An Alternative Phenomenological Explanation of the Binary Pulsar PSR 1913+16 P. A. Murad1 Vienna, Virginia 22182
[email protected] Abstract. Claims suggest that the PSR 1913+16 binary pulsar is indirect evidence concerning the existence of gravitational waves. The pulsar is postulated to have a neutron star and a companion body of equal mass that rotate in separate elliptical orbits that have the same period anchored to a common focus. The perihelion of the pulsar’s decaying orbits changes rapidly suggesting that orbital energy is dissipated as gravitational radiation. The pulsar is reexamined concerning these two separate orbits should either coalesce into a single elliptical or circular orbit such as observed for some binary white dwarf systems or that these two bodies may be in orbit about: A third larger rotating body, A rotating black hole whose signature is also obscured, or A torsion/spin or gravitational-like field vortex singularity. The decaying orbits can be partially explained by the gravitational attraction due to a third celestial object that would partially dissipate each orbiting body’s momentum. This implies the pulsar may not be a binary but possibly a tertiary system. The high rotational rate of the object located at the common focus may create through frame dragging to produce the observed perihelion changes. This suggests reconsidering the amount of radiation supposedly converted to gravitational waves and that this pulsar may have more interesting celestial phenomenon than initially considered. Keywords: gravity, anomalies, model, torsion field, angular momentum and gravitational waves. PACS: 91.10.Op, 91.25.Rt, 91.60.Qr, 96.25.De, 96.25.Nc
Introduction Hulse and Taylor performed unusual groundbreaking work in discovery of a unique binary pulsar (Huguenin et al 1968, Taylor 1972, Hulse and Taylor 1974, 1975a, 1975b, Rawley et al 1988, Hulse 1994) amongst the 1400 plus known pulsars found in the last 35 years. Pulsars contain highly magnetized rotating neutron stars, which are very dense objects that emit a beam of detectable electromagnetic radiation in the form of radio waves (Goldreich and Julian 1969). The radiation is observed when this beam of emissions point toward the Earth. This is called the lighthouse effect (Hewish et al 1968). Neutron stars have a stable rotation period where observed pulses are very regular and gives rise to the pulsed nature that gives pulsars their name. Pulsars make exceptional clocks ( Richards and Comella 1969, Ryba and Taylor 1991, Stineberg and Kaspi 1992, Kaspi, Taylor, and Ryba 1994), which enable a number of unique astronomical experiments. Some very old pulsars "spin up" to speeds of over 600 rotations per second by material flowing onto them from a companion star, appear to be rotating so smoothly that they may even "keep time" more accurately than the best atomic clocks on Earth. The neutron star binary system is one of these systems, with an orbit that is decaying more rapidly than any previously discovered. Using the Arecibo 305m antenna, Hulse and Taylor detected pulsed radio emissions and thus identified the source as a pulsar with a rapidly rotating, highly magnetized neutron star. The neutron star rotates on its axis 17 times per second; thus the pulse period is 59 milliseconds. After timing the radio pulses for some time, Hulse and Taylor noticed that there was a systematic variation in the arrival time of the pulses. 1
“Views expressed in this article are those of the author and do not reflect the official policy or position of the U.S. Government.”
