Binary Black Holes, Gravitational Lensing and Proving Einstein Right through The Precession of Mercury’s Perihelion

There are things that happen in our universe which create bizarre observable affects, confirming them scientifically though can often be a challenge because they don't occur regularly, by their very nature, they are weird, seldom witnessed phenomena - But their existence is predicted as a result of the mathematics and physics at work, things like Binary Black-Holes, which until recently was only ever once recorded within our observable-detectable universe, but the black holes were 7.3 parsecs apart. That's roughly one and a half million times the distance between us and our own sun, so not exactly a tight orbit.

The biggest and brightest galaxy of the Hickson 96 compact group, Markarian 533 / NGC 7674. Image Credit: NASA, ESA, the Hubble Heritage Team (STScI/AURA)-ESA/Hubble Collaboration and A. Evans (University of Virginia, Charlottesville/NRAO/Stony Brook University)
The biggest and brightest galaxy of the Hickson 96 compact group, Markarian 533 / NGC 7674. Image Credit: NASA, ESA, the Hubble Heritage Team (STScI/AURA)-ESA/Hubble Collaboration and A. Evans (University of Virginia, Charlottesville/NRAO/Stony Brook University)
In a paper featured in Nature: Astronomy; titled: "A candidate sub-parsec binary black hole in the Seyfert galaxy NGC 7674" published just recently. Scientists used imaging in the infrared spectrum to record data from two orbiting black holes in the first ever sub-parsec Binary Black Hole with a separation of less than 5% the total distance of the previously observed occurrence, meaning they're close, very close to each other in terms of space distances.

For the brightest member of a group galaxies, experiencing a small merger with another galaxy near by, they are seeing emission spectra (light wavelengths) most commonly associated with mass accreteion onto super massive black holes, - the blackhole is sucking the galaxies in and friction is burning them up. Researchers were not expecting to find this binary black hole system in such a cluster of spiral galaxies, when spiral galaxies merge they tend to form elliptical galaxies which are more likely to have supermassive black holes.

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Never the less, they recorded data from a radio jet emission (Hawking's) stream blasting 700 parsecs out into space at almost a third the speed of light, that shows two separate cores of supermassive black holes orbiting each other around a central locus. They infer the mass to be approximately 3.63 x 10^7 Solar Masses, which is 7.218x10^37 kg and they estimate the source to be 8200 years old. Supermassive is a term reserved only for those blackholes which exceed the Schwarzschild limit, our own Milky Way galaxy is known to have a single supermassive blackhole in the center, from earth it's in the direction of Sagittarius.

Contour image map showing Jet 1 and Jet 2, associated with Binary Black Holes from the paper in Nature
Contour image map showing Jet 1 and Jet 2, associated with Binary Black Holes from the paper in Nature
The researchers also noted something interesting, that the spin-axis of the jet is off and is slowly moving, precessing, and suggested it could be a result of the Lense-Thirring Effect. This is basically where one orbiting supermassive black hole 'drags' the gravity around it and throws off the other supermassive black hole - think about the scale we're talking about here, these are extremely massive objects measurably sheering the space-time fabric we exist on. The Gravito-Lense/Gravitational Lense effect is one of the most interesting observable phenomena resulting from Einstein's General Relativity. So interesting in fact, that while I was doing my undergrad I happened to write a research paper about exactly that, so, presented below, for those interested in more details:

The Precession of Mercury’s Perihelion

We start with a review of Kepler’s laws in order to gain a better understanding of how the planet Mercury is supposed to behave. Then we look at the planet itself, its characteristics of orbit, and how an anomaly arose, and what it means. Then we consider a mathematical analysis of the situation, and realize the classical models will not work in this situation. The problem is approached through General Relativity, and then after explained qualitatively through the Lense-Thirring Effect, or “Gravito-Magnetism”.

Introduction: In order to fully comprehend and gain a true understanding of the perihelion advance of Mercury, it is important also to understand the planet’s history, current state, and the laws that govern its motion. For these reasons exactly we will begin with a brief explanation of Kepler’s three laws of planetary motion, so we can see how they were supposed to govern the motion of the planet mercury, and then we will proceed to look at why they just didn’t seem to be entirely right.

