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tech / sci.physics.relativity / Re: Analytic GR solutions of S2-star orbits and precession 732” per revolution

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* Re: Analytic GR solutions of S2-star orbits and precHannu Poropudas
`* Re: Analytic GR solutions of S2-star orbits and precHannu Poropudas
 `- Re: Analytic GR solutions of S2-star orbits and precHannu Poropudas

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Re: Analytic GR solutions of S2-star orbits and precession 732” per revolution

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Subject: Re:_Analytic_GR_solutions_of_S2-star_orbits_and_prec
ession_732”_per_revolution
From: haporop...@gmail.com (Hannu Poropudas)
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 by: Hannu Poropudas - Wed, 16 Nov 2022 08:58 UTC

torstai 17. syyskuuta 2020 klo 10.05.28 UTC+3 hanp...@luukku.com kirjoitti:
> On Monday, September 14, 2020 at 2:05:10 PM UTC+3, Hannu Poropudas wrote:
> > On Sunday, September 13, 2020 at 12:34:54 PM UTC+3, Hannu Poropudas wrote:
> > > On Thursday, September 10, 2020 at 10:10:49 AM UTC+3, Hannu Poropudas wrote:
> > > > Analytic GR solutions of S2-star orbits and precession 732” per revolution
> > > >
> > > > New initial data 2020 is used. My earlier initial data was 2016 in my
> > > > postings in this sci.physics.relativity.
> > > >
> > > > Please also take a look my two postings about OJ287 in sci.astro. I have got
> > > > no comments there due so heavy traffic of unbssinesslike postings from many posters.
> > > >
> > > > (Only command lines of Maple 9 program is below and command line mark is >. This can be copy-pasted in Maple 9 program and run it.
> > > > Remark that for example e-2 below means 10^(-2)
> > > > or e2 below means 10^2.)
> > > >
> > > > Best Regards,
> > > >
> > > > Hannu Poropudas
> > > > Finland
> > > >
> > > > ---COPY of the Maple 9 program below------
> > > >
> > > > > # RIGHT: Particle in Schwarzschild metrics H.P. 09.09.2020
> > > > > # S2 Orbit around Sagittarius A*
> > > > > # Perihelion precession of S2.
> > > > > #OK=COMPARISION to Post-Newtonian approx of Abuter R et al 2020, A&A 636,L5(2020)
> > > > > # Abuter R. et al. 2020. Detection of the Schwarzschild precession in the
> > > > > # orbit of the star S2 near the Galactic centre massive black hole.
> > > > > # Astronomy Astrophysics, 636, L5 (2020).
> > > > > # In this article: 12'/rev=720"/rev,12.1'/rev=726"/rev, 12.3'/rev=738"/rev
> > > > > # My GR calculation below, result : 731.8543470"/rev OR -676.7436912"/rev
> > > > > # (I use not rounded numbers). Approximately 732"/rev OR -677"/rev.
> > > > > # New starting values from Table E.1 Best-fit Orbit Parameters above article
> > > > > # and other constants of physics are taken from Wikipedia date 9.9.2020.
> > > > > # NEW CALCULATION x and y are roots. E and J little changed.
> > > > > # (analytic solutions defined between roots a2..a1 OR a4..a3)
> > > > >Restart;
> > > > >with(plots):with(plottools):
> > > > > # GEOMETRIZED UNIT SYSTEM IS PROBLEMATIC.
> > > > > # Ref. Formulae for E and J from Weinberg's book. (THESE FORMULAE ARE NOT USED.)
> > > > > # Reference: Weinberg, S. 1972. Gravitation and Cosmology: Principles
> > > > > # and Applications of the General Relativity. Wiley, New York. pp.. 179-210.
> > > > > # (units c = 1, and c.g.s in Weinberg's book).
> > > > > # J = r^2*dPhi/dp = constant (angular momentum per unit mass), p. 186.
> > > > > # E = constant (energy per unit mass), p. 186.
> > > > > # E > 0 for material particles, E = 0 for photons. p. 186.
> > > > > # Integration limits must be determined from the problem to which apply these.
> > > > >#Int(1/(r^4*(1/J^2-E/J^2)+r^3*(2*M*G*E/J^2)-r^2+2*M*G*r)^(1/2),r);
> > > > >AU := 149597870700*10^2;
> > > > ># 125.058 mas, 8246.7 pc, 1 pc =3.26 ly
> > > > >ap:= 1030.799324*AU;
> > > > >em := 0.884649;
> > > > >M := 4.261*10^6*1.98847*10^30*10^3;
> > > > ># v = c
> > > > >v := 2.99792458*10^8*10^2;
> > > > >G := 6.6743015*10^(-11)*10^3;
> > > > ># BH mass*G geometric units (cm) and v = c.
> > > > >M*G/v^2;
> > > > >MG := 0.6292090968e12;
> > > > > # EXAMPLE. Weinberg formulae E and J (THESE FORMULAE ARE NOT ANY MORE USED.)
> > > > > # Perihelion distance
> > > > >x := ap*(1-em);
> > > > ># Aphelion distance
> > > > >y := ap*(1+em);
> > > > > # NEW CALCULATION (NEW FORMULAE OF MY OWN) x and y are roots
> > > > > #***********************************************************
> > > > >E := (2*x*y*MG+2*y^2*MG+2*x^2*MG-x^2*y-x*y^2)/((-y+2*MG)*(-2*x*MG-2*y*MG+x^2+x*y));
> > > > >J := 2^(1/2)*(-(-y+2*MG)*MG*(-2*x*MG-2*y*MG+x^2+x*y))^(1/2)*y*x/(-x^2*y+2*x^2*MG-4*x*MG^2+4*x*y*MG-x*y^2+2*y^2*MG-4*y*MG^2);
> > > > > #*****************************************************************
> > > > >
> > > > > # Weinberg's formula (NOT USED ANY MORE)
> > > > > #J := sqrt((1/(1-2*MG/y)-(1/(1-2*MG/x)))/(1/y^2-1/x^2));
> > > > > # NEW CALCULATION x and y are roots
> > > > >J := -0.4594478956e14;
> > > > > # Weinberg's formula (NOT USED ANY MORE)
> > > > > #E := (y^2/(1-2*MG/y)-x^2/(1-2*MG/x))/(y^2-x^2);
> > > > > # NEW CALCULATION x and y are roots
> > > > >E := 1.000040803;
> > > > >solve(r^4*(1/J^2-E/J^2)+r^3*(2*MG*E/J^2)-r^2+2*MG*r=0,r);
> > > > >a1 := 0.2906262559e17;
> > > > >a2 := 0.1778773310e16;
> > > > >a3 := 0.1259363679e13;
> > > > >a4 := 0;
> > > >
> > > > ># Primitive function, definition areas 0<=P<=Pi/2 and a2 <= r <= a1.
> > > >
> > > > >e := P -> 2.000794538*EllipticF(sin(P),0.2578161452e-1);
> > > > >r := P -> (0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17);
> > > >
> > > > > #******************************
> > > > > # P which corresponds to the perihelion distance (TWO VALUES)
> > > > >solve((0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17)=0.1778774525e16,P);
> > > > >e(0.8532953281e-3);
> > > > >e(-0.8532953281e-3);
> > > > ># Solution's definition area limit is a2-root (TWO VALUES)
> > > > >solve((0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17)=0.1778773310e16,P);
> > > > >e(0.6516324491e-5);
> > > > >e(-0.6516324491e-5);
> > > > ># P which corresponds to the aphelion distance (TWO VALUES)
> > > > >solve((0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17)=0.2906230228e17,P);
> > > > >e(1.569945054);
> > > > >e(-1.569945054);
> > > > > # Solution's correct definition limit is a1-root (TWO VALUES)
> > > > > # COMPLEX NUMBER (SMALL COMPLEX PARTS ARE LITTLE OUTSIDE ORBIT)
> > > > >solve((0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17)=0.2906262559e17,P);
> > > > >e(1.570796327-0.1136598102e-4*I);
> > > > >e(-1.570796327+0.1136598102e-4*I);
> > > >
> > > > > # Angle change, definition areas 0<=P<=Pi/2 ja a2 <= r <= a1.
> > > > > # + sign perihelion in last term
> > > >
> > > > >e2 := P -> 2*abs(2.000794538*EllipticF(sin(P),0.2578161452e-1)-2..000794538*EllipticF(sin(0.8532953281e-3),0.2578161452e-1))-2*Pi;
> > > > >r2 := P -> (0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17);
> > > >
> > > > > # Calculation of different combinations of + and - signs
> > > > > # - sign perihelion in last term
> > > >
> > > > >e2B := P -> 2*abs(2.000794538*EllipticF(sin(P),0.2578161452e-1)-2.000794538*EllipticF(sin(-0.8532953281e-3),0.2578161452e-1))-2*Pi;
> > > > >r2B := P -> (0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17);
> > > >
> > > > > # Angle change in radians per one revolution. (TWO e2 and e2B functions)
> > > > > # FOR e2
> > > > >evalf(e2(1.569945054));
> > > > ># Angle change in degrees per one revolution.
> > > > >evalf(-0.3280946e-2*180/Pi);
> > > > ># Angle change in arc seconds per one revolution.
> > > > >-0.1879843586*60*60;
> > > > >###(-676.7436912)
> > > > ># Second root
> > > > >evalf(e2(-1.569945054));
> > > > >evalf(0.3548130e-2*180/Pi);
> > > > >0.2032928741*60*60;
> > > > >###(731.8543470)
> > > > ># FOR e2B
> > > > >evalf(e2B(1.569945054));
> > > > ># Angle change in degrees per one revolution.
> > > > >evalf(0.3548130e-2*180/Pi);
> > > > ># Angle change in arc seconds per one revolution.
> > > > >0.2032928741*60*60;
> > > > >###(731.8543470)
> > > > ># Second root
> > > > >evalf(e2B(-1.569945054));
> > > > >evalf(-0.3280946e-2*180/Pi);
> > > > >-0.1879843586*60*60;
> > > > >####(-676.7436912)
> > > > > # + and - signs for e2 or e2B does not have different results due abs-values
> > > > > # Second Primitive function, definition areas 0<=P<=Pi/2 and a4=0 <= r <= a3.
> > > > > # This has form of spiral (+ and - signs) which leads to the origin.
> > > > >
> > > > > # SUMMARY of S2 precession: 731.8543470"/revolution OR -676.7436912"/revolution.
> > > > >
> > > > > #******************************
> > > >
> > > > ># Second Primitive function, definition areas 0<=P<=Pi/2 and a4=0 <=
> > > > r <= a3.
> > > >
> > > > >ee := P-> 2.000794538*EllipticF(sin(P),0.9996675989);
> > > > >rr := P-> (0.3660041508e29*sin(P)^2)/(0.1259363679e13*sin(P)^2+0..2906136623e17);
> > > >
> > > > > #*********************************************************
> > > > >
> > > > > # Plottings only on definition area a2..a1.
> > > >
> > > > > # First side of the solution (+,- solution)
> > > > >plot([r(P),e(P),P=0..Pi/2],coords=polar);
> > > >
> > > > ># Second side of the solution (+,- solution)
> > > > >plot([r(P),-e(P),P=0..Pi/2],coords=polar);
> > > >
> > > > ># Angle change picture has no other meaning than above calculated precession
> > > > >plot([r2(P),e2(P),P=0..Pi/2],coords=polar);
> > > >
> > > > ># Angle change picture has no other meaning than above calculated precession
> > > > >plot([r2B(P),e2B(P),P=0..Pi/2],coords=polar)
> > > >
> > > > > # Plottings only on definition area a4..a3.