Sometimes, the pulses were received a little sooner than expected; sometimes, later than expected. These variations changed in a smooth and repetitive manner, with a period of 7.75 hours. They realized that such behavior is predicted if the neutron star were in a binary orbit with another companion star. The pulsar and its companion both follow elliptical orbits around the barycenter or their common center of mass. Each star moves in its orbit according to Kepler's Laws; at all times the two stars are found on opposite sides of a plane perpendicular to a line joining both stars and passing through the center of mass. The stars are believed to be nearly equal in mass, about 1.4 solar masses. As an explanation for the decay, Hulse-Taylor suggested that orbital energy of the pulsar PSR 1913-16 is converted into gravitational energy that was being radiated away in the form of gravitational waves. Einstein speculated about the existence of gravitational waves as a means of explaining observational delays of a Nova over considerable distances. This explanation regarding the pulsar’s orbit decay is viewed as indirect evidence concerning the existence of gravitational waves. According to Taylor, Hulse, Fowler, Gullahorn and Rankin (1976): “… on the binary pulsar PSR 1913+16. They have found a systematic decrease of the orbital period of the system that is consistent with energy loss due to gravitational radiation as predicted by Einstein's general theory of relativity… These observations represent the first tests of general relativity outside the solar system and also constitute the first convincing experimental evidence, though indirect, for gravitational waves.” ”The general theory of relativity predicts radiation that, in the lowest order, is proportional to the third derivative of the quadrupole moment of the mass-energy distribution. It is a consequence of the conservation equations that the first derivative of the monopole moment and the second derivative of the dipole moment are zero, so that radiation is first seen in the quadrupole term.”
Discussion A sketch of the pulsar is shown in figure 1 (Schrewe and Stern 1993) that shows a neutron star rotating in an elliptical orbit about a second body of supposedly equivalent mass in a different elliptical orbit. In the Author’s perception of space mechanics, there may be a basic problem with such orbits. If two bodies are attracted or captured by each other, they would orbit in either a single elliptical or circular orbit as observed by several white dwarfs systems such as the eccentric pulsar- white dwarf binary PSR J1141-6545 (Stairs 2003), which, over considerable time and if the masses are equal, results in a circular orbit about each other. If this occurs, which is in contrast to figure 1, a means of explaining the observed behavior may be due to the presence of a third body that may possibly be obscured and is located somewhere between the pulsar and the observer on the Earth. Gravitational lensing may provide a possible explanation for such lack of observations.
Figure 1. The orbits of the PSR 1913-16 pair and characteristics radiation pattern of a neutron star as shown in the 1993 Nobel Prize announcement. There are some issues with the basic model postulated by Hulse and Taylor regarding the barycenter of the two orbiting bodies where these separate elliptical orbits coincide with the common focus point. What is the probability that two orbiting bodies would exist that have both the same weight and the same orbital period in such a configuration? The probability would be very low. Throw in the possibility that one of the two bodies is a neutron star that represents a collapsed star and, although compressed, would still have the same mass as the second body. Moreover, the orbits are not only coplanar but the major axes of both elliptical orbits have to be collinear for the barycenter to remain at the foci location. If these conditions were not meet, the barycenter would oscillate and if these oscillations are small, create the perihelion problem. Considering further that this pulsar is one in 1400, the probability of these ‘conditions’ existing is fortuitous, which could be a major reason why this pulsar is so unique. These facts are too coincidental to be real and that another explanation may be required to explain the signature emitted by and observed from this pulsar. If on the other hand, all of these conditions are feasible, then there is no reason to continue this evaluation. If PSR 1913-16 decays because it is a binary system, then wouldn’t one expect other binary systems to decay releasing gravitational waves in the same process? Why would this occur only for a neutron star in lieu of a white dwarf system? These conditions are extremely unusual. Consider further that if there was once a star with a single planet, one would assume that the planet would rotate about the star, which would possess the larger mass. However, if the star collapses to a neutron star, changes in energy and mass distribution would be extremely unpredictable. This includes additional uncertainties about the final configuration. There are other possibilities that could possibly explain the data if these orbits are correct. The trajectory configuration would possible require the presence of a gravitational influence located at the mutual foci or barycenter. The form of this influence may include: A solid body located at the barycenter- The signature of the neutron star could be so overwhelming that it blots out any signature from this third body. If this body rotates extremely fast, it could create some of the observed perihelion effects due to frame dragging. The orbital decay could be due to attraction by this third body and implies there is no need for creation of gravitational waves beyond what is expected, A black hole may be located at the barycenter- Again, the signature may be difficult to detect. Moreso, rotation of the black hole could also be a mechanism for explaining the perihelion
problem. The black hole attraction may also explain the orbital decay and, again, there would be no need for the use of gravitational waves as a hypothesis, or A gravitational vortex may be located at the barycenter- The rotation of the bodies in their orbit as well as the presence of a vortex may explain the changing high rotational rates. Here, without a mass-like source, the presence of a gravitational vortex would need gravitational waves to explain the decay.
