We compute the long-term orbital variation of a test particle orbiting a central body acted upon by normal incident of plane gravitational wave. We use the tools of celestial mechanics to give the first order solution...We compute the long-term orbital variation of a test particle orbiting a central body acted upon by normal incident of plane gravitational wave. We use the tools of celestial mechanics to give the first order solution of canonical equations of long-period and short-period terms of the perturbed Hamiltonian of gravitational waves. We consider normal incident of plane gravitational wave and characteristic size of bound—two body system (earth’s satellite or planet) is much smaller than the wavelength of the wave and the wave’s frequency nw is much smaller than the particle’s orbital np. We construct the Hamiltonian of the gravitational waves in terms of the canonical variables (l,g,h,L,G,H)?and we solve the canonical equations numerically using Runge-Kutta fourth order method using language MATHEMATICA V10. Taking Jupiter as practical example we found that there are long period perturbations on ω,Ωand i?and not changing with revolution and the short period perturbations on a, e and M?changing with revolution during the interval of time (t−t0 ) which is changing from 0→4π.展开更多
A frozen orbit is beneficial for observation owing to its stationary apsidal line.The traditional gravitational field model of frozen orbits only considers the main zonal harmonic terms J_(2) and limited high-order te...A frozen orbit is beneficial for observation owing to its stationary apsidal line.The traditional gravitational field model of frozen orbits only considers the main zonal harmonic terms J_(2) and limited high-order terms,which cannot meet the stringent demands of all missions.In this study,the gravitational field is expanded to J_(15) terms and the Hamiltonian canonical form described by the Delaunay variables is used.The zonal harmonic coefficients of the Earth are chosen as the sample.Short-periodic terms are eliminated based on the Hori-Lie transformation.An algorithm is developed to solve all equilibrium points of the Hamiltonian function.A stable frozen orbit with an argument of perigee that equals neither 90°nor 270°is first reported in this paper.The local stability and topology of the equilibrium points are obtained from their eigenvalues.The bifurcations of the equilibrium points are presented by drawing their global long-term evolution of frozen orbits and their orbital periods.The relationship between the terms of the gravitational field and number of frozen points is addressed to explain why only limited frozen orbits are found in the low-order term case.The analytical results can be applied to other Earth-like planets and asteroids.展开更多
This work deals with the numerical solution of the gravitational waves effects on the orbital elements of the planets in case of commensurability between the wave’s frequency ng and the planet’s mean motion np. Taki...This work deals with the numerical solution of the gravitational waves effects on the orbital elements of the planets in case of commensurability between the wave’s frequency ng and the planet’s mean motion np. Taking Mercury and Pluto as practical examples for low frequency and high frequency, the variations of the orbital elements of Mercury due to resonance of gravitational wave are different and small than the perturbation on Pluto. The amount of changing in the orbital elements under the effects of gravitational waves is different from planet to planet according to the planet’s mean motion np. For low frequency ng, the secular variation in orbital elements will be negative (i.e. decreasing) in the inclination, semi-major axis and the eccentricity (i, a, e) like as Pluto. For high frequency ng like Mercury, the secular variation in all the orbital elements will be positive (i.e. increasing). The perturbation on all the orbital elements of two planets is changing during each revolution except the eccentricity e of Mercury and the mean anomaly M of Mercury and Pluto during the time.展开更多
A set of parameters called relative orbital elements were defined to describe the relative motion of the satellites in the formation flying. With the help of these parameters, the effect of the perturbations on the re...A set of parameters called relative orbital elements were defined to describe the relative motion of the satellites in the formation flying. With the help of these parameters, the effect of the perturbations on the relative orbit trajectory and geometric properties of satellite formation can be easily analyzed. First, the relative orbital elements are derived, and pointed out： if the eccentricity of the leading satellite is a small value, the relative orbit trajectory is determined by the intersection between an elliptic cylinder and a plane in the leading satellite orbit frame reference; and the parameters that describe the elliptic cylinder and the plane can be used to obtain the relative orbit trajectory and the relative orbital elements. Second, by analyzing the effects of gravitational perturbations on the relative orbit using the relative orbital elements,it is found that the propagation of a relative orbit consists of two parts ： one is the drift of the elliptic cylinder; and the other is the rotation of the plane resulted from the rotation of the normal of the plane. Meanwhile, the analytic formulae for the drift and rotation rates of a relative trajectory under gravitational perturbations are presented. Finally, the relative orbit trajectory and the corresponding changes were analyzed with respect to the J2 perturbation.展开更多