> > > >
> > > > > # First side of the Second solution (+,- solution)
> > > > >plot([rr(P),ee(P),P=0..Pi/2],coords=polar);
> > > >
> > > > ># Second side of the Second solution (+,- solution)
> > > > >plot([rr(P),-ee(P),P=0..Pi/2],coords=polar);
> > > *** 1. CORRECTION: Formula of J has + and - signs. Positive sign should be selected.
> > > Previous - sign selection does not influence due J^2 was always used.
> > >
> > > J := 0.4594478956e14
> > >
> > > *** 2. -Pi/2 .. Pi/2 plots gives both sides of both solutions.
> > > This is due
> > > sin(-P)=-sin(P)
> > > and
> > > EllipticF(sin(-P),q)=EllipticF(-sin(P),q)=-EllipticF(sin(P),q).
> > >
> > > 3. I calculaled below total coordinate velocity, proper velocity and
> > > total proper acceleration for S2-star:
> > > (These are used only in definition areas of above analytic solutions
> > > a2<=r<=a1 or a4<r<=a3,
> > > a1 := 0.2906262559e17,
> > > a2 := 0.1778773310e16,
> > > a3 := 0.1259363679e13,
> > > a4 := 0)
> > >
> > > (REMARK: > is command line mark of Maple 9 program)
> > > ># Total coordinate velocity and total proper velocity S2-star
> > > ># at perihelion and at aphelion HP 12092020
> > >
> > > > ### total coordinate velocity v^2=(dr/dt)^2+r^2*(dPhi/dt)^2
> > > > sqrt((1-2*MG/r)^2*(1-(E+J^2/r^2)*(1-2*MG/r))+(J^2/r^2)*(1-2*MG/r)^2)
> > > > sqrt((1-2*MG/x)^2*(1-(E+J^2/x^2)*(1-2*MG/x))+(J^2/x^2)*(1-2*MG/x)^2);
> > > 0.2581119858e-1
> > > > # km/s
> > > > 0.2997924580e11*0.2581119858e-1/10^5;
> > > 7738.002666
> > > > sqrt((1-2*MG/y)^2*(1-(E+J^2/y^2)*(1-2*MG/y))+(J^2/y^2)*(1-2*MG/y)^2);
> > > 0.1581028031e-2
> > > > # km/s
> > > > 0.2997924580e11*0.1581028031e-2/10^5;
> > > 473.9802796
> > > > ###
> > > > ### total proper velocity v^2=(dr/dtau)^2+r^2*(dPhi/dtau)^2
> > > > sqrt((1/E)*(1-(E+J^2/r^2)*(1-2*MG/r))+J^2/r^2)
> > > > sqrt((1/E)*(1-(E+J^2/x^2)*(1-2*MG/x))+J^2/x^2);
> > > 0.2582947199e-1
> > > > # km/s
> > > > 0.2997924580e11*0.2582947199e-1/10^5;
> > > 7743.480897
> > > > sqrt((1/E)*(1-(E+J^2/y^2)*(1-2*MG/y))+J^2/y^2);
> > > 0.001581096486
> > > > # km/s
> > > > 0.2997924580e11*0.1581096486e-2/10^5;
> > > 474.0008019
> > > > ###
> > >
> > > ># Total proper acceleration S2 star HP 13092020
> > >
> > > > ### Total proper acceleration (- + sign d^2Phi/dta^2)(+ - sign dr/da)
> > >
> > > > ### A^2=(d^2r/dta^2-r*(dPhi/dta)^2)^2+(r*d^2Phi/dta^2+2*(dr/dta)*(dPhi/dta))^2
> > >
> > > > #sqrt(((J^2/(sqrt(E)*r^3))*(1-2*MG/r)-(MG/(sqrt(E)*r^2*(1-2*MG/r)))+(MG/(sqrt(E)*r^2))*(1-(E+J^2/r^2)*(1-2*MG/r))-r*(J/r^2)^2)^2+(r*((2*J/(r^3))*sqrt(1-(E+J^2/r^2)*(1-2*MG/r)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/r^2)*(1-2*MG/r)))*(J/r^2))^2)
> > >
> > > > ### total proper acceleration at perihelion (+ sign selected for both)
> > >
> > > > sqrt(((J^2/(sqrt(E)*x^3))*(1-2*MG/x)-(MG/(sqrt(E)*x^2*(1-2*MG/x)))+(MG/(sqrt(E)*x^2))*(1-(E+J^2/x^2)*(1-2*MG/x))-x*(J/x^2)^2)^2+(x*((2*J/(x^3))*sqrt(1-(E+J^2/x^2)*(1-2*MG/x)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/x^2)*(1-2*MG/x)))*(J/x^2))^2);
> > > 1.992765885*10^(-19)
> > > > # cm/s^2
> > > > (2.99792458*10^8*10^2)^2*0.1992765885e-18;
> > > 179.1008659
> > > > ### total proper acceleration at aphelion (+ sign selected for both)
> > >
> > > > sqrt(((J^2/(sqrt(E)*y^3))*(1-2*MG/y)-(MG/(sqrt(E)*y^2*(1-2*MG/y)))+(MG/(sqrt(E)*y^2))*(1-(E+J^2/y^2)*(1-2*MG/y))-y*(J/y^2)^2)^2+(y*((2*J/(y^3))*sqrt(1-(E+J^2/y^2)*(1-2*MG/y)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/y^2)*(1-2*MG/y)))*(J/y^2))^2);
> > > 7.450050462*10^(-22)
> > > > # cm/s^2
> > > > (2.99792458*10^8*10^2)^2*0.7450050462e-21;
> > > 0.6695771434
> > > > ### total proper acceleration at event horizon z = MG(+ sign selected for both)
> > >
> > > > # ONE NOTICE: pure imaginary number acceleration for example at 3*MG , this is due 3*MG is over definition area of the second analytic solution.
> > > > # Different selections of +, - signs (- + sign d^2Phi/dta^2)(+ - sign dr/da) are not treated here , + sign selections is made for both cases.
> > >
> > > > z := MG;
> > > 6.292090968*10^11
> > > > sqrt(((J^2/(sqrt(E)*z^3))*(1-2*MG/z)-(MG/(sqrt(E)*z^2*(1-2*MG/z)))+(MG/(sqrt(E)*z^2))*(1-(E+J^2/z^2)*(1-2*MG/z))-z*(J/z^2)^2)^2+(z*((2*J/(z^3))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))*(J/z^2))^2);
> > > 3.494379988*10^(-8)
> > > > # cm/s^2
> > > > (2.99792458*10^8*10^2)^2*0.3494379988e-7;
> > > 3.140592111*10^13
> > > > ###
> > > Best Regards,
> > > Hannu Poropudas
> > > Finland
> > CORRECTION (event horizon is was not MG ):
> > Interesting OPEN QUESTIONS would be if S2-star somehow happens to go to my second analytic solution orbit,
> > which definition area is 0 < r <= 1.259363679*10^12 cm.
> > At r = 2*MG = = 1.258418194*10^12 cm, event horizon of the SgrA* black hole seems to
> > have following (OPEN QUESTION OF INTERPRETATION of the total proper velocity and the total proper acceleration):
> >
> > total coordinate velocity = 1.460945827*10^(-8) , (geometric units, Weinberg 1972),
> > total coordinate velocity = 437.9805405 cm/s, (c.g.s units, Weinberg 1972),
> > total proper velocity = 36.52364512 (geometric units, Weinberg 1972),
> > total proper velocity = 1.094951335*10^12, cm/s (c.g.s units, Weinberg 1972)
> > and
> > total proper acceleration = 8,927256908*10^17 cm/s^2 , (c.g.s units, Weinberg 1972)
> >
> > I have used definitions of Weinberg S. 1972. Gravitation and Cosmology book
> > and Becker 1954 Introduction to Theoretical Mechanics book
> > when I calculated above total coordinate velocity, total proper velocity and total proper acceleration.
> > (Formulae of these GR calculations are in above posting of mine).
> >
> > Best Regards,,
> > Hannu Poropudas
> > Finland
> Preliminary formulae and few important numerical points calculated below:
>
> ># S2-star’s two analytic solutions of orbits around SgrA* Black Hole
> ># Total coordinate velocity, total proper velocity and
> ># Total coordinate acceleration, total proper acceleration
> ># formulae and numerical calculation for some points.
> ># OPEN QUESTIONS OF PHYSICAL INTERPRETATIONS:
> ># second analytic solution cases r=2*MG and
> ># INSIDE Black Hole 0<r<= 2*MG
> ># This is copy part of my Maple 9 program
> ># where other symbols have their numerical values H.P. 15.9.2020
> > ### total coordinate velocity v^2=(dr/dt)^2+r^2*(dPhi/dt)^2
> > #sqrt((1-2*MG/r)^2*(1-(E+J^2/r^2)*(1-2*MG/r))+(J^2/r^2)*(1-2*MG/r)^2)
> > # total coordinate velocity at perihelion
> > sqrt((1-2*MG/x)^2*(1-(E+J^2/x^2)*(1-2*MG/x))+(J^2/x^2)*(1-2*MG/x)^2);
>
> > # 0.2581119858e-1
> > # km/s
> > 0.2997924580e11*0.2581119858e-1/10^5;
>
> > # 7738.002666
> > # total coordinate velocity at aphelion
> > sqrt((1-2*MG/y)^2*(1-(E+J^2/y^2)*(1-2*MG/y))+(J^2/y^2)*(1-2*MG/y)^2);
>
> > # 0.1581028031e-2
> > # km/s
> > 0.2997924580e11*0.1581028031e-2/10^5;
>
> > # 473.9802796
> > ###
> > ### total proper velocity v^2=(dr/dtau)^2+r^2*(dPhi/dtau)^2
> > #sqrt((1/E)*(1-(E+J^2/r^2)*(1-2*MG/r))+J^2/r^2)
> > # total proper velocity at perihelion
> > sqrt((1/E)*(1-(E+J^2/x^2)*(1-2*MG/x))+J^2/x^2);
>
> > # 0.2582947199e-1
> > # km/s
> > 0.2997924580e11*0.2582947199e-1/10^5;
>
> > # 7743.480897
> > # total proper velocity at aphelion
> > sqrt((1/E)*(1-(E+J^2/y^2)*(1-2*MG/y))+J^2/y^2);
>
> > # 0.1581096486e-2
> > # km/s
> > 0.2997924580e11*0.1581096486e-2/10^5;
>
> > # 474.0008019
>
> >###
> > ### Total proper acceleration (- + sign d^2Phi/dta^2)(+ - sign dr/da)
> > ### A^2=(d^2r/dta^2-r*(dPhi/dta)^2)^2+(r*d^2Phi/dta^2+2*(dr/dta)*(dPhi/dta))^2
> > #sqrt(((J^2/(sqrt(E)*r^3))*(1-2*MG/r)-(MG/(sqrt(E)*r^2*(1-2*MG/r)))+(MG/(sqrt(E)*r^2))*(1-(E+J^2/r^2)*(1-2*MG/r))-r*(J/r^2)^2)^2+(r*((2*J/(r^3))*sqrt(1-(E+J^2/r^2)*(1-2*MG/r)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/r^2)*(1-2*MG/r)))*(J/r^2))^2)
> >
> > ### total proper acceleration at perihelion (+ sign selected for both)
> > # total proper acceleration at perihelion
> > sqrt(((J^2/(sqrt(E)*x^3))*(1-2*MG/x)-(MG/(sqrt(E)*x^2*(1-2*MG/x)))+(MG/(sqrt(E)*x^2))*(1-(E+J^2/x^2)*(1-2*MG/x))-x*(J/x^2)^2)^2+(x*((2*J/(x^3))*sqrt(1-(E+J^2/x^2)*(1-2*MG/x)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/x^2)*(1-2*MG/x)))*(J/x^2))^2);
>
> > # 0.1992765885e-18
> > # cm/s^2
> > (2.99792458*10^8*10^2)^2*0.1992765885e-18;
>
