# Planetary orbits, binary star orbits and orbits around black holes

Ardra S.

Shantanu Rastogi

Department of Physics, DDU Gorakhpur University, Gorakhpur, UP 273009

## Abstract

In the current study, planetary orbits are examined using Kepler's law of planetary motion. Applying Newton's laws and central force solutions, equations for elliptical orbits are derived. The trajectory followed by the celestial bodies in terms of their energy and masses is studied. This will be further extended to understand the motion of bound binary stars and orbits of celestial bodies around black holes. In binary stars, the concept is used to understand the position of Lagrange points and its applications e.g. the Aditya solar observatory is scheduled to be placed in L1 point between the Sun and Earth. The relation between time period and major axis length for different eccentricity orbits is modelled. In the second phase extreme gravity situations like those near black holes, are studied, where the concepts of classical orbits break down and one needs to consider general relativity. The wrapping of the space-time fabric in general relativity can be visualized using a stretched membrane with weight at the centre. The path of an object near a black hole changes from its classical elliptical orbit and it spirals into the central massive object. In this phase there is a large amount of energy released by the object. The recent first direct observation of a black hole in the centre of galaxy M87 shows the photons being accreted into the central object in a spiral path. The radio emission by these accelerated photons is observed in the photograph. On a stretched membrane the path of a marble can be considered nearly like that what an object near a black hole will follow. The amount of space - time wrapping varies according to the mass of the central black hole and the orbit depends on the initial energy of the trapped object. An experiment was designed to study this phenomenon on a stretched membrane (sheet of cloth). When a heavy object is at its centre, it mimics space time curvatures near massive objects. Thus, the study is useful in understanding and visualizing complex astrophysical phenomenon.

Keywords: Kepler's laws, spacetime curvature, black holes, stretched membrane

## INTRODUCTION

The Newton's laws of motion as Universal law of gravitation remained true until the advent of Einstein’s theory of special relativity. Newton's theory depends on the concept that mass, time and distance are constant. But the theory of relativity treats time, space and mass as changeable according to observer's frame of reference. Within a fixed frame of reference, the laws of classical physics, i.e. Newton's law hold true. But to explain the motion of object from two different frames of reference. Einstein's theory of relativity is required, where space and time are no longer separate entities, instead they are just different direction in single object called space - time. This space time is usually flat but can be warped and curved by the matter and energy in it. Several experiments have already demonstrated that Einstein's theory of general relativity describes the way gravity behaves, especially for high speeds and large masses. But the theory is enshrouded in complex mathematical symbols and conceptual visualization is difficult.

The laws of Physics are same everywhere in the universe, this is the Lorentz symmetry statement, which is an important postulate in Einstein’s relativity theory. Planetary orbits can be best tests for studying any violation of Lorentz symmetry. Orbits are calculated using Newton's law with a correction for general relativity. If Lorentz symmetry is violated, then the orbits for far away objects will differ by a measurable amount. In the present project study of orbits is performed in different scales. All the planets are moving around the sun in ellipse rather than perfect circle and the long axis of the ellipse slowly turns as the planets tug on each other. These orbits can be explained by Kepler’s laws and classical physics. In the galaxy, however, most stars come in pairs or multiplies. It is estimated that more than half of all the stars in the sky have other stars orbiting them as companions. These binary stars orbit around a common centre of mass. The binary orbits create interesting potential variations around them that may influence satellites going around. In the extreme situation of very massive object, like a black hole, space time is very distorted as per the requirements of general relativity. As a result, near the black hole orbits are unstable. To visualize the spiral inflow of matter after it is trapped in the black hole gravitational field, experiment on sheet of cloth is performed. In the following sections these studies are reported.

## PLANETARY ORBITS

The curved trajectory of any object under the influence of a central force, like gravity, is the orbit of the body. More specifically paths that are indefinitely extended or repetitive are termed as orbits, e.g. the trajectory of a planet around a star or a satellite around a planet. Gravitational influence of point masses or of spherical stars/planets lead to three distinct types of paths​[1]​. The closed path is usually like an ellipse and open path is hyperbolic. This depends on the initial energy or velocity of the object as it encounters gravity. For velocity smaller than escape velocity the path is elliptical and velocities greater than escape velocity lead to hyperbolic trajectory. In between the two, when velocity is equal to escape velocity, the object takes parabolic path. With the central mass (e.g. a star) at the focus the path of a trapped body (e.g. a planet) is described by Kepler's laws of planetary motion. A comet in a parabolic or hyperbolic orbit about a barycentre is not gravitationally bound to the central star. When the two masses are comparable in order, they orbit each other around a barycentre. The periapsis is that point at which the two objects are closest to each other and the apoapsis is that point at which they are the farthest. More specific terms used for specific bodies include perigee and apogee for the lowest and highest parts of an orbit around Earth, while perihelion and aphelion are the closest and farthest points of an orbit around the Sun.