Kepler’s Laws
Johannes Kepler is one of the most renowned astronomers of antiquity, and for good reason. After careful deliberation and studying of Tycho Brahe’s stellar observations, Kepler came to several conclusions, which spawned the three laws that govern planetary motion, known as “Kepler’s Laws of Planetary Motion”, and they are as follows:

  • Every planet follows an orbital path of an ellipse, with the sun centered at one of the two foci.
  • A straight line connecting the planet to the sun, will sweep out equal areas over an equal time interval.
  • The period of a planet’s orbit squared is directly proportional to the cube of its semi-major axis, a.

r = p / (1 + εCosθ) is the equation for an ellipse, perihelion (the point of closest approach to the sun) occurs when θ = 0˚, which will result in the smallest r value. Aphelion (the distance farthest from the sun) occurs when   θ = 180˚

α= p / (1-ε²) Determines the semi-major axis, and the latter equation determines the semi-minor axis b = 1 /(√(1-ε²)), where ε is the eccentricity of the ellipse[1].

Characteristics of the Planet Mercury
Mercury is a planet unlike any other in our solar system, its extreme nature, proximity to the sun, and unique characteristics make it a genuine focal point for any modern research topic. Mercury is the inner most planet in our solar system, because of this we are afforded unique possibilities to test theories with Mercury that may not be possible anywhere else, it also moves with an extremely high velocity through the Sun’s gravitational field, which provides a canvas for even more unique features to be observed[2]. It is very important to understand about in particular the four most inner planets of our solar system, the history, and the current states, because it will help to understand the future of life and help in the search for possible extra-terrestrial life[5] (SETI project¨). Being the planet closest to the sun, has unfortunately – until the recent Mariner 10 mission – provided mercury with a veil, it is a planet which is very difficult to observe from earth bound positions because it has only a 28˚ separation from the sun at its maximum, Which means for us that we are only able to observe Mercury at sunrise, or sunset.

Mercury has an orbital period of 87.969[3] days, and a rotational period of 58.646 days, which puts it in an exact ratio of 2:3 for rotational versus orbital periods[4]. One Mercury day lasts for an astonishing 176 Earth days, which means that the sunlit surface - because of the duration it spends exposed to the sun and the planets proximity to the sun - can reach a surface temperature of more than 400˚C, and on the contrary, the night-side of the planet will experience temperatures plummeting well below -100˚C[5]. Mercury has a magnetic field surrounding it, although it is much smaller than earths magnetic field, the source is speculated to be similar to earths; an at least partially molten iron core as large as 75% of the planets’ entire radius, which creates a dynamo, and in turn a magnetic field[6]. Mercury has a semi-major axis (α) of Km[3,7], an orbital eccentricity of 0.20563069[3,7], an inclination of 7.00487[3,7], and a mass of Kg[7].

An Observation not easily explained
In 1859 Le Verrier noticed that there was an anomaly occurring when calculating the perihelion of Mercury’s orbit[8]. After accounting for perturbing effects from other planets, and the effect of the precession of the equinoxes, he found there to be an unexplained advance in the perihelion position of Mercury’s orbit. This advance could not be explained by the classical Newtonian theory of Gravitation[9]. There are many things known which will affect the perihelion of Mercury’s’ orbit:
  • The perturbation from the general precession of the equinoxes accounts for about 5000 arc-seconds per century.
  • The effect of perturbing other planets, about 280 arc-seconds per century from Venus, 150 arc-seconds per century from Jupiter, and 100 arc-seconds per century from the remainder of the planets[10].
Realizing the persistence of the problem a number of, left field style solutions were proposed to correct this:
  • The addition of a new planet called “Vulcan” near the sun.
  • A ring of planetoids.
  • The existence of a solar quadrupole moment.
  • Something other than the inverse square law of gravitation[8].
As later discovered in the early 1900’s, this shift in the perihelion of Mercury’s orbit was to be attributed to a purely relativistic effect, from the theory of General Relativity,  and the sun’s quadrupole moment, which accounted precisely for the extra 43 arc-seconds per century. This is now considered living proof of Einstein’s Theory of General Relativity[8,9,10].

