which are, at the most, 100–200 times the solar mass. The first signal of gravitational wave, about which we will talk later, that was detected in September 2015, came from a merger of two black holes, 36 and 29 times heavier than the Sun. But astronomers have reasons to believe that most of the big galaxies in the universe, either spiral or elliptic, have a supermassive black hole at the centre. By supermassive, we mean something of the order of a million solar masses. Maybe it has started as a small black hole, but the density of stars is high in the central region, and so the black hole, like a Jurassic carnivore, ate up more and more stars over billions of years and became so massive. The supermassive black holes at the centre of the galaxies are typically of a few million solar masses, but there can be some real giants. Astronomers have found very recently—the paper was published in Nature Astronomy in February 2024—a monster black hole of 17 billion solar masses at the centre of a galaxy 12 billion light-years away. This black hole gobbles up almost one solar mass every day, its accretion disk is 7 light-years wide, and the tremendous amount of X-ray coming out of this galaxy makes it the brightest known object in the universe [12].

The first image of such a supermassive black hole was published in April 2019. This black hole is at the centre of a galaxy called M87 (the 87th object of the catalogue made by Charles Messier) and was imaged by a wonderful machine called Event Horizon Telescope (EHT) ¹⁴; the image can be seen in Fig. 4.16. But no light can ever come out of a black hole, how do we photograph it? The answer is simple: we do not photograph the black hole. We detect the radiation coming out of the gas that is being sucked by the black hole, and that radiation is a telltale sign that there is some massive object sucking the nearby matter in.

Fig. 4.16

Credit EHT Collaboration

Image of the supermassive black hole at the centre of M87 galaxy.

The natural question is whether there is such a similar huge black hole at the centre of our own galaxy. The centre lies in the direction of the constellation Sagittarius. Unfortunately, it is not easy to look at the centre, even with the best telescopes. There is a huge amount of dust—the matter particles that failed to coalesce into a star—in the central region which absorbs most of the light of the stars there, and the centre looks like a huge dust cloud.

A bright radio source in Sagittarius was discovered in 1954. Being the brightest radio source, this was named Sagittarius A. In 1982, astronomer Robert Brown added an asterisk and called it Sagittarius A*; the asterisk means the source is exciting. Brown was right, the source turned out to be a very compact object. The object has been observed in the radio range only; the optical part is too dim because of the absorption by the dust. 

People started looking at stars orbiting Sagittarius A*. The dust cloud made their task very hard, and they had to watch for years. Two groups did pioneering works on the orbits of several stars. The European group was led by Reinhard Genzel and the American group by Andrea Ghez, and they were both awarded the Nobel Prize in 2020. They confirmed that the mass of Sagittarius A* is about 4.15 million solar masses. Considering its spatial extent (which can be measured from radio emission), it can be nothing but a supermassive black hole.

This must be one of the most dramatic rhythmic motions in the sky!

Finally, in 2022, the Event Horizon Telescope published the first picture of this giant, presented in Fig. 4.17 something similar to that found in M87. There are reasons to believe that they are not static but rotating on their axes, so the Schwarzschild radius is not exactly the event horizon.

Fig. 4.17

Credit EHT Collaboration

The first image of the supermassive black hole at the centre of our galaxy.

4.7 Summary

Early attempts to calculate orbits of solar system bodies naturally assumed a geocentric system. Ptolemy used epicycles to describe the motion of planets, the Moon and the Sun in the sky and was successful in explaining the apparent retrograde motion of planets. Copernicus was the first to propose a heliocentric system which worked better than the model of Ptolemy. Tycho Brahe, although he did not believe in the heliocentric system, built an extensive database based on superb naked-eye observations, which helped Kepler to propose the three by observation through telescope, in the case of the former, and mathematics, in the case of the latter. Galileo’s discovery of the Jovian moons not only strongly supported the idea of a moving Earth but also was instrumental in the first measurement of the velocity of light. Newton proposed the law of universal gravitation which could explain the laws of planetary motion. Newton showed that the law predicts an oblate shape for the Earth, that could explain the precession of the equinoxes. His idea was verified by measuring the shape of the Earth at different latitudes. The size of the solar system was measured by observing the transit of Venus. Newton’s law was used brilliantly by Le Verrier and Adams independently to propose a new planet, which was subsequently discovered and named Neptune. The solar system also contains numerous comets, asteroids and dwarf planets; these are still being detected and their orbits are being computed. However, Newton’s law failed to describe the orbit of the planet Mercury, a task brilliantly carried out by the more successful theory of gravitation, the general theory of relativity.

Advances in technology allowed us to observe binary stars and exoplanets. from measurements of motions, masses of stars and planets could be measured, significantly advancing astronomy. The motion of the stars of the galaxies as a whole showed the existence of some completely new type of matter, yet to be understood by us. Measurements of orbits of stars in the galactic core proved what has been long assumed, the supermassive black hole at the centre of our galaxy.