## Kepler's Laws of Planetary Motion

From observations and derived empirical relations Kepler gave the following three laws for the orbit of planets:

• Kepler's first law or law of orbits - The orbit of a planet forms an ellipse with the sun at one focus.
• Kepler's second law or law of area - The sun-planet radius vector sweeps out equal areas in equal time.
• Kepler's third law or law of period - The square of the period of revolution of a planet is proportional to the cube of the semi major axis of its elliptical orbit.

Newton showed that the empirical laws of Kepler can be derived from inverse square law of gravity. Yet the laws are approximate to the extent that the planets and the sun can be idealized as perfect spheres and that the mass of the sun is much greater than that of plan. It is also approximate to the extent that relativistic and quantum-mechanical effects are negligible. All these approximations are valid in the actual solar system. By using the Kepler's laws the path travelled by a celestial body in the central force field can be calculated and also the energy change that happens to the body​[2]​.

## Path of a Celestial Body in Central Force Field

The gravitational force exerted by a large mass (star or planet) on another celestial body is inverse square central force. To find the path followed by the body under such central force, consider the radial component of the force F(r) acting on the system. The angular component of force F(θ) is zero as the angular momentum L is constant i.e.

$\displaystyle L\;=\mu\;r^2\;\frac{\operatorname d\theta}{\operatorname dt}$

where μ is the reduced mass and θ is the angular displacement. The radial component of force is resultant of the central force and centrifugal force (neglecting Coriolis force),

$\text{Unexpected text node: '  '}$

But from (1)

$\text{Unexpected text node: '  '}$

Therefore, F(r) can be written as,

$\displaystyle F(r)=\mu\frac{d^2r}{\operatorname dt^2}-\frac{L^2}{\mu r^3}$

To have an idea about the path travelled by the particle, we study the variation of (1⁄r). For convenience we substitute (1⁄r) with 'u'. Therefore, the first and second derivatives of r are -

$\displaystyle \frac{\operatorname dr}{\operatorname dt}=-\frac1{{\displaystyle u}^2}.\frac{\partial u}{\partial\theta}.\frac{\operatorname d\theta}{\operatorname dt}=-\frac L\mu\frac{\partial u}{\partial\theta}$
$\displaystyle \frac{\operatorname d^2r}{\operatorname dt^2}=-\frac L\mu\frac{d^2u}{\operatorname d\theta^2}\frac L{\mu r^2}=\frac{L^2u^2}{\mu^2}\frac{\operatorname d^2u}{\operatorname d\theta^2}$

On substituting the values of dr/dt and d²r/dt² in equation (4) it becomes:

$\displaystyle F(\frac1u)=\mu(-\frac{L^2u^2}{\mu^2}\frac{d^2u}{\operatorname d\theta^2})-\frac{L^2u^3}\mu$

But we have

$\displaystyle F(r)=\frac{-Gm_1m_2}{r^2}=-\frac k{r^2}=-ku^2$

On equating equations (4) and (8) we get -

$\displaystyle -ku^2\;=\;\mu(-\frac{L^2u^2}{\mu^2}\frac{d^2u}{\operatorname d\theta^2})-\frac{L^2u^3}\mu$

or

$\displaystyle \frac{d^2u}{\operatorname d\theta^2}+u=\frac{\mu k}{L^2}$

The solution for this differential equation, giving variation of u with ϴ, is

$\displaystyle u=A\;\cos\left(\theta-\theta_0\right)+\frac{\mu k}{L^2}$

in terms of r this can be written as

$\text{Unexpected text node: '  '}$

Comparing this with the equation of conic section having eccentricity ε, implies

$\displaystyle \varepsilon=\frac{AL^2}{\mu k}$

The shape of the conic section or the path of the object (i.e. the orbit) depends on the value of ε as:

If ε > 1, the path is open and hyperbolic

If ε = 1, the path is open and parabolic, while

If ε = 0, the path is colsed circular and for the conditions 0 ≤ ε < 1 the path is closed ellipse. For ellipse the r_maximum (semi major axis) and r_minimum (semi minor axis) can be given as -