It may also be conceivable that for a binary system, the neutron star could be in an elliptical orbit where a larger mass companion body resides at one of the foci of the ellipse. This is plausible. If the orbital configuration for the neutron star is correct as postulated, there must be an additional body at the focal point such that the requirement for equal masses is no longer needed. This could be a very large celestial body or to be exotic, a black hole, at the common focus point of the elliptical orbit. If, as Jefimenko (Murad 2007a, 2007b, 2008) suggests, the period of a satellite depends upon the rotation rate of a larger primary body, what impact would this have upon a satellite in an elliptical orbit? Moreover, what is the impact if the primary body if it rotates with a significantly high rate, on the neutron star’s orbital period? If the neutron star rotates at a high rate, then one has to assume that by association, or frame dragging, there is a preferential direction of the orbits as well as rotation rates of other bodies. In other words, if a satellite performs a fly-by of the earth and travels in a direction with the rotation of the earth it picks up an increase in velocity whereas if it goes in contrast to the Earth’s rotation, the satellite would lose speed. For this pulsar, the rotation would act to increase the satellite’s energy and if in an opposite direction, the satellite should lose energy and the orbit should decay. This effect could also explain the perihelion decay on PSR 1913-16. There is a need to examine the expected Doppler shift data under the influence of a strong gravitational gradient that may influence measurements and subsequent conclusions. For example if the speed of light depends upon the gravitational field potential per Poincare, how does one evaluate effects due to a gravitational gradient? This may affect a data stream from a radiating object to an observer on Earth that may pass near a body possessing a large gravitational field. Multiple bodies will also create gravity gradients. If either a large planetary body or a black hole is not at this common focus, one could use a space-time singularity of undetermined origin. This could be due to a torsion, spin field, or a gravitational phenomenon. If we think of a singularity, how do we describe it and its behavior? Obviously we are not talking about known phenomenon but something new and different. For example vortices can be described differently. One can talk about micro-vortices in a superconductor that describes rotation of Cooper pairs that reduces the electrical resistance and increases magnetic strength. In the fluid dynamic perspective, a fluid dynamic vortex adds rotationality to the flow. If a cork floats within such a vortex, it will rotate about the spin axis and an arrow on the cork will also rotate about the cork’s axis as a consequence of rotationality. The issue is one of scale. The wake behind a submarine is a vortex created by the rotationality induced upon the water by a propeller. A tornado to the weatherman represents a vortical flow whereas it differs considerably because of rotationality that does not occur in a hurricane. Also the scale of a tornado is also different. The scale here under consideration is considerably larger… of the scale of planets. Regarding the last possibility of a gravitational vortex, in the words of Mazur (2007), critical bifurcation singularities can include magnetic monopoles and vortex singularities. Bifurcation singularities are similar to the effects of monopoles in electromagnetism similar to the way that Dirac predicted that are recursively excited by a coupling field. Moreover, the notion of a gravitational vortex is not new but goes back to 1931 (Hilgenberg 1931). For the argument used here, the issue of scale is also of importance. Murad (Murad and Baker 2003, Murad 2003, 2007a, 2007b, 2008) in several references discusses the existence of a gravitational vortex as an admissible solution because of the mathematics used in developing several different gravitational laws. However, although the mathematics is defined, that does not necessarily mean that the physical phenomena actually exist or not. This is an area for further cosmological research and discovery. The question is how could one detect a singularity such as a torsion/spin or gravitational vortex. Such a vortex would be a necessity to create the large measured perihelion of the neutron star’s orbit. Let us assume for argument sake that we first look at a gravitational vortex. How does one detect such
singularities that could arise as Mazur suggests? What would be used to characterize this phenomenon? Gravity affects mass and light. If there is such a thing as a gravitational vortex, it could be detected by insertion of mass particles that would depend upon their distance from the rotational axis. This will, because the subsequent rotational field causes rotation about this axis and about the mass particles axis. The latter axis is parallel to the rotational axis. If light is used, assuming the vortex has a very high intensity, the light should bend in a direction toward the axis. The amount of light deflection or bending depends upon the vortex’s strength and the distance of the light ray from the vortex’s axis. To detect a torsion or spin field phenomenon we need to determine what would interact with such an event for the purposes of detection. This also raises an interesting possibility. Stars provide gravitational lensing because light is bent due to the star’s gravitational field. No one has reported of similar events occurring due to the presence of a black hole in the path of light that goes from a star to an observer on Earth. If this too is a realistic capability, then gravitational lensing should also be valid for detecting such a gravitational singularity or vortex.