Observed by radar
Not long after the birth of General Relativity, experiments were underway to determine accurately the perihelion advance of Mercury. Between the years of 1966 and 1971 two sites were used to measure within an uncertainty of 0.02 the perihelion precession of mercury. From Haystack, Massachusetts, 200 time delay measurements at an operating frequency of 7840 MHz were made. From Arecibo, Porto-Rico, 150 time delay measurements were made at an operating frequency of 430 MHz. both experiments incurred only a 5 to 20 microsecond individual error in reading[11].

Mathematical analysis
Finally the first mathematical explanation for the observed shift in Mercury’s Perihelion can be attempted. The following calculation will include contributions of the relativistic nature, Newtonian nature, and from a possible solar quadrupole moment.
Where:  m = m1 + m2 and µ = (m1m2/m) the total mass and the reduced mass of the two body system. p = a(1- ε²) Where “a” is the semi-major axis, ε is the eccentricity of orbit. R is the average radius of the oblate body.  J2 = [(C - A)/m1R²] Is a measure of the quadrupole moment, and C and A are moments of inertia about the rotation, and equatorial axes. The first term of the equation accounts for the classical relativistic perihelion shift. The second term depends on the ratio of mass, and should be zero in any conservative gravitational theory. The solar quadrupole moment is estimated at 1x10^-7.[12]
When we assume the solar quadrupole moment is zero, these results agree with the general relativistic prediction[11].

Closing remarks on Mercury
Mercury is certainly a unique planet indeed; it is also a great planet for testing many other important theories, such as a time variation of the gravitational constant G, the Nordvedt violation of the strong equivalence principle, or the De-Sitter geodetic precession (which is responsible for the precession of Mercury’s orbital plane by about 0.2 arc-seconds per year)[13].

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Discussion of General Relativity and its effects
The theory of General Relativity really became solidified with the discovery of the precession of Mercury’s perihelion. But this was actually only the first of two confirmations for the theory. The second came with the discovery of the deflection of light[14]. When ever we are considering an observation to be a test of General Relativity, it is important to remember that there are really only 2 principle tests of the theory. The first is the measurement of precession of perihelia of gyroscopic systems’ spin axes’ in a gravity probe relative to the distant surroundings. The second is the predictions from the solutions to Einstein’s equations, for the solar system in Minkowski-space, which would not contain any form of distant surroundings[15]. General Relativistic effects are largely dependent on mass; this is demonstrated experimentally in the following ways:

  • The apparent Red shift of a photon moving away from a gravitating body.
  • The increase in energy between levels in atoms with their increase in distance from the body.
  • The retardation from radar echo from planets.
  • The deflection of starlight during a solar eclipse[16].

To see how much the dependence on mass can affect the gravitational potential of a system, we need only to look at Mercury as it moves in its path around the sun with a precession of 14.326 arc-second per century[16].

The Bending of light
Further support for the theory of General Relativity came in 1964 when Irwin Shapiro made the original discovery demonstrating the time delay of light in support of the theory[17]. The bending of light around the sun[18], or the deflection of starlight is also considered a discovery in favor of the theory of General Relativity. This is when a photon gains energy and blue-shifts as it approaches a massive body, then loses energy and red-shifts as it leaves the body’s field. The gain in energy can also be equated to a gain in mass by Einstein’s famous equation[19].

The precession of mercury’s perihelion by various methods
The precession of Mercury’s perihelion in orbit around the sun was an extremely large part of the prediction of General relativity, much like the similar effect seen in a binary pulsar star system[¡], or a satellite orbiting earth[20]. When considering the first proof of Einstein’s theory, the Precession of perihelia, the sources of advances in the perihelia are often thought to be the Schwarzschild Gravito-electric part of the solar gravitational field, and the Quadra-polar mass moment of the sun[21]. In particular another common way to deal with the advance of the perihelia problem comes from “Schwarzschild spherically symmetric, solution of Einstein’s field equations”, how ever this will lead to a prediction that the orbit is an ellipse and is situated in a rotating co-ordinate system. So while the ellipse rotates, the periapsis appears to precess[22]. When again considering the dependence on mass of gravitational potential the end result turns out to be the same as for the above Schwarzschild proposed solution, except now there will be a dw/dt = -14.326 arc-second per century for mercury, this further solidifies the dependence on mass[23].