$\displaystyle \frac1{r_{max}}=\frac{\mu k}{L^2}(1-\frac{AL^2}{\mu k})\;=\;\frac{\mu k}{L^2}(1-\varepsilon)$
$\displaystyle \frac1{r_{min}}=\frac{\mu k}{L^2}(1+\frac{AL^2}{\mu k})\;=\;\frac{\mu k}{L^2}(1+\varepsilon)$

Equation for conservation of energy is

$\displaystyle E=\frac12\mu(\frac{\operatorname dr}{\operatorname dt})^2+\frac{L^2}{2\mu r^2}+U=0$

Here U is potential energy. At turning points dr/dt=0, therefore,

$\displaystyle E=\frac{L^2}{2\mu r^2}+U=\frac{L^2}{2\mu}(\frac1{r^2})+k(\frac1r)$
$\displaystyle \frac{L^2}{2\mu}(\frac1{r^2})+k(\frac1r)-E=0$

This form of quadratric equation has solution (-b±√(b²-4ac))/2a and we can obtain

$\displaystyle \frac1r=\frac{\mu k}{L^2}\pm\frac\mu{L^2}(k^2+\frac{2L^2E}\mu)^\frac12$

On comparing with equations (14) and (15) we get the value of A,

$\displaystyle A^2=\frac{\mu^2k^2}{L^4}+\frac{2\mu E}{L^2}\cos\left(\theta-\theta_0\right)$

and the square of eccentricity can be written as

$\displaystyle \varepsilon^2=(1+\frac{L^2+2E}{\mu k^2}\cos\left(\theta-\theta_0\right))$

From this energy E is

$\displaystyle E=\frac{(\varepsilon^2-1)\mu k^2}{2L^2}\sec\left(\theta-\theta_0\right)$

Change in energy can be calculated by taking the first partial derivative of the above equation,

$\displaystyle \operatorname dE=\frac{\mu k^2}{2L^2}(\varepsilon^2-1)\sec\left(\theta\right)\tan\left(\theta\right)\operatorname d\theta+\frac{\mu k^2\mu}{L^2}\sec(\theta)d\varepsilon$

Also change in energy with respect to eccentricity can be represented as -

$\displaystyle \frac{\operatorname dE}{\operatorname d\varepsilon}=\frac{\varepsilon\mu k^2}{L^2}\sec\left(\theta\right),Since\;\frac{\operatorname d\theta}{\operatorname d\varepsilon}=0$

This equation gives the variation ​​of 1 ⁄r with θ.

Thus in terms of energy E if E > 0, the path will be a hyperbola; for E < 0, a closed ellipse is formed; E = 0 leads to a parabola and for E = −μk2 ⁄2L2, a circle is traced​[3]​​​​​​. When due to friction the energy is lost the orbit size becomes smaller and the resultant track will be similar to a convolute.

## BINARY STAR ORBITS

A binary star system consists of two stars which orbit around their common centre of mass. A large proportion of stars belong to such system. Simplest binary star systems are considered to satisfy the conditions – stars have equal or comparable mass; their orbits lie on the same plane; their planets orbit on the same plane as the stars; planets are of negligible mass and there are no tidal effects. In astronomy, Binary stars are of immense importance as they allow testing of gravitational theories and determining the masses of stars.

Some binary systems are observable, with components well separated, through a telescope. These too are governed by Kepler’s laws, as modified by Newton to account for the centre of mass. Each body executes an elliptical orbit such that at any instant the two bodies are on opposite sides of the centre of mass. Orientation of the binary system is usually inclined with respect to the line of sight. Thus, observations are only the projection of that orbit on the celestial sphere. For example, a circular orbit may look flattened and therefore elliptical. Geometry of observation governs the properties that can be derived from binaries and they are accordingly classified​[4]​​​.

Orbit of a binary star system

## Visual binaries

A visual binary is a system in which the component stars can be individually resolved through a telescope. Long-term observations can be made to plot the relative positions of the members and obtain the orbits of the stars. Examples of visual binaries in the southern sky include α Crucis, β Crucis, γ Centauri and Castor in Gemini. Some stars appear close together through a telescope but may not be gravitationally bound, and can be hundreds of parsecs apart. Such line-of-sight pairs are not true binaries but "optical pairs."

## Spectroscopic binaries

For very distant binary systems or where the two stars are very close, the stars cannot be resolved visually. These binary systems are detected by the Doppler shifts in their spectral lines and are called spectroscopic binaries. As the stars orbit each other one star, A, may be moving towards us whilst the other, B, may be moving away. The spectrum from A will, therefore, be blue-shifted to higher frequencies (shorter wavelengths) while the spectrum from B will be red-shifted. The resulting spectrum will this have a double peak or a doppler broadened peak.