Analysis In space mechanics, analysis is performed based upon some limiting assumptions that simplify a complex problem. For example in the two-body problem, a very small body orbits about a larger body in terms of mass. The large body influences the smaller body but the effects of the smaller body upon the larger body are considered inconsequential and some terms within a formulation can be ignored to produce a viable solution. A. Equations of Motion for a Third Body- Modeling a Binary Pulsar The issue is how to model this binary pulsar. If we look at the restricted three-body problem (Baker 1967, Broucke and Prado 1993, and Battin 1987), there are many assumptions in using this model to include that the mass of the third body is minimal compared to the two major bodies; the major bodies are at a constant distance apart; the barycenter of the rotating coordinate system is located at the origin; the rotation rate is assumed to be constant; and Newtonian gravitation is assumed. If the neutron star moves in an elliptical orbit for the two-body problem with equal masses, we can assume that the two bodies oscillate about their nominal positions along the x-axis that rotates about the barycenter. This would produce two elliptical trajectories as prescribed by Hulse-Taylor. If indeed a body exists at the focus point, the equations of motion are of the form: V x 2 y 2 x , x x
V , (1) y y V z . z z This is for an orthogonal x, y, z coordinate system where differentiation is with respect to time; V is the Newtonian gravitational potential. The first term on the RHS of equation (1) is the acceleration due to centrifugal forces while the second term on the LHS is Coriollis acceleration based upon the orbit’s path, ω is the rotation rate of this coordinate system. The gravity potential Ω for a rotating coordinate system has the form: y 2 x 2 y
1 2 x 2 y 2 . 2 r1 r2
( 2)
The variable μ is the mass ratio of one body divided by the total weight. The integral of motion has the form: V 2 V V 1 2 x y 2 z 2 x 2 y 2 dx dy dz . ( 3) 2 2 y z x For a constant rotation rate, this results in:
JEC
1 2 2 1 x y 2 z 2 x2 y2 . 2 2 r1 r2
( 4)
where E is the energy, C is the angular momentum, r1 is the distance to one primary while r2 is the distance to the second primary. B. Realistic Modeling Considerations To use these equations to investigate this pulsar, we realize that almost all of these assumptions are violated. Thus the model should be modified to make these equations relevant. For the mass terms, the mass of the neutron star will decrease as it is converted into energy suggesting that μ should be a function of time. The radius of the third mass to either of the major bodies, the neutron star and its companion is the same value. Since each of these bodies move in elliptical trajectories with a common focus point, the trajectories in polar coordinates are: p r1 and r 2 r1 . ( 5) 1 e cos f Let f, the true anomaly, be the angle that the vector r makes with the major axis joining both primaries, e is less than one for an elliptical trajectory, and p is a distance based upon a constant of integration. Where r1 is equal to r2 and μ = 0.5 for the neutron star and the companion having the same mass. The value ω is the rotation rate. In using the restricted three-body problem, the distances between the two primaries varies as a function of time and the distance along the x axis for one primary should vary per orbit period from (1 – e) a (periastron) to (1 + e) a (apastron) where a is the elliptical orbit’s major axis. The period of this motion where the distance to the common focus varies is when the true anomaly goes from 0 to 2 π, which is the rotation of the coordinate system based upon ω. This sinusoidal-like variation of the distance (eq (5)) separating the neutron star from its companion oscillates as a gravitational dipole resonator. There is some work that indicates the gravity potential should also include a variation as a function of time (Murad 2007a, 2007b). This assumes the rotation rate is a function of time resulting in a constant during the integration. This changes eq (3) when the rotation rate varies with time resulting in: 1 d 2 2 2 2 d 2 V V V 2 d 2 2 V x y z x y2 x y z x y . 2 dt 2 dt x y z 2 dt t