Another possible way to address the precession of perihelion in Mercury’s orbit is to calculate it using the Laplace – Range – Lenz vector; this will provide an approximation for the value of precession, to within 4.4%. The only problem with this method is that it assumes circular orbits from all contributing, perturbing planets, and they must be co-planar with Mercury’s orbit. In any case the precession due to a perturbing planet by this method is found to be proportional to the mass of the planet over the semi-major axis cubed[24].
Finally we arrive at the proper method for determination of this phenomena, it is strictly a General Relativistic effect.

General Relativity Equation
Where R is the Schwarzschild radius of the sun, Ms is the mass of the sun, Rs is the radius of the sun, “i” is the inclination of mercury’s orbit, J2 is the quadrupole moment of the sun (2.0±0.4x10^-7), a is the semi-major axis of Mercury’s orbit, and e is the eccentricity[25]. In General Relativity it is quite clear that the properties of planetary orbits are very dependent on the structure of the surrounding space-time[26]. According to the theory, the motion of a planet is actually a time-like geodesic in a Schwarzschild space-time surround the sun[27], and the entire concept of a curved spaced-time is based around Einstein’s principle of equivalence:

The weak equivalence principle is valid, secondly; the out-come of any local non-gravitational experiment is independent of the velocity of the free-falling reference frame in which is it performed. Finally, the out-come of any local non-gravitational experiment is independent of where and when in the universe it is performed[28]”

Closing remarks on general relativity
With so much in favor of General Relativity, there is still some against. One issue is that the theory can not be quantified, which makes it very difficult to unify the fundamental interactions. Also, experimentally, General Relativity alone can not reproduce the flat velocity distributions in the near by areas of galaxies. The Newtonian potential would predict a decrease in distribution. How ever a solution to this problem is the existence of dark matter[29]. Also it is difficult to ignore the fact that we do not know for certain the interior of Mercury or the sun, so the precession of mercury is not a perfect test of relativity, in fact most of the observed precession could even be attributed to a non-uniform, unique, planetary density distribution[30]. The final thing to note about General Relativity is that it predicts that a body such as earth will drag the inertial frames surrounding it, while it rotates. This effect is measurable by satellites currently in orbit around the earth with an accuracy of 99±5%[31], this effect also goes by the name of Gravito-magnetism.

Gravito-magnetism and the Lense-Thirring effect
De-Sitter first deduced this effect in the equatorial plane of the central body, and then it was studied by Lense-Thirring. The basic idea is that a rotating mass drags the space-time around it like a very thick liquid[32], which would lead us to envision space-time to be much like very thick viscous oil. The Lense-Thirring effect, (also called Gravito-Magnetism, Co-Gravity), caused by a rotation of mass demonstrates the General Relativity phenomena that mass currents will generate a curvature in space-time, which can actually be measured by an orbiting satellite with a gyroscope[31]. When he considered the effect this would have on Mercury, in 1916 De-Sitter became the first to calculate the perihelion advance, unfortunately he assumed the suns’ angular momentum to be zero[33]. Thus a new derivation for the advance of Mercury’s perihelion was born, based on the theory of Gravito-Magnetism, which makes use of the theory established by Oliver Heaviside and Oleg Jefimenko that there is some sort of apparent link between electromagnetism and the gravitation forces[34]. Although this effect is relatively small for anything earth related, and the field has never been detected in any earth lab situation, (because the force of gravity is much greater than the force of co-gravity[35]), it becomes abundantly apparent when considering the accretion disk of black holes, or jets in active galactic nuclei[36]. According to General Relativity, any rotating matter will produce a gravitational field, similar to how any moving charge will produce a magnetic field in the study of electrodynamics[37]. Basically, Gravito-Magnetism is the analog for the force of gravity, of electromagnetism. It’s similar to comparing Newton’s law to Coulombs law, in fact Maxwell proposed the possibility of a “Gravito-magnetic field” in 1865[38]. Another Theory associated with frame dragging, is the theory of Holonomy, this theory promotes the idea that the main important thing in a time shift is the fact that something is rotating.

“Given a parallel transported vector around an equatorial circle in the exterior Schwarzschild space-time, the vector returns with a shift in its time component, the rotation introduces a Gravito-Magnetic clock effect. For a four velocity-vector, this Holonomy is a Lorentz boost, and the local observer would measure a time dilation.[39]”

There are several other Gravito-Magnetic effects which will affect the perihelion advance:

  • The impact of solar oblateness[40].
  • Post Newtonian-Gravito-electric precessions[41].
  • The impact of nearby asteroids[41].
  • The possibility of the multi-dimensional membrane world gravity model Dvali, Gabadadze, and porrati (2000)[42].