In this paper, we describe quasinormal modes（QNMs） for gravitational perturbations of Einstein-GaussBonnet black holes（BHs） in higher dimensional spacetimes, and derive the corresponding parameters of such black h...In this paper, we describe quasinormal modes（QNMs） for gravitational perturbations of Einstein-GaussBonnet black holes（BHs） in higher dimensional spacetimes, and derive the corresponding parameters of such black holes in three types of spacetime（flat, de Sitter（d S） and anti-de Sitter（Ad S））. Our attention is concentrated on discussing the（in）stability of Einstein-Gauss-Bonnet Ad S BHs through the temporal evolution of all types of gravitational perturbation fields（tensor, vector and scalar）. It is concluded that the potential functions in vector and scalar gravitational perturbations have negative regions, which suppress quasinormal ringing. Furthermore,the influences of the Gauss-Bonnet coupling parameter α, the number of dimensions n and the angular momentum quantum number l on the Einstein-Gauss-Bonnet Ad S BHs quasinormal spectrum are analyzed. The QNM frequencies have greater oscillation and lower damping rate with the growth of α. This indicates that QNM frequencies become increasingly unstable with large α. Meanwhile, the dynamic evolutions of the perturbation field are compliant with the results of computation from the Horowitz and Hubeny method. Because the number of extra dimensions is connected with the string scale, the relationship between α and properties of Einstein-Gauss-Bonnet Ad S BHs might be beneficial for the exploitation of string theory and extra-dimensional brane worlds.展开更多
We review the theory of inflationary perturbations. Perturbations at both linear and nonlinear orders are reviewed. We also review a variety of inflation models, emphasizing their signatures on cosmic perturbations.
In this paper, we study the gravitational quasi-normal modes（QNMs） for a static R^2 black hole（BH） in Anti-de Sitter（AdS） spacetime. The corresponding master equation of odd parity is derived and the QNMs are ev...In this paper, we study the gravitational quasi-normal modes（QNMs） for a static R^2 black hole（BH） in Anti-de Sitter（AdS） spacetime. The corresponding master equation of odd parity is derived and the QNMs are evaluated by the Horowitz and Hubeny method. Meanwhile the stability of such BH is also discussed through the temporal evolution of the perturbation field. Here we mainly consider the coefficient λ, which is related to the radius of AdS black hole, on the QNMs of the R^2 AdS BH. The results show that the Re（ω） and ｜Im（ω）｜ of the QNMs increase together as ｜λ｜ increases for a given angular momentum number l. That indicates with a larger value of ｜λ｜ the corresponding R^2 AdS BH returns to stable much more quickly. The dynamic evolution of the perturbation field is consistent with the results derived by the Horowitz and Hubeny method. Since in the conformal field theory the QNMs can reflect its approach to equilibrium, so our related results could be referential to studies of the AdS/CFT conjecture. The relationship between λ and the properties of the static R^2 BH might be helpful for the development of R^2 gravitational theory.展开更多
A recently proposed orbital dynamics model in the close proximity of an asteroid,which is called“attitude-restricted orbital dynamics”,includes the perturbation caused by the spacecraft’s gravitational orbit-attitu...A recently proposed orbital dynamics model in the close proximity of an asteroid,which is called“attitude-restricted orbital dynamics”,includes the perturbation caused by the spacecraft’s gravitational orbit-attitude coupling.This orbital model improves the precision of classical point-mass orbital model with only the non-spherical gravity.Equatorial equilibrium points have been investigated in the previous paper.In this paper,the inplane non-equatorial equilibrium points,which are outside the asteroid’s equatorial plane but within its longitudinal principal plane,are further studied for a uniformly-rotating asteroid.These non-equatorial equilibrium points are more diverse than those in the classical point-mass orbital dynamics without gravitational orbit-attitude coupling perturbation(GOACP).Two families of them have been found.The equatorial equilibrium points studied before and the non-equatorial ones studied here give a complete map of equilibrium points in the asteroid’s principal planes.Compared with the classical point-mass orbital dynamics without GOACP,the equatorial equilibrium points have extended the longitude range of equilibrium points around an asteroid,while the non-equatorial ones studied here will extend the latitude range.These equatorial and non-equatorial equilibrium points provide natural hovering positions for the asteroid close-proximity operations.展开更多
The first-order perturbations of the energy levels of a hydrogen atom in central internal gravitational field are investigated. The internal gravitational field is produced by the mass of the atomic nucleus. The energ...The first-order perturbations of the energy levels of a hydrogen atom in central internal gravitational field are investigated. The internal gravitational field is produced by the mass of the atomic nucleus. The energy shifts are calculated for the relativistic 1S, 2S, 2P, 3S, 3P, 3D, 4S, and 4P levels with Schwarzschild metric. The calculated results show that the gravitational corrections are sensitive to the total angular momentum quantum number.展开更多