## Eclipsing binaries

Based on the brightness variation and spectroscopic observations unresolved stars can be identified as being binary systems. The variations in light intensity is due to periodic eclipsing of one star with another. In these binary systems the orbital plane lies edge-on to the observer and one star comes in front of the other periodically causing variations in luminosity. These systems are called eclipsing binaries. The first eclipsing binary detected was Algol, β Perseus, also known as the Demon star, possibly due to its changing brightness.

## Astrometric binaries

Some stars, when observed repeatedly over time, show a perturbation or "wobble" in their proper motion. A periodic wobble implies the gravitational influence of an unseen or dimmer companion star. The wobble is due to the orbit around the common centre of mass. Such systems are called astrometric binaries.

## "Exotic" pairs

In late 2003, the Parkes radio telescope discovered a pulsar, PSR J0737-3039, having a 23-millisecond pulsar ‘A’ and a 2.8 second companion pulsar ‘B’. These orbit around each other with a period of just 2.4 hours. This system is an eclipsing binary system and due to their large masses in fast orbits, is an amazing test bed for General Relativity and gravitational waves.

## Lagrange Points

Two massive objects orbit around their common barycenter. It is interesting to consider their gravitational influence on any light mass object or a planet for binary star system. Considering any instantaneous situation the force on the light mass can be given as:

$\displaystyle \overset\rightharpoonup F\;=\;-\;\frac{GM_1m}{\left|\overset\rightharpoonup r\;-\;{\overset\rightharpoonup r}_1\right|^3}(\overset\rightharpoonup r\;-\;{\overset\rightharpoonup r}_1)\;-\;\frac{GM_2m}{\left|\overset\rightharpoonup r\;-\;{\overset\rightharpoonup r}_2\right|^3}(\overset\rightharpoonup r\;-\;{\overset\rightharpoonup r}_2)$
Three body description

One needs to solve the equation for r by first assuming results of r1 and r2 in the two body solution.

In a moving frame, where the light mass is orbiting the two objects, there is need to consider Centrifugal and Coriolis forces. For the frame rotating with angular velocity Ω the force equation in the orbital plane becomes -

As the derivative of potential is force, one can look for stationary points where force is zero. There are five such points called Lagrange Points​[5].

At these five points the light mass can orbit in a constant pattern with the two larger masses. In other words, the Lagrange Points are positions where the gravitational pull of two large masses precisely equals the centripetal force required for a small object to move with them. Their positions are depicted in Figure 3. The three Lagrange points along the line joining the large masses L1, L2 and L3 are potential saddle points and dynamically unstable. The other two Lagrange points L4 and L5 in the transverse direction provide stable equilibrium.

The five Lagrange points

The Lagrange points are important in the Sun – Earth system as they can be used for putting satellites. L1 is balanced between the gravitational pull of the two objects. The proposed Indian Solar observation satellite ‘Aditya L1’ is scheduled to be kept in this Sun-Earth L1 position. The Sun-Earth L2 point will fall behind the Earth and is a good location for uninterrupted view of the sky. The NASA James Webb Space Telescope is planned to be kept in this L2 position.

## ORBITS AROUND A BLACK HOLE

Black hole is considered to be a cosmic body of intense gravity from which nothing, not even light, can escape. A black hole is formed by the collapse of a massive star, when the star has exhausted its fusion fuel. The internal core gravitationally collapses upon itself, and the star’s outer layers get blown away. The constituent matter falling in from all sides compresses the star to a point of zero volume and infinite density called the singularity​​[6][7]​. There are also supermassive black holes at the centre of galaxies, wherein the mass density may be smaller but the gravitational effect is strong.

A black hole has a boundary, called the event horizon. Here the gravity is just strong enough to drag light back, and prevent it from escaping. As nothing travels faster than light, everything else gets dragged back into it. There is no way to learn about any phenomenon occurring beyond this point. Any object when trapped in the black hole gravitational field experiences strong tidal effect breaking it into small fragments.

The highly accelerated mass will loose some energy via radiation and its orbit will become smaller and smaller in each period, until finally it crosses the event horizon and is lost forever. General relativity considers space time curvature for this capturing of mass into the black hole. In order to visualize the space curvature and orbits during the capture stage, experiments on a membrane sheet are performed. Though the sheet model works on the principle of classical gravitational field of earth and shape of the wrapped sheet, yet it provides interesting analogies and helps in visual understanding of orbits near a black hole.