Conclusions on the Gravito-Magnetic theory
Although this theory seems to be an alternative to the general relativistic point of view, it is important to remember that the theory arises, partially as a result of general relativity. And although general relativity does a superb job of predicting the perihelion advance, it doesn’t offer much of an explanation. The Gravito-Magnetic theory offers a qualitative explanation, to provide a better understanding of what is actually happening physically to cause this precession in Mercury’s Perihelion. No one has ever achieved a correct derivation based on this theory with out having to substitute a different value for the speed of light[43].

There are currently a number of ongoing tests for Gravito-Magnetism right now:

  • April 2004 GP-B spacecraft launched, with 4 super conducting gyroscopes, aimed at measuring the Gravito-Magnetic effect, the “precession of spins”.
  • Analyzed data from the LAGEOS (I & II) satellites for measurement of the Lense-Thirring effect in the orbit of a test particle (2004).
  • A Lense-Thirring test on the orbit of mars global surveyor in the gravitational field of mars (2006)[44].

The Precession of Mercury’s Perihelion can be attributed mathematically and fundamentally to General Relativity, but the true explanation is with the expansion of General Relativity into the Gravito-Magnetic effect. The frame dragging is precisely what’s causing the effect to occur.  In the future more accurate data will be needed from the sun and from Mercury to determine more accurately this effect, but there are on-going missions such as SoHO, and GONG, aimed and acquiring such data to within a few percent of accuracy, and this is necessary to determine whether or not the precession of Mercury’s perihelion is a good test for General Relativity or not[45]. It is surprising that in a field of such precise work, there is so much speculation around such an important topic.

  • Balogh, André. Giampieri, Giacomo. Mercury: the planet and its orbit March 20th, 2002, PII: S0034-4885(02)12697-2
  • Behera, Harihar. Naik, P. C. A flat space-time relativistic explanation for the perihelion advance of Mercury. June 29th 2003, 0306611 v1.
  • Ciufolini, I.  Pavlis, E. C. A confirmation of the general relativistic prediction of the Lense–Thirring effect. October 21st 2004.  Nature vol 431.
  • Harihar, Behera. The contribution to the perihelion advance of Mercury arising out of the dependence of mass on gravitational potential. October 17th 2004, 0410401 v1
  • Iorio, L. First preliminary tests of the general relativistic gravitomagnetic field of the Sun and new constraints on a Yukawa-like fifth force from planetary data. March 27th 2007. 0507041 v11
  • Iorio, L. Is it possible to measure the Lense-Thirring effect on the orbits Of the planets in the gravitational field of the Sun?, September 14th, 2004, 0407047 v6.
  • Kraniotis, G. V. Whitehouse, S. B. Compact calculation of the Perihelion Precession of Mercury in General Relativity, the Cosmological Constant and Jacobi’s Inversion problem. May 15th, 2006, 0305181 v4.
  • Life on Other Worlds, The Universe: The Infinite Frontier. 1994 Film Series V26
  • Maartens, Roy. Mashhoon, Bahram. Matravers, David R. Holonomy and gravitomagnetism. January 2nd 2002, 195–201, PII: S0264-9381(02)30677-4
  • Moore, Patrick and Nicolson, Lain. Black Holes in Space. New York W.W. Norton & Company Inc. 1976.
  • Prof. Astronomy, Class Notes on Kepler’s laws, Fall 2005, 
  • Pireaux, Sophie. Rozelot, Jean-Pierre. Godier, Stephanie. Solar quadrupole moment and purely relativistic gravitation contributions to Mercury’s perihelion Advance. September 3rd, 2001, 0109032 v1.
  • Sagan, Carl. CETI, Communication with extraterrestrial intelligence. MIT Press, Cambridge Massachusetts, USA 02147
  • Schmid, Christoph. Cosmological gravitomagnetism and Mach’s principle. August 28th 2006, D 74, 044031
  • Shapiro, Irwin, I. Pettengill, Gordon, H. Ash, Michael, E. Ingalls, Richard, P. Campbell, D. B. Dyce, R. B. Mercury’s Perihelion Advance: Determination by Radar. June 12th, 1972, Volume 28, #24.
  • Stewart, M. G. Precession of the perihelion of Mercury’s orbit. May 20th 2005, Vol 73 # 8
  • Tajmar ,M. De Matos, C. J. Advance of Mercury Perihelion Explained by Co gravity
  • Will, Clifford M. The Confrontation between General Relativity and Experiment. April 4th, 2006, 0510072 v2