> > # 179.1008659
> > ### total proper acceleration at aphelion (+ sign selected for both)
> > # total proper acceleration at aphelion
> > sqrt(((J^2/(sqrt(E)*y^3))*(1-2*MG/y)-(MG/(sqrt(E)*y^2*(1-2*MG/y)))+(MG/(sqrt(E)*y^2))*(1-(E+J^2/y^2)*(1-2*MG/y))-y*(J/y^2)^2)^2+(y*((2*J/(y^3))*sqrt(1-(E+J^2/y^2)*(1-2*MG/y)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/y^2)*(1-2*MG/y)))*(J/y^2))^2);
>
> > # 0.7450050462e-21
> > # cm/s^2
> > (2.99792458*10^8*10^2)^2*0.7450050462e-21;
>
> > # 0.6695771434
> > ####
> > ### Total coordinate acceleration (- + sign d^2Phi/dt^2)(+ - sign dr/dt)
> > ### A^2=(d^2r/dt^2-r*(dPhi/dt)^2)^2+(r*d^2Phi/dt^2+2*(dr/dt)*(dPhi/dt))^2
> > # First component of the sqrt formula
> > #((J^2/r^3)*(1-2*MG/r)^2-MG/r^2+MG*( (1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3 )^2/(r^2*(1-2*MG/r))-r*((J/r^2)*(1-2*MG/r))^2)^2:
> > # Second component of the sqrt formula
> > #(r*((-2/(r*(1-2*MG/r)))*((J/r^2)*(1-2*MG/r))*((1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3))-2*sqrt((1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3)*((J/r^2)*(1-2*MG/r)))^2:
> > #sqrt(((J^2/r^3)*(1-2*MG/r)^2-MG/r^2+MG*( (1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3 )^2/(r^2*(1-2*MG/r))-r*((J/r^2)*(1-2*MG/r))^2)^2+(r*((-2/(r*(1-2*MG/r)))*((J/r^2)*(1-2*MG/r))*((1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3))-2*sqrt((1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3)*((J/r^2)*(1-2*MG/r)))^2):
>
> > # total coordinate acceleration at perihelion
> > sqrt(((J^2/x^3)*(1-2*MG/x)^2-MG/x^2+MG*( (1-2*MG/x)^2-(E+J^2/x^2)*(1-2*MG/x)^3 )^2/(x^2*(1-2*MG/x))-x*((J/x^2)*(1-2*MG/x))^2)^2+(x*((-2/(x*(1-2*MG/x)))*((J/x^2)*(1-2*MG/x))*((1-2*MG/x)^2-(E+J^2/x^2)*(1-2*MG/x)^3))-2*sqrt((1-2*MG/x)^2-(E+J^2/x^2)*(1-2*MG/x)^3)*((J/x^2)*(1-2*MG/x)))^2);
>
> > # 0.1988634824e-18
> > # cm/s^2
> > (2.99792458*10^8*10^2)^2*0.1988634824e-18;
>
> > # 178.7295847
> > # total coordinate acceleration at aphelion
> > sqrt(((J^2/y^3)*(1-2*MG/y)^2-MG/y^2+MG*( (1-2*MG/y)^2-(E+J^2/y^2)*(1-2*MG/y)^3 )^2/(y^2*(1-2*MG/y))-x*((J/y^2)*(1-2*MG/y))^2)^2+(y*((-2/(y*(1-2*MG/y)))*((J/y^2)*(1-2*MG/y))*((1-2*MG/y)^2-(E+J^2/y^2)*(1-2*MG/y)^3))-2*sqrt((1-2*MG/y)^2-(E+J^2/y^2)*(1-2*MG/y)^3)*((J/y^2)*(1-2*MG/y)))^2);
>
> > # 0.6642424264e-21
> > # cm/s^2
> > (2.99792458*10^8*10^2)^2*0.6642424264e-21;
>
> > # 0.5969913206
> > #####################################################
> > ####
> > K := 2*MG;
>
> > # K := 0.1258418194e13
> > # K = 2*MG = 0.1258418194e13 cm EVENT HORIZON of SgrA* Black Hole
>
> > # total coordinate acceleration at event horizon K=2*MG(+ sign selected for both)
> > sqrt(((J^2/K^3)*(1-2*MG/K)^2-MG/K^2+MG*( (1-2*MG/K)^2-(E+J^2/K^2)*(1-2*MG/K)^3 )^2/(K^2*(1-2*MG/K))-K*((J/K^2)*(1-2*MG/K))^2)^2+(K*((-2/(K*(1-2*MG/K)))*((J/K^2)*(1-2*MG/K))*((1-2*MG/K)^2-(E+J^2/K^2)*(1-2*MG/K)^3))-2*sqrt((1-2*MG/K)^2-(E+J^2/K^2)*(1-2*MG/K)^3)*((J/K^2)*(1-2*MG/K)))^2);
>
> > # 0.3973241981e-12
> > # cm/s^2
> > (2.99792458*10^8*10^2)^2*0.3973241981e-12;
>
> > # 357097180.7
> > ### total proper acceleration at event horizon K = 2*MG(+ sign selected for both)
> > # ONE NOTICE: pure imaginary number acceleration for example at 3*MG, this is due 3*MG is over definition area of the second analytic solution.
> > # Different selections of +, - signs (- + sign d^2Phi/dta^2)(+ - sign dr/da) are not treated here, + sign selections is made for both cases.
> > ####
> > # K = 2*MG = 0.1258418194e13 EVENT HORIZON of SgrA* Black Hole
> > K := 2*MG;
>
> > # K := 0.1258418194e13
> > ### total proper acceleration
> > sqrt(((J^2/(sqrt(E)*K^3))*(1-2*MG/K)-(MG/(sqrt(E)*K^2*(1-2*MG/K)))+(MG/(sqrt(E)*K^2))*(1-(E+J^2/K^2)*(1-2*MG/K))-K*(J/K^2)^2)^2+(K*((2*J/(K^3))*sqrt(1-(E+J^2/K^2)*(1-2*MG/K)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/K^2)*(1-2*MG/K)))*(J/K^2))^2);
>
> > # 0.9932912900e-3
> > # cm/s^2
> > (2.99792458*10^8*10^2)^2*0.9932912900e-3;
>
> > # 0.8927256908e18
> > ### total coordinate velocity
> > sqrt((1-2*MG/K)^2*(1-(E+J^2/K^2)*(1-2*MG/K))+(J^2/K^2)*(1-2*MG/K)^2);
>
> > # 0.1460945827e-7
> > # cm/s
> > 2.99792458*10^8*10^2*0.1460945827e-7;
>
> > # 437.9805405
> > ### total proper velocity
> > sqrt((1/E)*(1-(E+J^2/K^2)*(1-2*MG/K))+J^2/K^2);
>
> > # 36.52364512
> > # cm/s
> > 2.99792458*10^8*10^2*36.52364512;
>
> > # 0.1094951335e13
> > ####
> > ##########################################################
> > ###### Inside SgrA* Black Hole if S2 follows somehow the second analytic solution
> > z := MG;
>
> > # z := 0.6292090968e12
> > ### total coordinate velocity
> > sqrt((1-2*MG/z)^2*(1-(E+J^2/z^2)*(1-2*MG/z))+(J^2/z^2)*(1-2*MG/z)^2);
>
> > # 103.2754259
> > # cm/s
> > 2.99792458*10^8*10^2*103.2754259;
>
> > # 0.3096119378e13
> > ### total proper velocity
> > sqrt((1/E)*(1-(E+J^2/z^2)*(1-2*MG/z))+J^2/z^2);
>
> > # 103.2743723
> > # cm/s
> > 2.99792458*10^8*10^2*103.2743723;
>
> > # 0.3096087792e13
> > ### total proper acceleration
> > sqrt(((J^2/(sqrt(E)*z^3))*(1-2*MG/z)-(MG/(sqrt(E)*z^2*(1-2*MG/z)))+(MG/(sqrt(E)*z^2))*(1-(E+J^2/z^2)*(1-2*MG/z))-z*(J/z^2)^2)^2+(z*((2*J/(z^3))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))*(J/z^2))^2);
>
> > # 0.3494379988e-7
> > # cm/s^2
> > (2.99792458*10^8*10^2)^2*0.3494379988e-7;
>
> > # 0.3140592111e14
> > ### total coordinate acceleration
> > sqrt(((J^2/(sqrt(E)*z^3))*(1-2*MG/z)-(MG/(sqrt(E)*z^2*(1-2*MG/z)))+(MG/(sqrt(E)*z^2))*(1-(E+J^2/z^2)*(1-2*MG/z))-K*(J/z^2)^2)^2+(z*((2*J/(z^3))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))*(J/z^2))^2);
>
> > # 0.3790004584e-7
> > # cm/s^2
> > (2.99792458*10^8*10^2)^2*0.3790004584e-7;
>
> > # 0.3406286247e14
> Best Regards,
> Hannu Poropudas
> Finland


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Re: Analytic GR solutions of S2-star orbits and precession 732” per revolution

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Subject: Re:_Analytic_GR_solutions_of_S2-star_orbits_and_prec
ession_732”_per_revolution
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 by: Hannu Poropudas - Thu, 24 Nov 2022 07:19 UTC

keskiviikko 16. marraskuuta 2022 klo 10.58.55 UTC+2 Hannu Poropudas kirjoitti:
> torstai 17. syyskuuta 2020 klo 10.05.28 UTC+3 hanp...@luukku.com kirjoitti:
> > On Monday, September 14, 2020 at 2:05:10 PM UTC+3, Hannu Poropudas wrote:
> > > On Sunday, September 13, 2020 at 12:34:54 PM UTC+3, Hannu Poropudas wrote:
> > > > On Thursday, September 10, 2020 at 10:10:49 AM UTC+3, Hannu Poropudas wrote:
> > > > > Analytic GR solutions of S2-star orbits and precession 732” per revolution
> > > > >
> > > > > New initial data 2020 is used. My earlier initial data was 2016 in my
> > > > > postings in this sci.physics.relativity.
> > > > >
> > > > > Please also take a look my two postings about OJ287 in sci.astro. I have got
> > > > > no comments there due so heavy traffic of unbssinesslike postings from many posters.
> > > > >
> > > > > (Only command lines of Maple 9 program is below and command line mark is >. This can be copy-pasted in Maple 9 program and run it.
> > > > > Remark that for example e-2 below means 10^(-2)
> > > > > or e2 below means 10^2.)
> > > > >
> > > > > Best Regards,
> > > > >
> > > > > Hannu Poropudas
> > > > > Finland
> > > > >
> > > > > ---COPY of the Maple 9 program below------
> > > > >
> > > > > > # RIGHT: Particle in Schwarzschild metrics H.P. 09.09.2020
> > > > > > # S2 Orbit around Sagittarius A*
> > > > > > # Perihelion precession of S2.
> > > > > > #OK=COMPARISION to Post-Newtonian approx of Abuter R et al 2020, A&A 636,L5(2020)
> > > > > > # Abuter R. et al. 2020. Detection of the Schwarzschild precession in the
> > > > > > # orbit of the star S2 near the Galactic centre massive black hole.
> > > > > > # Astronomy Astrophysics, 636, L5 (2020).
> > > > > > # In this article: 12'/rev=720"/rev,12.1'/rev=726"/rev, 12.3'/rev=738"/rev
> > > > > > # My GR calculation below, result : 731.8543470"/rev OR -676.7436912"/rev
> > > > > > # (I use not rounded numbers). Approximately 732"/rev OR -677"/rev.
> > > > > > # New starting values from Table E.1 Best-fit Orbit Parameters above article
> > > > > > # and other constants of physics are taken from Wikipedia date 9.9.2020.
> > > > > > # NEW CALCULATION x and y are roots. E and J little changed.