## Experiment - Wrapped Space

Probably the best qualitative classical visualization of a relativistic phenomenon is the two-dimensional stretched membrane or sheet with a heavy ball at the centre. The weight of the ball curves the sheet which can be corelated with space curvature in presence of a mass​[8]​​​. Since central force like gravity leads to planar orbits this two-dimensional sheet works as a good example. The demonstration describes gravity as a geometric property of spacetime. Although it must be pointed out and made clear that the picture is far from real. The motion of any smaller marble on this sheet is still under the influence of earth's gravity. The central heavy ball is only curving the sheet to restrict the path.

The elastic fabric was draped over a circular frame and fastened securely by clips. Once the fabric was secured and flattened, a mass is hung at the bottom near the centre of the fabric. Due to the weight at the centre the fabric stretches and takes a curved shape, acting like warped space near a central massive black hole. Now when a smaller mass marble is rolled across it falls into the slope following a trajectory of an object spiraling into a black hole. The path of the marble depends on the initial energy imparted to the marble. When the marble is imparted a high velocity it only bends ending at the other edge - demonstrating bending of light towards the massive object. This bending of light near massive objects lead to formation of multiple images of any astrophysical object through a process known as gravitational lensing​[9]​.

The marble demonstrates elliptical orbit of an object under gravitational influence. With each orbit the marble continuously loses its energy due to friction. As it falls along the fabric curve it also gains some kinetic energy. In general, the path followed by the marble is an involute of an ellipse (Figure 4).

Involute of ellipse

An Involute is a curve that is traced by the end of a stretched thread unwinding from a circle or ellipse in a way such that the thread is always tangential to the circle or ellipse ​[10]​. Figure 4 shows the involute of an ellipse. Involutes in differential geometry have many applications e.g. in gears between revolving parts in machines. It is also interesting as in central force too the orbiting object is moving at all times tangential to the radial direction. For an ellipse expressed as

x = a cos t and y = b sin t

the tangent vector is T = [-a sin t, b cos t]

and the arc length is,

$\displaystyle S\;=\;a\int\sqrt{1-\varepsilon^2\sin^2t}\;dt$

with 'ε' being the eccentricity of the ellipse. ​

In the sheet experiment, the path of the marble varies with initial speed. If the object is slow it gets trapped rather fast and goes almost straight into the centre. While for faster marbles there are several orbits with gradually decreasing major axis lengths before it reaches the centre. This is similar to an object that is trapped in the strong field of a massive object viz. a black hole and getting sucked into it. While doing so the accelerated particles or even photons emit radiation that helped recent direct observation of radio waves from a black hole at the centre of the galaxy M87 ​[11]​.

To vary the initial energy or speed of entry for the marble a tube with holes is taken to drop the marble. The full length of the tube used is 24 inch and holes are made at every 3 inches with smallest length being 9 inches. Thus, there are six possible speeds that can be studied. The tube was then fixed on a very small incline and the kinetic energy of entry is proportional to the length of tube (i.e. 9, 12, 15, 18, 21 or 24 inch) from which the marble is dropped. This is as depicted in Figure 5.

The 24 inch tube with holes every 3 inches apart

A mobile phone camera capable of taking video clips was mounted through a selfie stick above the curved fabric. The camera itself was out of reach by hand so the recording could be started through the selfie stick. Thus, the orbital track of the marble from each of the six holes is recorded. The six video clips were then displayed on a graph sheet frame by frame to trace the path of the marble. The experimental setup and a sample video for the 24-inch drop experiment is given here:-

Experimental setup and marble path for drop from 24 inch hole. <https://www.youtube.com/watch?v=aUKWlHYpY9c>

The corresponding curve of the marble path that is traced by playing the video frame by frame is given in the adjoining Figure 6.

The marble path (24inch drop)

The experiment demonstrates several aspects of orbits in high gravitational fields and also helps visualize accretion of matter by a Black Hole. The trapped objects may move into any number of elliptical-like orbits around the black hole and their radiations can be a means of observing or detecting black holes. The planets in our solar system tend to orbit in nearly circular orbits, comets take on long elliptical orbits, faster objects may go across in parabolic or hyperbolic path. All these paths are depicted by the marble in our model depending on the initial energy of entry.