[1] Prof. Astronomy, Class Notes on Kepler’s laws.
[2] Kraniotis, G. V. Whitehouse, S. B page 2.
¨ SETI is a project, in search of extra-terrestrial intelligent life forms; one way to search for life outside of our own planet was founded by Frank Drake. In 1960, Frank Drake made the first radio observation in the SETI (Search for Extra Terrestrial Intelligence); he used radio band wavelengths from the spectrum to search the cosmos for a response from beyond and came upon the background noise of space. By using microwave wavelengths, he found the background noise was quietest. Certain frequencies were used more then others, such as the “magic” frequency of the Hydrogen atom: 1420MHz, during observing, signals were always received but then never found again, so they were dismissed as interference from a nearby electronic device (Life on other worlds, Film). We use electromagnetic waves to try and contact other beings because they have the longest range, and are the most economical, (Sagan, 231).
[3] Balogh, André. Giampieri, Giacomo page 5.
[4] Balogh, André. Giampieri, Giacomo page 3.
[5] Balogh, André. Giampieri, Giacomo page 6.
[6] Balogh, André. Giampieri, Giacomo page 4.
[7] Iorio, L. page 3.
[8] Will, Clifford M page 38.
[9] Balogh, André. Giampieri, Giacomo page 3.
[10] Pireaux, Sophie. Rozelot, Jean-Pierre. Godier, Stephanie page 1.
[11] Shapiro, Irwin, I. Pettengill, Gordon, H. Ash, Michael, E. Ingalls, Richard, P. Campbell, D. B. Dyce, R. B. page 1.
[12] Will, Clifford M. page 38.
[13] Balogh, André. Giampieri, Giacomo. Page 16.
[14] Will, Clifford M. page 2.
[15] Schmid, Christoph. Page 1.
[16] Harihar, Behera. Page 1.
[17] Will, Clifford M. page 36.
[18] Will, Clifford M. page 33.
[19] Kurucz, Robert L. page 3.
Gamma note: Modern astronomers, make use of the Binary star system in the search for black holes because the best way to find a black hole is to observe the gravitational effects on the surrounding environment of a suspected black hole region, (Moore and Nicolson 97). The most popular case is a binary star system, or rather what appears to be a binary star system. A star will appear to be revolving around a central mass; the next thing to look for is the drizzling of matter from the star into the “central mass”, the friction created from all this drizzling of material will create heat, UV, and X-radiation, which can be detected via specialized telescopes (Moore and Nicolson 89-96).
[20] Ciufolini , I. Pavlis, E. C. page 4.
[21] Iorio, L. page 10.
[22] Harihar, Behera. Page 1.
[23] Harihar, Behera. Page 4.
[24] Stewart, M. G. page 1.
[25] Pireaux, Sophie. Rozelot, Jean-Pierre. Godier, Stephanie. Page 5.
[27] Kraniotis, G. V. Whitehouse, S. B. page 12.
[28] Will, Clifford M. page 4.
[29] Pireaux, Sophie. Rozelot, Jean-Pierre. Godier, Stephanie. Page 10.
[30] Kurucz, Robert L. p age 1.
[31] Ciufolini , I. Pavlis, E. C. page 1.
[33] Iorio, L. page 3.
[34] Tajmar ,M. De Matos, C. J. page 2.
[35] Tajmar ,M. De Matos, C. J. page 4
[36] Ciufolini , I. Pavlis, E. C page 1.
[37] Will, Clifford M. page 42.
[38] Behera, Harihar. Naik, P. C.
[39] Maartens, Roy. Mashhoon, Bahram. Matravers, David R. page 1-2.
[40] Iorio, L. page 10.
[41] Iorio, L. page 13.
[42] Iorio, L. page 14.
[43] Tajmar ,M. De Matos, C. J. page 2.
[44] Iorio, L. page 4.
[45] Iorio, L page 3.