> > > > > > # (analytic solutions defined between roots a2..a1 OR a4..a3)
> > > > > >Restart;
> > > > > >with(plots):with(plottools):
> > > > > > # GEOMETRIZED UNIT SYSTEM IS PROBLEMATIC.
> > > > > > # Ref. Formulae for E and J from Weinberg's book. (THESE FORMULAE ARE NOT USED.)
> > > > > > # Reference: Weinberg, S. 1972. Gravitation and Cosmology: Principles
> > > > > > # and Applications of the General Relativity. Wiley, New York. pp. 179-210.
> > > > > > # (units c = 1, and c.g.s in Weinberg's book).
> > > > > > # J = r^2*dPhi/dp = constant (angular momentum per unit mass), p. 186.
> > > > > > # E = constant (energy per unit mass), p. 186.
> > > > > > # E > 0 for material particles, E = 0 for photons. p. 186.
> > > > > > # Integration limits must be determined from the problem to which apply these.
> > > > > >#Int(1/(r^4*(1/J^2-E/J^2)+r^3*(2*M*G*E/J^2)-r^2+2*M*G*r)^(1/2),r);
> > > > > >AU := 149597870700*10^2;
> > > > > ># 125.058 mas, 8246.7 pc, 1 pc =3.26 ly
> > > > > >ap:= 1030.799324*AU;
> > > > > >em := 0.884649;
> > > > > >M := 4.261*10^6*1.98847*10^30*10^3;
> > > > > ># v = c
> > > > > >v := 2.99792458*10^8*10^2;
> > > > > >G := 6.6743015*10^(-11)*10^3;
> > > > > ># BH mass*G geometric units (cm) and v = c.
> > > > > >M*G/v^2;
> > > > > >MG := 0.6292090968e12;
> > > > > > # EXAMPLE. Weinberg formulae E and J (THESE FORMULAE ARE NOT ANY MORE USED.)
> > > > > > # Perihelion distance
> > > > > >x := ap*(1-em);
> > > > > ># Aphelion distance
> > > > > >y := ap*(1+em);
> > > > > > # NEW CALCULATION (NEW FORMULAE OF MY OWN) x and y are roots
> > > > > > #***********************************************************
> > > > > >E := (2*x*y*MG+2*y^2*MG+2*x^2*MG-x^2*y-x*y^2)/((-y+2*MG)*(-2*x*MG-2*y*MG+x^2+x*y));
> > > > > >J := 2^(1/2)*(-(-y+2*MG)*MG*(-2*x*MG-2*y*MG+x^2+x*y))^(1/2)*y*x/(-x^2*y+2*x^2*MG-4*x*MG^2+4*x*y*MG-x*y^2+2*y^2*MG-4*y*MG^2);
> > > > > > #*****************************************************************
> > > > > >
> > > > > > # Weinberg's formula (NOT USED ANY MORE)
> > > > > > #J := sqrt((1/(1-2*MG/y)-(1/(1-2*MG/x)))/(1/y^2-1/x^2));
> > > > > > # NEW CALCULATION x and y are roots
> > > > > >J := -0.4594478956e14;
> > > > > > # Weinberg's formula (NOT USED ANY MORE)
> > > > > > #E := (y^2/(1-2*MG/y)-x^2/(1-2*MG/x))/(y^2-x^2);
> > > > > > # NEW CALCULATION x and y are roots
> > > > > >E := 1.000040803;
> > > > > >solve(r^4*(1/J^2-E/J^2)+r^3*(2*MG*E/J^2)-r^2+2*MG*r=0,r);
> > > > > >a1 := 0.2906262559e17;
> > > > > >a2 := 0.1778773310e16;
> > > > > >a3 := 0.1259363679e13;
> > > > > >a4 := 0;
> > > > >
> > > > > ># Primitive function, definition areas 0<=P<=Pi/2 and a2 <= r <= a1.
> > > > >
> > > > > >e := P -> 2.000794538*EllipticF(sin(P),0.2578161452e-1);
> > > > > >r := P -> (0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17);
> > > > >
> > > > > > #******************************
> > > > > > # P which corresponds to the perihelion distance (TWO VALUES)
> > > > > >solve((0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17)=0.1778774525e16,P);
> > > > > >e(0.8532953281e-3);
> > > > > >e(-0.8532953281e-3);
> > > > > ># Solution's definition area limit is a2-root (TWO VALUES)
> > > > > >solve((0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17)=0.1778773310e16,P);
> > > > > >e(0.6516324491e-5);
> > > > > >e(-0.6516324491e-5);
> > > > > ># P which corresponds to the aphelion distance (TWO VALUES)
> > > > > >solve((0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17)=0.2906230228e17,P);
> > > > > >e(1.569945054);
> > > > > >e(-1.569945054);
> > > > > > # Solution's correct definition limit is a1-root (TWO VALUES)
> > > > > > # COMPLEX NUMBER (SMALL COMPLEX PARTS ARE LITTLE OUTSIDE ORBIT)
> > > > > >solve((0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17)=0.2906262559e17,P);
> > > > > >e(1.570796327-0.1136598102e-4*I);
> > > > > >e(-1.570796327+0.1136598102e-4*I);
> > > > >
> > > > > > # Angle change, definition areas 0<=P<=Pi/2 ja a2 <= r <= a1.
> > > > > > # + sign perihelion in last term
> > > > >
> > > > > >e2 := P -> 2*abs(2.000794538*EllipticF(sin(P),0.2578161452e-1)-2.000794538*EllipticF(sin(0.8532953281e-3),0.2578161452e-1))-2*Pi;
> > > > > >r2 := P -> (0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17);
> > > > >
> > > > > > # Calculation of different combinations of + and - signs
> > > > > > # - sign perihelion in last term
> > > > >
> > > > > >e2B := P -> 2*abs(2.000794538*EllipticF(sin(P),0.2578161452e-1)-2.000794538*EllipticF(sin(-0.8532953281e-3),0.2578161452e-1))-2*Pi;
> > > > > >r2B := P -> (0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17);
> > > > >
> > > > > > # Angle change in radians per one revolution. (TWO e2 and e2B functions)
> > > > > > # FOR e2
> > > > > >evalf(e2(1.569945054));
> > > > > ># Angle change in degrees per one revolution.
> > > > > >evalf(-0.3280946e-2*180/Pi);
> > > > > ># Angle change in arc seconds per one revolution.
> > > > > >-0.1879843586*60*60;
> > > > > >###(-676.7436912)
> > > > > ># Second root
> > > > > >evalf(e2(-1.569945054));
> > > > > >evalf(0.3548130e-2*180/Pi);
> > > > > >0.2032928741*60*60;
> > > > > >###(731.8543470)
> > > > > ># FOR e2B
> > > > > >evalf(e2B(1.569945054));
> > > > > ># Angle change in degrees per one revolution.
> > > > > >evalf(0.3548130e-2*180/Pi);
> > > > > ># Angle change in arc seconds per one revolution.
> > > > > >0.2032928741*60*60;
> > > > > >###(731.8543470)
> > > > > ># Second root
> > > > > >evalf(e2B(-1.569945054));
> > > > > >evalf(-0.3280946e-2*180/Pi);
> > > > > >-0.1879843586*60*60;
> > > > > >####(-676.7436912)
> > > > > > # + and - signs for e2 or e2B does not have different results due abs-values
> > > > > > # Second Primitive function, definition areas 0<=P<=Pi/2 and a4=0 <= r <= a3.
> > > > > > # This has form of spiral (+ and - signs) which leads to the origin.
> > > > > >
> > > > > > # SUMMARY of S2 precession: 731.8543470"/revolution OR -676.7436912"/revolution.
> > > > > >
> > > > > > #******************************
> > > > >
> > > > > ># Second Primitive function, definition areas 0<=P<=Pi/2 and a4=0 <=
> > > > > r <= a3.
> > > > >
> > > > > >ee := P-> 2.000794538*EllipticF(sin(P),0.9996675989);
> > > > > >rr := P-> (0.3660041508e29*sin(P)^2)/(0.1259363679e13*sin(P)^2+0.2906136623e17);
> > > > >
> > > > > > #*********************************************************
> > > > > >
> > > > > > # Plottings only on definition area a2..a1.
> > > > >
> > > > > > # First side of the solution (+,- solution)
> > > > > >plot([r(P),e(P),P=0..Pi/2],coords=polar);
> > > > >
> > > > > ># Second side of the solution (+,- solution)
> > > > > >plot([r(P),-e(P),P=0..Pi/2],coords=polar);
> > > > >
> > > > > ># Angle change picture has no other meaning than above calculated precession
> > > > > >plot([r2(P),e2(P),P=0..Pi/2],coords=polar);
> > > > >
> > > > > ># Angle change picture has no other meaning than above calculated precession
> > > > > >plot([r2B(P),e2B(P),P=0..Pi/2],coords=polar)
> > > > >
> > > > > > # Plottings only on definition area a4..a3.
> > > > >
> > > > > > # First side of the Second solution (+,- solution)
> > > > > >plot([rr(P),ee(P),P=0..Pi/2],coords=polar);
> > > > >
> > > > > ># Second side of the Second solution (+,- solution)
> > > > > >plot([rr(P),-ee(P),P=0..Pi/2],coords=polar);
> > > > *** 1. CORRECTION: Formula of J has + and - signs. Positive sign should be selected.
> > > > Previous - sign selection does not influence due J^2 was always used.
> > > >
> > > > J := 0.4594478956e14
> > > >
> > > > *** 2. -Pi/2 .. Pi/2 plots gives both sides of both solutions.
> > > > This is due
> > > > sin(-P)=-sin(P)
> > > > and
> > > > EllipticF(sin(-P),q)=EllipticF(-sin(P),q)=-EllipticF(sin(P),q).