There is also the precession of the perihelion of elliptical-like orbits. Varying the initial input speeds of the marble lead to different types of paths and with varying eccentricities but similar precession angles. Study of the orbital paths of the marble for all six input speeds shows that the angle of the semi-major axis rotates by almost the same angle −90° per revolution. Possibly the precession is independent of energy but is due to the curvature of the sheet, which is kept constant in the present experiment.

## Analysis of the Observations

The orbits, made by the marble rolling on the warped fabric, are examined closely. The actual orbits depend on the shape of the curvature produced in the fabric. This is a complex problem reported in earlier studies e.g. White and Walker (2002)​[12]​. There are several features that are observable in a semi-quantitative way. As we do not change the central object mass the shape of the fabric remains the same. This is in no way similar to the wrapping of space in gravity, therefore, only a qualitative study is sufficient.

To see how the period of orbits varies as the marble spirals in to the centre, time for each half period in the revolution is noted. The computed time period is plotted in the graph below. The marble initially loses energy due to friction and gains potential energy due to the fall in height. Near the centre the fall is sharper and more kinetic energy is gained. So, we see a variation where time period initially falls and then becomes nearly constant.

Variation of orbital period as the marble rolls in

There are some other interesting features in the graph to be noted. The initial period is large for higher speed of entry. This is probably due to the larger size of the orbit and as the marble is far away from the centre the slope of the fabric is also smaller. Generally, the periods taper off to a constant value of about 0.6 second. The slope is steeper near the centre so this could be akin to the marble reaching a terminal speed and for small size of the orbit periods becomes stationary.

There are about six complete orbits in all of the six cases. Although the orbits spiral into the centre but they appear to follow elliptical path. Each ellipse gets smaller and if an approximate semi-major axis is identified and measured, its variation can be studied. This is as shown in the following graph

Semi major axis for each orbit.

The length of the semi major axis tapers down in an exponential way. For all initial speeds the last orbits become nearly equal. The Energy of ellipse orbit under gravity has energy proportional to 1/a where a is semi major axis. The variation of 1/a, if taken proportional to the average energy of that orbit, shows a linear fall in all the six cases. But, the energy of inner or the last orbits is nearly equal and seems to be independent of the initial energy of insertion.

In conclusion it is observed that the sheet experiment is actually very illustrative of the possible orbits taken by an object in a restricted fall under gravity. Nevertheless, it also helps to visualize the wrapped space and orbits around an exotic astrophysical object - the black hole.

## ACKNOWLEDGEMENTS

Primarily I would like to thank IASc for giving me such a golden opportunity to do fellowship in DDU Gorakhpur University. Then I would like to give my sincere thanks to my guide Professor Shantanu Rastogi, whose valuable guidance has been the one that helped me to patch this project. His suggestions and instructions served as the major contributor towards the completion of project. I also thank him for managing the difficult and erratic Author Cafe writing.

I would like to convey my gratitude to Dr. Prabhunath Prasad, Dr. Anju Maurya, Rahul Anand, Prayagraj Singh and Rashmi Singh who helped me with valuable suggestions and support in the various phases of my stay and the project.

#### References

• Daniel Kleppner, Robert J. Kolenkow, 2009, An Introduction to Mechanics, McGraw Hill Publication.

• Leon Blitzer, 1971, Satellite Orbit Paradox: A General View, American Journal of Physics, vol. 39, no. 8, pp. 882-886.

• Edward P. Chatters IV, Bryan Eberhardt and Michael S. Warner, 2009, Orbital Mechanics (Chap 6) in Air University Space Primer, USAF.

• Bradley W. Corroll, 2006, An Introduction to Modern Astrophysics, Cambridge University Press.

• Simon Edgeworth, 2001, Theoretical Orbits of Planets in Binary Star Systems, Academia pdf.

• Frank Shu, 1984, The Physical Universe, University Science Books.

• Chad A. Middleton and Dannyl Weller, 2016, Elliptical-like orbits on a warped spandex fabric: A theoretical/experimental undergraduate research project, American Journal of Physics, vol. 84, pp. 284-292.

• Gravitational Lensing <http://w.astro.berkeley.edu/~jcohn/lens.html>.

• Eric W. Weisstein, "Ellipse Involute." From MathWorld--A Wolfram Web Resource. <http://mathworld.wolfram.com/EllipseInvolute.html>.

• Event Horizon Telescope Collaboration, 2018, First M87 Event Horizon Telescope Results, Astrophysical Journal Letters, vol. 875, pp. L1-L7.

• G.D. White and M. Walker, 2002, The shape of “the Spandex” and orbits upon its surface, American Journal of Physics, vol. 70, pp. 48 - 52.

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