By considering the spacecraft as an extended,rigid body with a prior known attitude instead of a point mass,the attitude-restricted orbital dynamics can improve the precision of the classical point-mass orbital dynami...By considering the spacecraft as an extended,rigid body with a prior known attitude instead of a point mass,the attitude-restricted orbital dynamics can improve the precision of the classical point-mass orbital dynamics in close proximity to an asteroid,because it includes the perturbation caused by the gravitational orbit–attitude coupling of the spacecraft(GOACP).The GOACP is defined as the difference between the gravity acting on a non-spherical,extended body(the real case of a spacecraft)and the gravity acting on a point mass(the approximation of a spacecraft in classical orbital dynamics).Inplane equilibrium points that are within the principal planes of the asteroid have been investigated for the attitude-restricted orbital dynamics in previous studies,including equatorial and in-plane non-equatorial equilibrium points.In this study,out-of-plane equilibrium points outside the principal planes of the asteroid were examined.Out-ofplane equilibrium points cannot exist in the classical point-mass orbital dynamics but do exist in the attitude-restricted orbital dynamics owing to the effects of the GOACP.The previously investigated in-plane equilibrium points and the out-of-plane ones examined in this study provide a complete map of the equilibrium points in close proximity to an asteroid with the GOACP.Equatorial and in-plane non-equatorial equilibrium points have extended the longitude and latitude ranges of the classical equilibrium points without the GOACP,respectively,while the out-of-plane ones examined in the present study extend both the longitude and latitude ranges.Additionally,the invariant manifolds of out-of-plane equilibrium points were calculated,and the results indicated that the attitude of spacecraft significantly affects the invariant manifolds.In practice,these equilibrium points can provide natural hovering positions for operations in proximity to asteroids,and their invariant manifolds can be used for transfers to or from the equilibrium points.展开更多
The prospect of using gravitational wave detections via the quasinormal modes(QNMs) to test modified gravity theories is exciting area of current research.Gravitational waves(GWs) emitted by a perturbed black hole(BH)...The prospect of using gravitational wave detections via the quasinormal modes(QNMs) to test modified gravity theories is exciting area of current research.Gravitational waves(GWs) emitted by a perturbed black hole(BH) will decay as a superposition of their QNMs of oscillations at the ringdown phase.In this work,we investigate the QNMs of the Einstein-Euler-Heisenberg(EEH) BH for both axial and polar gravitational perturbations.We obtain master equations with the tetrad formalism,and the quasinormal frequencies of the EEH BH are calculated in the 6 th order Wentzel-Kramers-Brill?uin approximation.It is interesting to note that the QNMs of the EEH BH would differ from those of the Reissner-Nordstr?m BH under the EH parameter,which indicates the EH parameter would affect the gravitational perturbations for the EEH BH.展开更多
A double-well potential model is proposed for the pionic deuterium that enables to calculate the energy split, the potential barrier height and estimate the pion tetrahedron edge length. We propose that pion tetrahedr...A double-well potential model is proposed for the pionic deuterium that enables to calculate the energy split, the potential barrier height and estimate the pion tetrahedron edge length. We propose that pion tetrahedrons, π<sup>Td</sup> = u<sub>d</sub>~</sup>dũ, play a central role in the Yukawa interaction by enabling quark exchange reactions between protons and neutrons by tunneling through a potential barrier. A vacuum polarization Feynman diagram is proposed for the π<sup>Td</sup> having chains of fermion loops for the two valence quarks and anti-quarks connected by gluons. With a higher order vacuum polarization diagram, the d and u quark loops are interleaved and the chiral symmetry is broken dynamically. The proposed π<sup>Td</sup> vacuum polarization integral does not diverge in both the IR and UV limits and vanishes in the limit of an infinite pion tetrahedron condensate. We propose a new Delbruck scattering Feynman diagram that includes d and u quark and anti-quark interleaved loops. We further propose that conversion of gravitons to photons may occur via quark and anti-quark loops that describe the pion tetrahedrons dynamics in the vacuum and may also transfer gravitational waves.展开更多
文摘We compute the long-term orbital variation of a test particle orbiting a central body acted upon by normal incident of plane gravitational wave. We use the tools of celestial mechanics to give the first order solution of canonical equations of long-period and short-period terms of the perturbed Hamiltonian of gravitational waves. We consider normal incident of plane gravitational wave and characteristic size of bound—two body system (earth’s satellite or planet) is much smaller than the wavelength of the wave and the wave’s frequency nw is much smaller than the particle’s orbital np. We construct the Hamiltonian of the gravitational waves in terms of the canonical variables (l,g,h,L,G,H)?and we solve the canonical equations numerically using Runge-Kutta fourth order method using language MATHEMATICA V10. Taking Jupiter as practical example we found that there are long period perturbations on ω,Ωand i?and not changing with revolution and the short period perturbations on a, e and M?changing with revolution during the interval of time (t−t0 ) which is changing from 0→4π.