> > > >
> > > > 3. I calculaled below total coordinate velocity, proper velocity and
> > > > total proper acceleration for S2-star:
> > > > (These are used only in definition areas of above analytic solutions
> > > > a2<=r<=a1 or a4<r<=a3,
> > > > a1 := 0.2906262559e17,
> > > > a2 := 0.1778773310e16,
> > > > a3 := 0.1259363679e13,
> > > > a4 := 0)
> > > >
> > > > (REMARK: > is command line mark of Maple 9 program)
> > > > ># Total coordinate velocity and total proper velocity S2-star
> > > > ># at perihelion and at aphelion HP 12092020
> > > >
> > > > > ### total coordinate velocity v^2=(dr/dt)^2+r^2*(dPhi/dt)^2
> > > > > sqrt((1-2*MG/r)^2*(1-(E+J^2/r^2)*(1-2*MG/r))+(J^2/r^2)*(1-2*MG/r)^2)
> > > > > sqrt((1-2*MG/x)^2*(1-(E+J^2/x^2)*(1-2*MG/x))+(J^2/x^2)*(1-2*MG/x)^2);
> > > > 0.2581119858e-1
> > > > > # km/s
> > > > > 0.2997924580e11*0.2581119858e-1/10^5;
> > > > 7738.002666
> > > > > sqrt((1-2*MG/y)^2*(1-(E+J^2/y^2)*(1-2*MG/y))+(J^2/y^2)*(1-2*MG/y)^2);
> > > > 0.1581028031e-2
> > > > > # km/s
> > > > > 0.2997924580e11*0.1581028031e-2/10^5;
> > > > 473.9802796
> > > > > ###
> > > > > ### total proper velocity v^2=(dr/dtau)^2+r^2*(dPhi/dtau)^2
> > > > > sqrt((1/E)*(1-(E+J^2/r^2)*(1-2*MG/r))+J^2/r^2)
> > > > > sqrt((1/E)*(1-(E+J^2/x^2)*(1-2*MG/x))+J^2/x^2);
> > > > 0.2582947199e-1
> > > > > # km/s
> > > > > 0.2997924580e11*0.2582947199e-1/10^5;
> > > > 7743.480897
> > > > > sqrt((1/E)*(1-(E+J^2/y^2)*(1-2*MG/y))+J^2/y^2);
> > > > 0.001581096486
> > > > > # km/s
> > > > > 0.2997924580e11*0.1581096486e-2/10^5;
> > > > 474.0008019
> > > > > ###
> > > >
> > > > ># Total proper acceleration S2 star HP 13092020
> > > >
> > > > > ### Total proper acceleration (- + sign d^2Phi/dta^2)(+ - sign dr/da)
> > > >
> > > > > ### A^2=(d^2r/dta^2-r*(dPhi/dta)^2)^2+(r*d^2Phi/dta^2+2*(dr/dta)*(dPhi/dta))^2
> > > >
> > > > > #sqrt(((J^2/(sqrt(E)*r^3))*(1-2*MG/r)-(MG/(sqrt(E)*r^2*(1-2*MG/r)))+(MG/(sqrt(E)*r^2))*(1-(E+J^2/r^2)*(1-2*MG/r))-r*(J/r^2)^2)^2+(r*((2*J/(r^3))*sqrt(1-(E+J^2/r^2)*(1-2*MG/r)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/r^2)*(1-2*MG/r)))*(J/r^2))^2)
> > > >
> > > > > ### total proper acceleration at perihelion (+ sign selected for both)
> > > >
> > > > > sqrt(((J^2/(sqrt(E)*x^3))*(1-2*MG/x)-(MG/(sqrt(E)*x^2*(1-2*MG/x)))+(MG/(sqrt(E)*x^2))*(1-(E+J^2/x^2)*(1-2*MG/x))-x*(J/x^2)^2)^2+(x*((2*J/(x^3))*sqrt(1-(E+J^2/x^2)*(1-2*MG/x)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/x^2)*(1-2*MG/x)))*(J/x^2))^2);
> > > > 1.992765885*10^(-19)
> > > > > # cm/s^2
> > > > > (2.99792458*10^8*10^2)^2*0.1992765885e-18;
> > > > 179.1008659
> > > > > ### total proper acceleration at aphelion (+ sign selected for both)
> > > >
> > > > > sqrt(((J^2/(sqrt(E)*y^3))*(1-2*MG/y)-(MG/(sqrt(E)*y^2*(1-2*MG/y)))+(MG/(sqrt(E)*y^2))*(1-(E+J^2/y^2)*(1-2*MG/y))-y*(J/y^2)^2)^2+(y*((2*J/(y^3))*sqrt(1-(E+J^2/y^2)*(1-2*MG/y)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/y^2)*(1-2*MG/y)))*(J/y^2))^2);
> > > > 7.450050462*10^(-22)
> > > > > # cm/s^2
> > > > > (2.99792458*10^8*10^2)^2*0.7450050462e-21;
> > > > 0.6695771434
> > > > > ### total proper acceleration at event horizon z = MG(+ sign selected for both)
> > > >
> > > > > # ONE NOTICE: pure imaginary number acceleration for example at 3*MG , this is due 3*MG is over definition area of the second analytic solution.
> > > > > # Different selections of +, - signs (- + sign d^2Phi/dta^2)(+ - sign dr/da) are not treated here , + sign selections is made for both cases..
> > > >
> > > > > z := MG;
> > > > 6.292090968*10^11
> > > > > sqrt(((J^2/(sqrt(E)*z^3))*(1-2*MG/z)-(MG/(sqrt(E)*z^2*(1-2*MG/z)))+(MG/(sqrt(E)*z^2))*(1-(E+J^2/z^2)*(1-2*MG/z))-z*(J/z^2)^2)^2+(z*((2*J/(z^3))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))*(J/z^2))^2);
> > > > 3.494379988*10^(-8)
> > > > > # cm/s^2
> > > > > (2.99792458*10^8*10^2)^2*0.3494379988e-7;
> > > > 3.140592111*10^13
> > > > > ###
> > > > Best Regards,
> > > > Hannu Poropudas
> > > > Finland
> > > CORRECTION (event horizon is was not MG ):
> > > Interesting OPEN QUESTIONS would be if S2-star somehow happens to go to my second analytic solution orbit,
> > > which definition area is 0 < r <= 1.259363679*10^12 cm.
> > > At r = 2*MG = = 1.258418194*10^12 cm, event horizon of the SgrA* black hole seems to
> > > have following (OPEN QUESTION OF INTERPRETATION of the total proper velocity and the total proper acceleration):
> > >
> > > total coordinate velocity = 1.460945827*10^(-8) , (geometric units, Weinberg 1972),
> > > total coordinate velocity = 437.9805405 cm/s, (c.g.s units, Weinberg 1972),
> > > total proper velocity = 36.52364512 (geometric units, Weinberg 1972),
> > > total proper velocity = 1.094951335*10^12, cm/s (c.g.s units, Weinberg 1972)
> > > and
> > > total proper acceleration = 8,927256908*10^17 cm/s^2 , (c.g.s units, Weinberg 1972)
> > >
> > > I have used definitions of Weinberg S. 1972. Gravitation and Cosmology book
> > > and Becker 1954 Introduction to Theoretical Mechanics book
> > > when I calculated above total coordinate velocity, total proper velocity and total proper acceleration.
> > > (Formulae of these GR calculations are in above posting of mine).
> > >
> > > Best Regards,,
> > > Hannu Poropudas
> > > Finland
> > Preliminary formulae and few important numerical points calculated below:
> >
> > ># S2-star’s two analytic solutions of orbits around SgrA* Black Hole
> > ># Total coordinate velocity, total proper velocity and
> > ># Total coordinate acceleration, total proper acceleration
> > ># formulae and numerical calculation for some points.
> > ># OPEN QUESTIONS OF PHYSICAL INTERPRETATIONS:
> > ># second analytic solution cases r=2*MG and
> > ># INSIDE Black Hole 0<r<= 2*MG
> > ># This is copy part of my Maple 9 program
> > ># where other symbols have their numerical values H.P. 15.9.2020
> > > ### total coordinate velocity v^2=(dr/dt)^2+r^2*(dPhi/dt)^2
> > > #sqrt((1-2*MG/r)^2*(1-(E+J^2/r^2)*(1-2*MG/r))+(J^2/r^2)*(1-2*MG/r)^2)
> > > # total coordinate velocity at perihelion
> > > sqrt((1-2*MG/x)^2*(1-(E+J^2/x^2)*(1-2*MG/x))+(J^2/x^2)*(1-2*MG/x)^2);
> >
> > > # 0.2581119858e-1
> > > # km/s
> > > 0.2997924580e11*0.2581119858e-1/10^5;
> >
> > > # 7738.002666
> > > # total coordinate velocity at aphelion
> > > sqrt((1-2*MG/y)^2*(1-(E+J^2/y^2)*(1-2*MG/y))+(J^2/y^2)*(1-2*MG/y)^2);
> >
> > > # 0.1581028031e-2
> > > # km/s
> > > 0.2997924580e11*0.1581028031e-2/10^5;
> >
> > > # 473.9802796
> > > ###
> > > ### total proper velocity v^2=(dr/dtau)^2+r^2*(dPhi/dtau)^2
> > > #sqrt((1/E)*(1-(E+J^2/r^2)*(1-2*MG/r))+J^2/r^2)
> > > # total proper velocity at perihelion
> > > sqrt((1/E)*(1-(E+J^2/x^2)*(1-2*MG/x))+J^2/x^2);
> >
> > > # 0.2582947199e-1
> > > # km/s
> > > 0.2997924580e11*0.2582947199e-1/10^5;
> >
> > > # 7743.480897
> > > # total proper velocity at aphelion
> > > sqrt((1/E)*(1-(E+J^2/y^2)*(1-2*MG/y))+J^2/y^2);
> >
> > > # 0.1581096486e-2
> > > # km/s
> > > 0.2997924580e11*0.1581096486e-2/10^5;
> >
> > > # 474.0008019
> >
> > >###
> > > ### Total proper acceleration (- + sign d^2Phi/dta^2)(+ - sign dr/da)
> > > ### A^2=(d^2r/dta^2-r*(dPhi/dta)^2)^2+(r*d^2Phi/dta^2+2*(dr/dta)*(dPhi/dta))^2
> > > #sqrt(((J^2/(sqrt(E)*r^3))*(1-2*MG/r)-(MG/(sqrt(E)*r^2*(1-2*MG/r)))+(MG/(sqrt(E)*r^2))*(1-(E+J^2/r^2)*(1-2*MG/r))-r*(J/r^2)^2)^2+(r*((2*J/(r^3))*sqrt(1-(E+J^2/r^2)*(1-2*MG/r)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/r^2)*(1-2*MG/r)))*(J/r^2))^2)
> > >
> > > ### total proper acceleration at perihelion (+ sign selected for both)
> > > # total proper acceleration at perihelion
> > > sqrt(((J^2/(sqrt(E)*x^3))*(1-2*MG/x)-(MG/(sqrt(E)*x^2*(1-2*MG/x)))+(MG/(sqrt(E)*x^2))*(1-(E+J^2/x^2)*(1-2*MG/x))-x*(J/x^2)^2)^2+(x*((2*J/(x^3))*sqrt(1-(E+J^2/x^2)*(1-2*MG/x)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/x^2)*(1-2*MG/x)))*(J/x^2))^2);
> >
> > > # 0.1992765885e-18
> > > # cm/s^2
> > > (2.99792458*10^8*10^2)^2*0.1992765885e-18;
> >
> > > # 179.1008659
> > > ### total proper acceleration at aphelion (+ sign selected for both)
> > > # total proper acceleration at aphelion
> > > sqrt(((J^2/(sqrt(E)*y^3))*(1-2*MG/y)-(MG/(sqrt(E)*y^2*(1-2*MG/y)))+(MG/(sqrt(E)*y^2))*(1-(E+J^2/y^2)*(1-2*MG/y))-y*(J/y^2)^2)^2+(y*((2*J/(y^3))*sqrt(1-(E+J^2/y^2)*(1-2*MG/y)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/y^2)*(1-2*MG/y)))*(J/y^2))^2);
> >
> > > # 0.7450050462e-21
> > > # cm/s^2
> > > (2.99792458*10^8*10^2)^2*0.7450050462e-21;
> >
> > > # 0.6695771434
> > > ####
> > > ### Total coordinate acceleration (- + sign d^2Phi/dt^2)(+ - sign dr/dt)
> > > ### A^2=(d^2r/dt^2-r*(dPhi/dt)^2)^2+(r*d^2Phi/dt^2+2*(dr/dt)*(dPhi/dt))^2
> > > # First component of the sqrt formula
> > > #((J^2/r^3)*(1-2*MG/r)^2-MG/r^2+MG*( (1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3 )^2/(r^2*(1-2*MG/r))-r*((J/r^2)*(1-2*MG/r))^2)^2:
> > > # Second component of the sqrt formula
> > > #(r*((-2/(r*(1-2*MG/r)))*((J/r^2)*(1-2*MG/r))*((1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3))-2*sqrt((1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3)*((J/r^2)*(1-2*MG/r)))^2:
> > > #sqrt(((J^2/r^3)*(1-2*MG/r)^2-MG/r^2+MG*( (1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3 )^2/(r^2*(1-2*MG/r))-r*((J/r^2)*(1-2*MG/r))^2)^2+(r*((-2/(r*(1-2*MG/r)))*((J/r^2)*(1-2*MG/r))*((1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3))-2*sqrt((1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3)*((J/r^2)*(1-2*MG/r)))^2):
> >
> > > # total coordinate acceleration at perihelion
> > > sqrt(((J^2/x^3)*(1-2*MG/x)^2-MG/x^2+MG*( (1-2*MG/x)^2-(E+J^2/x^2)*(1-2*MG/x)^3 )^2/(x^2*(1-2*MG/x))-x*((J/x^2)*(1-2*MG/x))^2)^2+(x*((-2/(x*(1-2*MG/x)))*((J/x^2)*(1-2*MG/x))*((1-2*MG/x)^2-(E+J^2/x^2)*(1-2*MG/x)^3))-2*sqrt((1-2*MG/x)^2-(E+J^2/x^2)*(1-2*MG/x)^3)*((J/x^2)*(1-2*MG/x)))^2);
> >
> > > # 0.1988634824e-18
> > > # cm/s^2
> > > (2.99792458*10^8*10^2)^2*0.1988634824e-18;
> >
> > > # 178.7295847
> > > # total coordinate acceleration at aphelion
> > > sqrt(((J^2/y^3)*(1-2*MG/y)^2-MG/y^2+MG*( (1-2*MG/y)^2-(E+J^2/y^2)*(1-2*MG/y)^3 )^2/(y^2*(1-2*MG/y))-x*((J/y^2)*(1-2*MG/y))^2)^2+(y*((-2/(y*(1-2*MG/y)))*((J/y^2)*(1-2*MG/y))*((1-2*MG/y)^2-(E+J^2/y^2)*(1-2*MG/y)^3))-2*sqrt((1-2*MG/y)^2-(E+J^2/y^2)*(1-2*MG/y)^3)*((J/y^2)*(1-2*MG/y)))^2);
> >
> > > # 0.6642424264e-21
> > > # cm/s^2
> > > (2.99792458*10^8*10^2)^2*0.6642424264e-21;
> >
> > > # 0.5969913206
> > > #####################################################
> > > ####
> > > K := 2*MG;
> >
> > > # K := 0.1258418194e13
> > > # K = 2*MG = 0.1258418194e13 cm EVENT HORIZON of SgrA* Black Hole
> >
> > > # total coordinate acceleration at event horizon K=2*MG(+ sign selected for both)
> > > sqrt(((J^2/K^3)*(1-2*MG/K)^2-MG/K^2+MG*( (1-2*MG/K)^2-(E+J^2/K^2)*(1-2*MG/K)^3 )^2/(K^2*(1-2*MG/K))-K*((J/K^2)*(1-2*MG/K))^2)^2+(K*((-2/(K*(1-2*MG/K)))*((J/K^2)*(1-2*MG/K))*((1-2*MG/K)^2-(E+J^2/K^2)*(1-2*MG/K)^3))-2*sqrt((1-2*MG/K)^2-(E+J^2/K^2)*(1-2*MG/K)^3)*((J/K^2)*(1-2*MG/K)))^2);
> >
> > > # 0.3973241981e-12
> > > # cm/s^2
> > > (2.99792458*10^8*10^2)^2*0.3973241981e-12;
> >
> > > # 357097180.7
> > > ### total proper acceleration at event horizon K = 2*MG(+ sign selected for both)
> > > # ONE NOTICE: pure imaginary number acceleration for example at 3*MG, this is due 3*MG is over definition area of the second analytic solution.