( 6)
In terms of determining how much energy is converted into gravitational waves, ignoring the change in mass used by the neutron star converted into radiated energy, is proportional to: 1 1 E C o2 f2 x 2 y 2 (7) 2 r 1 r 2 This yields an interesting conclusion. Since gravity waves are based upon changes in acceleration or jerk, the fact that the rotation rate is not constant is the correct trend. The rotation rate, ω is the time rate of the true anomaly and differences between initial and final values of ω represents the perihelion decay. This raises another issue. If we look at the traditional three-body problem, there are five libration points; two stable triangular points and three unstable collinear points. Mass should collect at these locations and should oscillate about their stable locations similar to the motion of the two primary bodies. Lagrangian points should also exist; however, they will vary as a function of time moving back and forth in a radial direction similar to the cycle provided by the distance functions for each of the two bodies. However, at
distances of interest, detection of such bodies would be extremely difficult although this would represent interesting phenomena. Thus all of these factors appear reasonable. The only item that requires definition is the reason for the perihelion decay in terms of cause and effect. Creation of gravitational waves is an effect and not a cause. B. Transforming the Governing Equations Murad (1975) identifies an approach to remove the time dependency in the restricted three-body problem by defining new potentials. Based upon defining the time derivative, the resulting equations are reduced either to a LaPlace equation or a Poisson equation where the potential acts like an elliptical function. This is demonstrated by the following substitutions: Let : x x , y y , z T h e n : x x
z , x
x
x
x
y
t x y xx x xy y
y y x z
zx
xz
x z z,
x
yy y
yz
z,
x
zy y
zz
z.
z
or:
Taking specific derivatives designed to eliminate terms yields: 1 xx yy Vxy y z 0. 2 xz 2 V
xz
o r : ( x, y , z )
( 8)
( 9)
1 V( x, z ) f ( y ) ...
In this evaluation, there are two givens: the neutron star in PSR 1913-16 moves in an elliptical orbit based upon Pulse-Taylor’s analysis of its emissions and that the orbit decays is observed by the perihelion as a function of time. This means that the energy described in equation (2) decreases based upon a decreasing function of ω as a function of time. This suggests that the coordinate system rotation rate is not a constant but should be a decreasing function of time. Based upon equation (9), the gravity potential effects upon the potential of motion will also change.
Results Let us examine what these equations mean with respect to modeling this pulsar. Equation (7) represents the energy difference due to the change in the coordinate system rotation rate. Since the rate is decreasing as a function of time, this should account for the amount of energy devoted to creating gravitational waves. The appearance of angular momentum may be the term that affects the perihelion changes. However, is this the primary cause for generating gravitational waves? The primaries are moving along the x-axis in a vibratory fashion as a function of the true anomaly making a complete cycle in a single rotation of the coordinate system. To match the perihelion information, the complete cycle of going from zero degrees to slightly less that 2 π. The motion along the x-axis should be a prime contribution to generating gravitational waves or that the change in radial acceleration or deceleration of the primaries creates a time rate of acceleration which, by Landau and Lifshitz should generate gravitational waves. A point was made earlier that the orbits of the two bodies was considered unusual and should eventually decay into a circular orbit. The way to do this is stop generating gravity waves by this approach and allow the eccentricity to go to zero as the radial distance becomes constant at a value of a. The two primary bodies then move in a circular orbit about each other.