基金supported in part by the National Natural Science Foundation of China(Nos.11772024 and 11432001)Qian Xuesen Youth Innovation Foundation of China Aerospace Science and Technology Corporation.
文摘A frozen orbit is beneficial for observation owing to its stationary apsidal line.The traditional gravitational field model of frozen orbits only considers the main zonal harmonic terms J_(2) and limited high-order terms,which cannot meet the stringent demands of all missions.In this study,the gravitational field is expanded to J_(15) terms and the Hamiltonian canonical form described by the Delaunay variables is used.The zonal harmonic coefficients of the Earth are chosen as the sample.Short-periodic terms are eliminated based on the Hori-Lie transformation.An algorithm is developed to solve all equilibrium points of the Hamiltonian function.A stable frozen orbit with an argument of perigee that equals neither 90°nor 270°is first reported in this paper.The local stability and topology of the equilibrium points are obtained from their eigenvalues.The bifurcations of the equilibrium points are presented by drawing their global long-term evolution of frozen orbits and their orbital periods.The relationship between the terms of the gravitational field and number of frozen points is addressed to explain why only limited frozen orbits are found in the low-order term case.The analytical results can be applied to other Earth-like planets and asteroids.
文摘This work deals with the numerical solution of the gravitational waves effects on the orbital elements of the planets in case of commensurability between the wave’s frequency ng and the planet’s mean motion np. Taking Mercury and Pluto as practical examples for low frequency and high frequency, the variations of the orbital elements of Mercury due to resonance of gravitational wave are different and small than the perturbation on Pluto. The amount of changing in the orbital elements under the effects of gravitational waves is different from planet to planet according to the planet’s mean motion np. For low frequency ng, the secular variation in orbital elements will be negative (i.e. decreasing) in the inclination, semi-major axis and the eccentricity (i, a, e) like as Pluto. For high frequency ng like Mercury, the secular variation in all the orbital elements will be positive (i.e. increasing). The perturbation on all the orbital elements of two planets is changing during each revolution except the eccentricity e of Mercury and the mean anomaly M of Mercury and Pluto during the time.
文摘A set of parameters called relative orbital elements were defined to describe the relative motion of the satellites in the formation flying. With the help of these parameters, the effect of the perturbations on the relative orbit trajectory and geometric properties of satellite formation can be easily analyzed. First, the relative orbital elements are derived, and pointed out： if the eccentricity of the leading satellite is a small value, the relative orbit trajectory is determined by the intersection between an elliptic cylinder and a plane in the leading satellite orbit frame reference; and the parameters that describe the elliptic cylinder and the plane can be used to obtain the relative orbit trajectory and the relative orbital elements. Second, by analyzing the effects of gravitational perturbations on the relative orbit using the relative orbital elements,it is found that the propagation of a relative orbit consists of two parts ： one is the drift of the elliptic cylinder; and the other is the rotation of the plane resulted from the rotation of the normal of the plane. Meanwhile, the analytic formulae for the drift and rotation rates of a relative trajectory under gravitational perturbations are presented. Finally, the relative orbit trajectory and the corresponding changes were analyzed with respect to the J2 perturbation.