> > > # Different selections of +, - signs (- + sign d^2Phi/dta^2)(+ - sign dr/da) are not treated here, + sign selections is made for both cases.
> > > ####
> > > # K = 2*MG = 0.1258418194e13 EVENT HORIZON of SgrA* Black Hole
> > > K := 2*MG;
> >
> > > # K := 0.1258418194e13
> > > ### total proper acceleration
> > > sqrt(((J^2/(sqrt(E)*K^3))*(1-2*MG/K)-(MG/(sqrt(E)*K^2*(1-2*MG/K)))+(MG/(sqrt(E)*K^2))*(1-(E+J^2/K^2)*(1-2*MG/K))-K*(J/K^2)^2)^2+(K*((2*J/(K^3))*sqrt(1-(E+J^2/K^2)*(1-2*MG/K)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/K^2)*(1-2*MG/K)))*(J/K^2))^2);
> >
> > > # 0.9932912900e-3
> > > # cm/s^2
> > > (2.99792458*10^8*10^2)^2*0.9932912900e-3;
> >
> > > # 0.8927256908e18
> > > ### total coordinate velocity
> > > sqrt((1-2*MG/K)^2*(1-(E+J^2/K^2)*(1-2*MG/K))+(J^2/K^2)*(1-2*MG/K)^2);
> >
> > > # 0.1460945827e-7
> > > # cm/s
> > > 2.99792458*10^8*10^2*0.1460945827e-7;
> >
> > > # 437.9805405
> > > ### total proper velocity
> > > sqrt((1/E)*(1-(E+J^2/K^2)*(1-2*MG/K))+J^2/K^2);
> >
> > > # 36.52364512
> > > # cm/s
> > > 2.99792458*10^8*10^2*36.52364512;
> >
> > > # 0.1094951335e13
> > > ####
> > > ##########################################################
> > > ###### Inside SgrA* Black Hole if S2 follows somehow the second analytic solution
> > > z := MG;
> >
> > > # z := 0.6292090968e12
> > > ### total coordinate velocity
> > > sqrt((1-2*MG/z)^2*(1-(E+J^2/z^2)*(1-2*MG/z))+(J^2/z^2)*(1-2*MG/z)^2);
> >
> > > # 103.2754259
> > > # cm/s
> > > 2.99792458*10^8*10^2*103.2754259;
> >
> > > # 0.3096119378e13
> > > ### total proper velocity
> > > sqrt((1/E)*(1-(E+J^2/z^2)*(1-2*MG/z))+J^2/z^2);
> >
> > > # 103.2743723
> > > # cm/s
> > > 2.99792458*10^8*10^2*103.2743723;
> >
> > > # 0.3096087792e13
> > > ### total proper acceleration
> > > sqrt(((J^2/(sqrt(E)*z^3))*(1-2*MG/z)-(MG/(sqrt(E)*z^2*(1-2*MG/z)))+(MG/(sqrt(E)*z^2))*(1-(E+J^2/z^2)*(1-2*MG/z))-z*(J/z^2)^2)^2+(z*((2*J/(z^3))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))*(J/z^2))^2);
> >
> > > # 0.3494379988e-7
> > > # cm/s^2
> > > (2.99792458*10^8*10^2)^2*0.3494379988e-7;
> >
> > > # 0.3140592111e14
> > > ### total coordinate acceleration
> > > sqrt(((J^2/(sqrt(E)*z^3))*(1-2*MG/z)-(MG/(sqrt(E)*z^2*(1-2*MG/z)))+(MG/(sqrt(E)*z^2))*(1-(E+J^2/z^2)*(1-2*MG/z))-K*(J/z^2)^2)^2+(z*((2*J/(z^3))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))*(J/z^2))^2);
> >
> > > # 0.3790004584e-7
> > > # cm/s^2
> > > (2.99792458*10^8*10^2)^2*0.3790004584e-7;
> >
> > > # 0.3406286247e14
> > Best Regards,
> > Hannu Poropudas
> > Finland
> IMPORTANT REMARK:
>
> I have recalculated proper time acceleration formulae and coordinate time
> acceleration formulae component by component.
>
> My calculation work is not ready at the moment,
>
> and
>
> I have some difficulties with these new acceleration formulae now.
>
> I don't recommend to use these above old
> acceleration formulae now.
>
> Best Regards, Hannu Poropudas


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Re: Analytic GR solutions of S2-star orbits and precession 732” per revolution

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Subject: Re:_Analytic_GR_solutions_of_S2-star_orbits_and_prec
ession_732”_per_revolution
From: haporop...@gmail.com (Hannu Poropudas)
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 by: Hannu Poropudas - Tue, 2 May 2023 08:34 UTC

torstai 24. marraskuuta 2022 klo 9.19.59 UTC+2 Hannu Poropudas kirjoitti:
> keskiviikko 16. marraskuuta 2022 klo 10.58.55 UTC+2 Hannu Poropudas kirjoitti:
> > torstai 17. syyskuuta 2020 klo 10.05.28 UTC+3 hanp...@luukku.com kirjoitti:
> > > On Monday, September 14, 2020 at 2:05:10 PM UTC+3, Hannu Poropudas wrote:
> > > > On Sunday, September 13, 2020 at 12:34:54 PM UTC+3, Hannu Poropudas wrote:
> > > > > On Thursday, September 10, 2020 at 10:10:49 AM UTC+3, Hannu Poropudas wrote:
> > > > > > Analytic GR solutions of S2-star orbits and precession 732” per revolution
> > > > > >
> > > > > > New initial data 2020 is used. My earlier initial data was 2016 in my
> > > > > > postings in this sci.physics.relativity.
> > > > > >
> > > > > > Please also take a look my two postings about OJ287 in sci.astro. I have got
> > > > > > no comments there due so heavy traffic of unbssinesslike postings from many posters.
> > > > > >
> > > > > > (Only command lines of Maple 9 program is below and command line mark is >. This can be copy-pasted in Maple 9 program and run it.
> > > > > > Remark that for example e-2 below means 10^(-2)
> > > > > > or e2 below means 10^2.)
> > > > > >
> > > > > > Best Regards,
> > > > > >
> > > > > > Hannu Poropudas
> > > > > > Finland
> > > > > >
> > > > > > ---COPY of the Maple 9 program below------
> > > > > >
> > > > > > > # RIGHT: Particle in Schwarzschild metrics H.P. 09.09.2020
> > > > > > > # S2 Orbit around Sagittarius A*
> > > > > > > # Perihelion precession of S2.
> > > > > > > #OK=COMPARISION to Post-Newtonian approx of Abuter R et al 2020, A&A 636,L5(2020)
> > > > > > > # Abuter R. et al. 2020. Detection of the Schwarzschild precession in the
> > > > > > > # orbit of the star S2 near the Galactic centre massive black hole.
> > > > > > > # Astronomy Astrophysics, 636, L5 (2020).
> > > > > > > # In this article: 12'/rev=720"/rev,12.1'/rev=726"/rev, 12.3'/rev=738"/rev
> > > > > > > # My GR calculation below, result : 731.8543470"/rev OR -676.7436912"/rev
> > > > > > > # (I use not rounded numbers). Approximately 732"/rev OR -677"/rev.
> > > > > > > # New starting values from Table E.1 Best-fit Orbit Parameters above article
> > > > > > > # and other constants of physics are taken from Wikipedia date 9.9.2020.
> > > > > > > # NEW CALCULATION x and y are roots. E and J little changed.
> > > > > > > # (analytic solutions defined between roots a2..a1 OR a4..a3)
> > > > > > >Restart;
> > > > > > >with(plots):with(plottools):
> > > > > > > # GEOMETRIZED UNIT SYSTEM IS PROBLEMATIC.
> > > > > > > # Ref. Formulae for E and J from Weinberg's book. (THESE FORMULAE ARE NOT USED.)
> > > > > > > # Reference: Weinberg, S. 1972. Gravitation and Cosmology: Principles
> > > > > > > # and Applications of the General Relativity. Wiley, New York.. pp. 179-210.
> > > > > > > # (units c = 1, and c.g.s in Weinberg's book).
> > > > > > > # J = r^2*dPhi/dp = constant (angular momentum per unit mass), p. 186.
> > > > > > > # E = constant (energy per unit mass), p. 186.
> > > > > > > # E > 0 for material particles, E = 0 for photons. p. 186.
> > > > > > > # Integration limits must be determined from the problem to which apply these.
> > > > > > >#Int(1/(r^4*(1/J^2-E/J^2)+r^3*(2*M*G*E/J^2)-r^2+2*M*G*r)^(1/2),r);
> > > > > > >AU := 149597870700*10^2;
> > > > > > ># 125.058 mas, 8246.7 pc, 1 pc =3.26 ly
> > > > > > >ap:= 1030.799324*AU;
> > > > > > >em := 0.884649;
> > > > > > >M := 4.261*10^6*1.98847*10^30*10^3;
> > > > > > ># v = c
> > > > > > >v := 2.99792458*10^8*10^2;
> > > > > > >G := 6.6743015*10^(-11)*10^3;
> > > > > > ># BH mass*G geometric units (cm) and v = c.