Going to the next two equations where the problem is transformed, the first equation in equation (9) can be considered as the space-time potential because it is a function of spatial dimensions; however, the spatial derivatives or slope or curvature gives the velocity distribution in the domain between the two primaries. This partial differential equation is elliptical in the potential and hyperbolic in the gravity potential. In other words one could find the velocity distribution based upon knowledge of the gravity potential and vice versa. Obviously these equations are greatly simplified for the two-dimensional case. The elliptical behavior provides another insight. A Laplace equation admits the use of complex variables. Since this is an inhomogeneous equation, one could use pseudo-analytical functions and a similar form for complex variables. In fluid dynamics, this allows the introduction of rotationality or vorticity that was one of the premises that was initially suggested. The suggestion was that no basic means could be established that would create decay in the true anomaly to produce the observed perihelion response. The rotation effect could be determined either through frame dragging for an extremely large body that is rotating at the barycenter. Jefimenko recently points out limitations and a shortcoming to Newtonian gravitation despite it accurately predicts the motion of planets and satellites. Here, Newtonian gravitation according to Jefimenko (Murad 2008) is incapable of explaining: –A fast-moving point mass passing a spherically symmetric body causes the latter to rotate, –A mass moving with rapidly decreasing velocity exerts both an attractive and repulsive force on neighboring bodies, –A fast-moving mass passing a stationary mass exerts an explosion-like force on the latter, –A rotating mass that is suddenly stopped causes neighboring bodies to rotate, and –The period of revolution of a planet or satellite is affected by the rotation of the central body. Of these Newtonian gravitational anomalies, the last one is of the most interest suggesting that the revolution of satellites depends upon the rotation rate of the primary body. Thus if a body was at the barycenter, changes in its rotation rate could replicate the perihelion behavior of this pulsar. To an extent, this is the effect that we are attributing to frame-dragging. Since there is decay, the rotation rate of the object at the barycenter should be directly opposite the orbital direction from that is traveled by the neutron star and its companion. Three choices were mentioned. In terms of these models, what can we say about these options? In looking at a possible third body located at the barycenter, the restricted three-body problem assumes that the mass of the third body is inconsequential compared to the two primaries. If a third body were there, changes would have to occur with respect to defining the gravity gradient to include an additional term. Since the rotation rate is the time derivative of the true anomaly, the presence of the third body could assume that each primary is located at a distance where their centrifugal force balances the gravity effects of the third body. Thus the coordinate rotation rate depends upon a, distance, and the masses of the bodies. Regarding a Black Hole, determining the mass of such a celestial body would be difficult. However, a similar expression would be used to define both the distance and coordinate rotation rate. If such a black hole, or for that matter a gravitational singularity such as a gravitational vortex, the same point holds. Moreover, existence of a Laplace equation is enough to support a vortex singularity.
Conclusions The data regarding this pulsar was reviewed and a model was proposed that replicates the dynamics of the neutron and its companion star. The premise that these two bodies are the same mass and orbit about a common focus are fortuitous. One should wonder if the same behavior could have resulted if one assumed the neutron star rotated about, say a super giant planet, which would have provided the elliptical equation behavior. However, since there are other binary systems, there needs to be an explanation for the unusual perihelion behavior unique to this pulsar. In other words, something else is going on to generate the perihelion decay, which is previously mentioned as an effect to generate gravity waves. Clearly the change in radial acceleration of the two bodies would clearly be enough to generate gravitational waves and an energy expression was determined that would make the gravitational energy directly a function of this decay. Another mechanism is required to create the rotational effect of the coordinate system. Three separate mechanisms were identified that may induce these effects. The issue is that this would require additional analysis and discovery to better understand the dynamics of this pulsar.
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