基金Supported by FAPESP(2012/08934-0)National Natural Science Foundation of China(11205254,11178018,11375279,11605015)+1 种基金the Natural Science Foundation Project of CQ CSTC(2011BB0052)the Fundamental Research Funds for the Central Universities(106112016CDJXY300002,106112017CDJXFLX0014,CDJRC10300003)
文摘In this paper, we describe quasinormal modes（QNMs） for gravitational perturbations of Einstein-GaussBonnet black holes（BHs） in higher dimensional spacetimes, and derive the corresponding parameters of such black holes in three types of spacetime（flat, de Sitter（d S） and anti-de Sitter（Ad S））. Our attention is concentrated on discussing the（in）stability of Einstein-Gauss-Bonnet Ad S BHs through the temporal evolution of all types of gravitational perturbation fields（tensor, vector and scalar）. It is concluded that the potential functions in vector and scalar gravitational perturbations have negative regions, which suppress quasinormal ringing. Furthermore,the influences of the Gauss-Bonnet coupling parameter α, the number of dimensions n and the angular momentum quantum number l on the Einstein-Gauss-Bonnet Ad S BHs quasinormal spectrum are analyzed. The QNM frequencies have greater oscillation and lower damping rate with the growth of α. This indicates that QNM frequencies become increasingly unstable with large α. Meanwhile, the dynamic evolutions of the perturbation field are compliant with the results of computation from the Horowitz and Hubeny method. Because the number of extra dimensions is connected with the string scale, the relationship between α and properties of Einstein-Gauss-Bonnet Ad S BHs might be beneficial for the exploitation of string theory and extra-dimensional brane worlds.
基金Supported by Fundings from the Kavli Institute for the Physics and Mathematics of the Universe(Kavli IPMU)the Japan Society for the Promotion of Science(JSPS)
文摘We review the theory of inflationary perturbations. Perturbations at both linear and nonlinear orders are reviewed. We also review a variety of inflation models, emphasizing their signatures on cosmic perturbations.
基金supported by the National Natural Science Foundation of China(Grant Nos.11205254,11178018,11375279,and 11605015)the Fundamental Research Funds for the Central Universities,China(Grant Nos.106112016CDJXY300002 and CDJRC10300003)+1 种基金the Chinese State Scholarship Fund,FAPESP(Grant No.2012/08934-0)the Natural Science Foundation Project of CQ CSTC(Grant No.2011BB0052)
文摘In this paper, we study the gravitational quasi-normal modes（QNMs） for a static R^2 black hole（BH） in Anti-de Sitter（AdS） spacetime. The corresponding master equation of odd parity is derived and the QNMs are evaluated by the Horowitz and Hubeny method. Meanwhile the stability of such BH is also discussed through the temporal evolution of the perturbation field. Here we mainly consider the coefficient λ, which is related to the radius of AdS black hole, on the QNMs of the R^2 AdS BH. The results show that the Re（ω） and ｜Im（ω）｜ of the QNMs increase together as ｜λ｜ increases for a given angular momentum number l. That indicates with a larger value of ｜λ｜ the corresponding R^2 AdS BH returns to stable much more quickly. The dynamic evolution of the perturbation field is consistent with the results derived by the Horowitz and Hubeny method. Since in the conformal field theory the QNMs can reflect its approach to equilibrium, so our related results could be referential to studies of the AdS/CFT conjecture. The relationship between λ and the properties of the static R^2 BH might be helpful for the development of R^2 gravitational theory.
基金This work has been supported by the National Natural Science Foundation of China under Grant Nos.11602009,11432001,and 11872007the Young Elite Scientist Sponsorship Program by China Association for Science and Technology under Grant No.2017QNRC001the Fundamental Research Funds for the Central Universities.
文摘A recently proposed orbital dynamics model in the close proximity of an asteroid,which is called“attitude-restricted orbital dynamics”,includes the perturbation caused by the spacecraft’s gravitational orbit-attitude coupling.This orbital model improves the precision of classical point-mass orbital model with only the non-spherical gravity.Equatorial equilibrium points have been investigated in the previous paper.In this paper,the inplane non-equatorial equilibrium points,which are outside the asteroid’s equatorial plane but within its longitudinal principal plane,are further studied for a uniformly-rotating asteroid.These non-equatorial equilibrium points are more diverse than those in the classical point-mass orbital dynamics without gravitational orbit-attitude coupling perturbation(GOACP).Two families of them have been found.The equatorial equilibrium points studied before and the non-equatorial ones studied here give a complete map of equilibrium points in the asteroid’s principal planes.Compared with the classical point-mass orbital dynamics without GOACP,the equatorial equilibrium points have extended the longitude range of equilibrium points around an asteroid,while the non-equatorial ones studied here will extend the latitude range.These equatorial and non-equatorial equilibrium points provide natural hovering positions for the asteroid close-proximity operations.