> > > > > > >M*G/v^2;
> > > > > > >MG := 0.6292090968e12;
> > > > > > > # EXAMPLE. Weinberg formulae E and J (THESE FORMULAE ARE NOT ANY MORE USED.)
> > > > > > > # Perihelion distance
> > > > > > >x := ap*(1-em);
> > > > > > ># Aphelion distance
> > > > > > >y := ap*(1+em);
> > > > > > > # NEW CALCULATION (NEW FORMULAE OF MY OWN) x and y are roots
> > > > > > > #***********************************************************
> > > > > > >E := (2*x*y*MG+2*y^2*MG+2*x^2*MG-x^2*y-x*y^2)/((-y+2*MG)*(-2*x*MG-2*y*MG+x^2+x*y));
> > > > > > >J := 2^(1/2)*(-(-y+2*MG)*MG*(-2*x*MG-2*y*MG+x^2+x*y))^(1/2)*y*x/(-x^2*y+2*x^2*MG-4*x*MG^2+4*x*y*MG-x*y^2+2*y^2*MG-4*y*MG^2);
> > > > > > > #*****************************************************************
> > > > > > >
> > > > > > > # Weinberg's formula (NOT USED ANY MORE)
> > > > > > > #J := sqrt((1/(1-2*MG/y)-(1/(1-2*MG/x)))/(1/y^2-1/x^2));
> > > > > > > # NEW CALCULATION x and y are roots
> > > > > > >J := -0.4594478956e14;
> > > > > > > # Weinberg's formula (NOT USED ANY MORE)
> > > > > > > #E := (y^2/(1-2*MG/y)-x^2/(1-2*MG/x))/(y^2-x^2);
> > > > > > > # NEW CALCULATION x and y are roots
> > > > > > >E := 1.000040803;
> > > > > > >solve(r^4*(1/J^2-E/J^2)+r^3*(2*MG*E/J^2)-r^2+2*MG*r=0,r);
> > > > > > >a1 := 0.2906262559e17;
> > > > > > >a2 := 0.1778773310e16;
> > > > > > >a3 := 0.1259363679e13;
> > > > > > >a4 := 0;
> > > > > >
> > > > > > ># Primitive function, definition areas 0<=P<=Pi/2 and a2 <= r <= a1.
> > > > > >
> > > > > > >e := P -> 2.000794538*EllipticF(sin(P),0.2578161452e-1);
> > > > > > >r := P -> (0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17);
> > > > > >
> > > > > > > #******************************
> > > > > > > # P which corresponds to the perihelion distance (TWO VALUES)
> > > > > > >solve((0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17)=0.1778774525e16,P);
> > > > > > >e(0.8532953281e-3);
> > > > > > >e(-0.8532953281e-3);
> > > > > > ># Solution's definition area limit is a2-root (TWO VALUES)
> > > > > > >solve((0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17)=0.1778773310e16,P);
> > > > > > >e(0.6516324491e-5);
> > > > > > >e(-0.6516324491e-5);
> > > > > > ># P which corresponds to the aphelion distance (TWO VALUES)
> > > > > > >solve((0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17)=0.2906230228e17,P);
> > > > > > >e(1.569945054);
> > > > > > >e(-1.569945054);
> > > > > > > # Solution's correct definition limit is a1-root (TWO VALUES)
> > > > > > > # COMPLEX NUMBER (SMALL COMPLEX PARTS ARE LITTLE OUTSIDE ORBIT)
> > > > > > >solve((0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17)=0.2906262559e17,P);
> > > > > > >e(1.570796327-0.1136598102e-4*I);
> > > > > > >e(-1.570796327+0.1136598102e-4*I);
> > > > > >
> > > > > > > # Angle change, definition areas 0<=P<=Pi/2 ja a2 <= r <= a1.
> > > > > > > # + sign perihelion in last term
> > > > > >
> > > > > > >e2 := P -> 2*abs(2.000794538*EllipticF(sin(P),0.2578161452e-1)-2.000794538*EllipticF(sin(0.8532953281e-3),0.2578161452e-1))-2*Pi;
> > > > > > >r2 := P -> (0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17);
> > > > > >
> > > > > > > # Calculation of different combinations of + and - signs
> > > > > > > # - sign perihelion in last term
> > > > > >
> > > > > > >e2B := P -> 2*abs(2.000794538*EllipticF(sin(P),0.2578161452e-1)-2.000794538*EllipticF(sin(-0.8532953281e-3),0.2578161452e-1))-2*Pi;
> > > > > > >r2B := P -> (0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17);
> > > > > >
> > > > > > > # Angle change in radians per one revolution. (TWO e2 and e2B functions)
> > > > > > > # FOR e2
> > > > > > >evalf(e2(1.569945054));
> > > > > > ># Angle change in degrees per one revolution.
> > > > > > >evalf(-0.3280946e-2*180/Pi);
> > > > > > ># Angle change in arc seconds per one revolution.
> > > > > > >-0.1879843586*60*60;
> > > > > > >###(-676.7436912)
> > > > > > ># Second root
> > > > > > >evalf(e2(-1.569945054));
> > > > > > >evalf(0.3548130e-2*180/Pi);
> > > > > > >0.2032928741*60*60;
> > > > > > >###(731.8543470)
> > > > > > ># FOR e2B
> > > > > > >evalf(e2B(1.569945054));
> > > > > > ># Angle change in degrees per one revolution.
> > > > > > >evalf(0.3548130e-2*180/Pi);
> > > > > > ># Angle change in arc seconds per one revolution.
> > > > > > >0.2032928741*60*60;
> > > > > > >###(731.8543470)
> > > > > > ># Second root
> > > > > > >evalf(e2B(-1.569945054));
> > > > > > >evalf(-0.3280946e-2*180/Pi);
> > > > > > >-0.1879843586*60*60;
> > > > > > >####(-676.7436912)
> > > > > > > # + and - signs for e2 or e2B does not have different results due abs-values
> > > > > > > # Second Primitive function, definition areas 0<=P<=Pi/2 and a4=0 <= r <= a3.
> > > > > > > # This has form of spiral (+ and - signs) which leads to the origin.
> > > > > > >
> > > > > > > # SUMMARY of S2 precession: 731.8543470"/revolution OR -676.7436912"/revolution.
> > > > > > >
> > > > > > > #******************************
> > > > > >
> > > > > > ># Second Primitive function, definition areas 0<=P<=Pi/2 and a4=0 <=
> > > > > > r <= a3.
> > > > > >
> > > > > > >ee := P-> 2.000794538*EllipticF(sin(P),0.9996675989);
> > > > > > >rr := P-> (0.3660041508e29*sin(P)^2)/(0.1259363679e13*sin(P)^2+0.2906136623e17);
> > > > > >
> > > > > > > #*********************************************************
> > > > > > >
> > > > > > > # Plottings only on definition area a2..a1.
> > > > > >
> > > > > > > # First side of the solution (+,- solution)
> > > > > > >plot([r(P),e(P),P=0..Pi/2],coords=polar);
> > > > > >
> > > > > > ># Second side of the solution (+,- solution)
> > > > > > >plot([r(P),-e(P),P=0..Pi/2],coords=polar);
> > > > > >
> > > > > > ># Angle change picture has no other meaning than above calculated precession
> > > > > > >plot([r2(P),e2(P),P=0..Pi/2],coords=polar);
> > > > > >
> > > > > > ># Angle change picture has no other meaning than above calculated precession
> > > > > > >plot([r2B(P),e2B(P),P=0..Pi/2],coords=polar)
> > > > > >
> > > > > > > # Plottings only on definition area a4..a3.
> > > > > >
> > > > > > > # First side of the Second solution (+,- solution)
> > > > > > >plot([rr(P),ee(P),P=0..Pi/2],coords=polar);
> > > > > >
> > > > > > ># Second side of the Second solution (+,- solution)
> > > > > > >plot([rr(P),-ee(P),P=0..Pi/2],coords=polar);
> > > > > *** 1. CORRECTION: Formula of J has + and - signs. Positive sign should be selected.
> > > > > Previous - sign selection does not influence due J^2 was always used.
> > > > >
> > > > > J := 0.4594478956e14
> > > > >
> > > > > *** 2. -Pi/2 .. Pi/2 plots gives both sides of both solutions.
> > > > > This is due
> > > > > sin(-P)=-sin(P)
> > > > > and
> > > > > EllipticF(sin(-P),q)=EllipticF(-sin(P),q)=-EllipticF(sin(P),q).
> > > > >
> > > > > 3. I calculaled below total coordinate velocity, proper velocity and
> > > > > total proper acceleration for S2-star:
> > > > > (These are used only in definition areas of above analytic solutions
> > > > > a2<=r<=a1 or a4<r<=a3,
> > > > > a1 := 0.2906262559e17,
> > > > > a2 := 0.1778773310e16,
> > > > > a3 := 0.1259363679e13,
> > > > > a4 := 0)
> > > > >
> > > > > (REMARK: > is command line mark of Maple 9 program)
> > > > > ># Total coordinate velocity and total proper velocity S2-star
> > > > > ># at perihelion and at aphelion HP 12092020
> > > > >
> > > > > > ### total coordinate velocity v^2=(dr/dt)^2+r^2*(dPhi/dt)^2
> > > > > > sqrt((1-2*MG/r)^2*(1-(E+J^2/r^2)*(1-2*MG/r))+(J^2/r^2)*(1-2*MG/r)^2)
> > > > > > sqrt((1-2*MG/x)^2*(1-(E+J^2/x^2)*(1-2*MG/x))+(J^2/x^2)*(1-2*MG/x)^2);
> > > > > 0.2581119858e-1
> > > > > > # km/s
> > > > > > 0.2997924580e11*0.2581119858e-1/10^5;
> > > > > 7738.002666
> > > > > > sqrt((1-2*MG/y)^2*(1-(E+J^2/y^2)*(1-2*MG/y))+(J^2/y^2)*(1-2*MG/y)^2);
> > > > > 0.1581028031e-2
> > > > > > # km/s
> > > > > > 0.2997924580e11*0.1581028031e-2/10^5;
> > > > > 473.9802796
> > > > > > ###
> > > > > > ### total proper velocity v^2=(dr/dtau)^2+r^2*(dPhi/dtau)^2
> > > > > > sqrt((1/E)*(1-(E+J^2/r^2)*(1-2*MG/r))+J^2/r^2)
> > > > > > sqrt((1/E)*(1-(E+J^2/x^2)*(1-2*MG/x))+J^2/x^2);
> > > > > 0.2582947199e-1
> > > > > > # km/s
> > > > > > 0.2997924580e11*0.2582947199e-1/10^5;
> > > > > 7743.480897
> > > > > > sqrt((1/E)*(1-(E+J^2/y^2)*(1-2*MG/y))+J^2/y^2);
> > > > > 0.001581096486
> > > > > > # km/s
> > > > > > 0.2997924580e11*0.1581096486e-2/10^5;
> > > > > 474.0008019
> > > > > > ###
> > > > >
> > > > > ># Total proper acceleration S2 star HP 13092020
> > > > >
> > > > > > ### Total proper acceleration (- + sign d^2Phi/dta^2)(+ - sign dr/da)
> > > > >
> > > > > > ### A^2=(d^2r/dta^2-r*(dPhi/dta)^2)^2+(r*d^2Phi/dta^2+2*(dr/dta)*(dPhi/dta))^2
> > > > >
> > > > > > #sqrt(((J^2/(sqrt(E)*r^3))*(1-2*MG/r)-(MG/(sqrt(E)*r^2*(1-2*MG/r)))+(MG/(sqrt(E)*r^2))*(1-(E+J^2/r^2)*(1-2*MG/r))-r*(J/r^2)^2)^2+(r*((2*J/(r^3))*sqrt(1-(E+J^2/r^2)*(1-2*MG/r)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/r^2)*(1-2*MG/r)))*(J/r^2))^2)
> > > > >
> > > > > > ### total proper acceleration at perihelion (+ sign selected for both)
> > > > >
> > > > > > sqrt(((J^2/(sqrt(E)*x^3))*(1-2*MG/x)-(MG/(sqrt(E)*x^2*(1-2*MG/x)))+(MG/(sqrt(E)*x^2))*(1-(E+J^2/x^2)*(1-2*MG/x))-x*(J/x^2)^2)^2+(x*((2*J/(x^3))*sqrt(1-(E+J^2/x^2)*(1-2*MG/x)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/x^2)*(1-2*MG/x)))*(J/x^2))^2);
> > > > > 1.992765885*10^(-19)
> > > > > > # cm/s^2
> > > > > > (2.99792458*10^8*10^2)^2*0.1992765885e-18;
> > > > > 179.1008659
> > > > > > ### total proper acceleration at aphelion (+ sign selected for both)
> > > > >
> > > > > > sqrt(((J^2/(sqrt(E)*y^3))*(1-2*MG/y)-(MG/(sqrt(E)*y^2*(1-2*MG/y)))+(MG/(sqrt(E)*y^2))*(1-(E+J^2/y^2)*(1-2*MG/y))-y*(J/y^2)^2)^2+(y*((2*J/(y^3))*sqrt(1-(E+J^2/y^2)*(1-2*MG/y)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/y^2)*(1-2*MG/y)))*(J/y^2))^2);
> > > > > 7.450050462*10^(-22)
> > > > > > # cm/s^2
> > > > > > (2.99792458*10^8*10^2)^2*0.7450050462e-21;
> > > > > 0.6695771434
> > > > > > ### total proper acceleration at event horizon z = MG(+ sign selected for both)
> > > > >
> > > > > > # ONE NOTICE: pure imaginary number acceleration for example at 3*MG , this is due 3*MG is over definition area of the second analytic solution.