基金The project supported by National Natural Science Foundation of China under Grant Nos. 10435080 and 10575123 and the Chinese Academy of Sciences Knowledge Innovation Project under Grant Nos. KJCX2-SW-N1b and KJC-SYW-N2
文摘The first-order perturbations of the energy levels of a hydrogen atom in central internal gravitational field are investigated. The internal gravitational field is produced by the mass of the atomic nucleus. The energy shifts are calculated for the relativistic 1S, 2S, 2P, 3S, 3P, 3D, 4S, and 4P levels with Schwarzschild metric. The calculated results show that the gravitational corrections are sensitive to the total angular momentum quantum number.
基金supported by the National Natural Science Foundation of China under Grant Nos.11602009,11432001 and 11872007the Fundamental Research Funds for the Central Universities.
文摘By considering the spacecraft as an extended,rigid body with a prior known attitude instead of a point mass,the attitude-restricted orbital dynamics can improve the precision of the classical point-mass orbital dynamics in close proximity to an asteroid,because it includes the perturbation caused by the gravitational orbit–attitude coupling of the spacecraft(GOACP).The GOACP is defined as the difference between the gravity acting on a non-spherical,extended body(the real case of a spacecraft)and the gravity acting on a point mass(the approximation of a spacecraft in classical orbital dynamics).Inplane equilibrium points that are within the principal planes of the asteroid have been investigated for the attitude-restricted orbital dynamics in previous studies,including equatorial and in-plane non-equatorial equilibrium points.In this study,out-of-plane equilibrium points outside the principal planes of the asteroid were examined.Out-ofplane equilibrium points cannot exist in the classical point-mass orbital dynamics but do exist in the attitude-restricted orbital dynamics owing to the effects of the GOACP.The previously investigated in-plane equilibrium points and the out-of-plane ones examined in this study provide a complete map of the equilibrium points in close proximity to an asteroid with the GOACP.Equatorial and in-plane non-equatorial equilibrium points have extended the longitude and latitude ranges of the classical equilibrium points without the GOACP,respectively,while the out-of-plane ones examined in the present study extend both the longitude and latitude ranges.Additionally,the invariant manifolds of out-of-plane equilibrium points were calculated,and the results indicated that the attitude of spacecraft significantly affects the invariant manifolds.In practice,these equilibrium points can provide natural hovering positions for operations in proximity to asteroids,and their invariant manifolds can be used for transfers to or from the equilibrium points.
基金Supported by the National Natural Science Foundation of China (11873001,12047564,12147102)the Fundamental Research Funds for the Central Universities(2021CDJZYJH-003)。
文摘The prospect of using gravitational wave detections via the quasinormal modes(QNMs) to test modified gravity theories is exciting area of current research.Gravitational waves(GWs) emitted by a perturbed black hole(BH) will decay as a superposition of their QNMs of oscillations at the ringdown phase.In this work,we investigate the QNMs of the Einstein-Euler-Heisenberg(EEH) BH for both axial and polar gravitational perturbations.We obtain master equations with the tetrad formalism,and the quasinormal frequencies of the EEH BH are calculated in the 6 th order Wentzel-Kramers-Brill?uin approximation.It is interesting to note that the QNMs of the EEH BH would differ from those of the Reissner-Nordstr?m BH under the EH parameter,which indicates the EH parameter would affect the gravitational perturbations for the EEH BH.
文摘A double-well potential model is proposed for the pionic deuterium that enables to calculate the energy split, the potential barrier height and estimate the pion tetrahedron edge length. We propose that pion tetrahedrons, π<sup>Td</sup> = u<sub>d</sub>~</sup>dũ, play a central role in the Yukawa interaction by enabling quark exchange reactions between protons and neutrons by tunneling through a potential barrier. A vacuum polarization Feynman diagram is proposed for the π<sup>Td</sup> having chains of fermion loops for the two valence quarks and anti-quarks connected by gluons. With a higher order vacuum polarization diagram, the d and u quark loops are interleaved and the chiral symmetry is broken dynamically. The proposed π<sup>Td</sup> vacuum polarization integral does not diverge in both the IR and UV limits and vanishes in the limit of an infinite pion tetrahedron condensate. We propose a new Delbruck scattering Feynman diagram that includes d and u quark and anti-quark interleaved loops. We further propose that conversion of gravitons to photons may occur via quark and anti-quark loops that describe the pion tetrahedrons dynamics in the vacuum and may also transfer gravitational waves.