> > > > > > # Different selections of +, - signs (- + sign d^2Phi/dta^2)(+ - sign dr/da) are not treated here , + sign selections is made for both cases.
> > > > >
> > > > > > z := MG;
> > > > > 6.292090968*10^11
> > > > > > sqrt(((J^2/(sqrt(E)*z^3))*(1-2*MG/z)-(MG/(sqrt(E)*z^2*(1-2*MG/z)))+(MG/(sqrt(E)*z^2))*(1-(E+J^2/z^2)*(1-2*MG/z))-z*(J/z^2)^2)^2+(z*((2*J/(z^3))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))*(J/z^2))^2);
> > > > > 3.494379988*10^(-8)
> > > > > > # cm/s^2
> > > > > > (2.99792458*10^8*10^2)^2*0.3494379988e-7;
> > > > > 3.140592111*10^13
> > > > > > ###
> > > > > Best Regards,
> > > > > Hannu Poropudas
> > > > > Finland
> > > > CORRECTION (event horizon is was not MG ):
> > > > Interesting OPEN QUESTIONS would be if S2-star somehow happens to go to my second analytic solution orbit,
> > > > which definition area is 0 < r <= 1.259363679*10^12 cm.
> > > > At r = 2*MG = = 1.258418194*10^12 cm, event horizon of the SgrA* black hole seems to
> > > > have following (OPEN QUESTION OF INTERPRETATION of the total proper velocity and the total proper acceleration):
> > > >
> > > > total coordinate velocity = 1.460945827*10^(-8) , (geometric units, Weinberg 1972),
> > > > total coordinate velocity = 437.9805405 cm/s, (c.g.s units, Weinberg 1972),
> > > > total proper velocity = 36.52364512 (geometric units, Weinberg 1972),
> > > > total proper velocity = 1.094951335*10^12, cm/s (c.g.s units, Weinberg 1972)
> > > > and
> > > > total proper acceleration = 8,927256908*10^17 cm/s^2 , (c.g.s units, Weinberg 1972)
> > > >
> > > > I have used definitions of Weinberg S. 1972. Gravitation and Cosmology book
> > > > and Becker 1954 Introduction to Theoretical Mechanics book
> > > > when I calculated above total coordinate velocity, total proper velocity and total proper acceleration.
> > > > (Formulae of these GR calculations are in above posting of mine).
> > > >
> > > > Best Regards,,
> > > > Hannu Poropudas
> > > > Finland
> > > Preliminary formulae and few important numerical points calculated below:
> > >
> > > ># S2-star’s two analytic solutions of orbits around SgrA* Black Hole
> > > ># Total coordinate velocity, total proper velocity and
> > > ># Total coordinate acceleration, total proper acceleration
> > > ># formulae and numerical calculation for some points.
> > > ># OPEN QUESTIONS OF PHYSICAL INTERPRETATIONS:
> > > ># second analytic solution cases r=2*MG and
> > > ># INSIDE Black Hole 0<r<= 2*MG
> > > ># This is copy part of my Maple 9 program
> > > ># where other symbols have their numerical values H.P. 15.9.2020
> > > > ### total coordinate velocity v^2=(dr/dt)^2+r^2*(dPhi/dt)^2
> > > > #sqrt((1-2*MG/r)^2*(1-(E+J^2/r^2)*(1-2*MG/r))+(J^2/r^2)*(1-2*MG/r)^2)
> > > > # total coordinate velocity at perihelion
> > > > sqrt((1-2*MG/x)^2*(1-(E+J^2/x^2)*(1-2*MG/x))+(J^2/x^2)*(1-2*MG/x)^2);
> > >
> > > > # 0.2581119858e-1
> > > > # km/s
> > > > 0.2997924580e11*0.2581119858e-1/10^5;
> > >
> > > > # 7738.002666
> > > > # total coordinate velocity at aphelion
> > > > sqrt((1-2*MG/y)^2*(1-(E+J^2/y^2)*(1-2*MG/y))+(J^2/y^2)*(1-2*MG/y)^2);
> > >
> > > > # 0.1581028031e-2
> > > > # km/s
> > > > 0.2997924580e11*0.1581028031e-2/10^5;
> > >
> > > > # 473.9802796
> > > > ###
> > > > ### total proper velocity v^2=(dr/dtau)^2+r^2*(dPhi/dtau)^2
> > > > #sqrt((1/E)*(1-(E+J^2/r^2)*(1-2*MG/r))+J^2/r^2)
> > > > # total proper velocity at perihelion
> > > > sqrt((1/E)*(1-(E+J^2/x^2)*(1-2*MG/x))+J^2/x^2);
> > >
> > > > # 0.2582947199e-1
> > > > # km/s
> > > > 0.2997924580e11*0.2582947199e-1/10^5;
> > >
> > > > # 7743.480897
> > > > # total proper velocity at aphelion
> > > > sqrt((1/E)*(1-(E+J^2/y^2)*(1-2*MG/y))+J^2/y^2);
> > >
> > > > # 0.1581096486e-2
> > > > # km/s
> > > > 0.2997924580e11*0.1581096486e-2/10^5;
> > >
> > > > # 474.0008019
> > >
> > > >###
> > > > ### Total proper acceleration (- + sign d^2Phi/dta^2)(+ - sign dr/da)
> > > > ### A^2=(d^2r/dta^2-r*(dPhi/dta)^2)^2+(r*d^2Phi/dta^2+2*(dr/dta)*(dPhi/dta))^2
> > > > #sqrt(((J^2/(sqrt(E)*r^3))*(1-2*MG/r)-(MG/(sqrt(E)*r^2*(1-2*MG/r)))+(MG/(sqrt(E)*r^2))*(1-(E+J^2/r^2)*(1-2*MG/r))-r*(J/r^2)^2)^2+(r*((2*J/(r^3))*sqrt(1-(E+J^2/r^2)*(1-2*MG/r)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/r^2)*(1-2*MG/r)))*(J/r^2))^2)
> > > >
> > > > ### total proper acceleration at perihelion (+ sign selected for both)
> > > > # total proper acceleration at perihelion
> > > > sqrt(((J^2/(sqrt(E)*x^3))*(1-2*MG/x)-(MG/(sqrt(E)*x^2*(1-2*MG/x)))+(MG/(sqrt(E)*x^2))*(1-(E+J^2/x^2)*(1-2*MG/x))-x*(J/x^2)^2)^2+(x*((2*J/(x^3))*sqrt(1-(E+J^2/x^2)*(1-2*MG/x)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/x^2)*(1-2*MG/x)))*(J/x^2))^2);
> > >
> > > > # 0.1992765885e-18
> > > > # cm/s^2
> > > > (2.99792458*10^8*10^2)^2*0.1992765885e-18;
> > >
> > > > # 179.1008659
> > > > ### total proper acceleration at aphelion (+ sign selected for both)
> > > > # total proper acceleration at aphelion
> > > > sqrt(((J^2/(sqrt(E)*y^3))*(1-2*MG/y)-(MG/(sqrt(E)*y^2*(1-2*MG/y)))+(MG/(sqrt(E)*y^2))*(1-(E+J^2/y^2)*(1-2*MG/y))-y*(J/y^2)^2)^2+(y*((2*J/(y^3))*sqrt(1-(E+J^2/y^2)*(1-2*MG/y)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/y^2)*(1-2*MG/y)))*(J/y^2))^2);
> > >
> > > > # 0.7450050462e-21
> > > > # cm/s^2
> > > > (2.99792458*10^8*10^2)^2*0.7450050462e-21;
> > >
> > > > # 0.6695771434
> > > > ####
> > > > ### Total coordinate acceleration (- + sign d^2Phi/dt^2)(+ - sign dr/dt)
> > > > ### A^2=(d^2r/dt^2-r*(dPhi/dt)^2)^2+(r*d^2Phi/dt^2+2*(dr/dt)*(dPhi/dt))^2
> > > > # First component of the sqrt formula
> > > > #((J^2/r^3)*(1-2*MG/r)^2-MG/r^2+MG*( (1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3 )^2/(r^2*(1-2*MG/r))-r*((J/r^2)*(1-2*MG/r))^2)^2:
> > > > # Second component of the sqrt formula
> > > > #(r*((-2/(r*(1-2*MG/r)))*((J/r^2)*(1-2*MG/r))*((1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3))-2*sqrt((1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3)*((J/r^2)*(1-2*MG/r)))^2:
> > > > #sqrt(((J^2/r^3)*(1-2*MG/r)^2-MG/r^2+MG*( (1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3 )^2/(r^2*(1-2*MG/r))-r*((J/r^2)*(1-2*MG/r))^2)^2+(r*((-2/(r*(1-2*MG/r)))*((J/r^2)*(1-2*MG/r))*((1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3))-2*sqrt((1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3)*((J/r^2)*(1-2*MG/r)))^2):
> > >
> > > > # total coordinate acceleration at perihelion
> > > > sqrt(((J^2/x^3)*(1-2*MG/x)^2-MG/x^2+MG*( (1-2*MG/x)^2-(E+J^2/x^2)*(1-2*MG/x)^3 )^2/(x^2*(1-2*MG/x))-x*((J/x^2)*(1-2*MG/x))^2)^2+(x*((-2/(x*(1-2*MG/x)))*((J/x^2)*(1-2*MG/x))*((1-2*MG/x)^2-(E+J^2/x^2)*(1-2*MG/x)^3))-2*sqrt((1-2*MG/x)^2-(E+J^2/x^2)*(1-2*MG/x)^3)*((J/x^2)*(1-2*MG/x)))^2);
> > >
> > > > # 0.1988634824e-18
> > > > # cm/s^2
> > > > (2.99792458*10^8*10^2)^2*0.1988634824e-18;
> > >
> > > > # 178.7295847
> > > > # total coordinate acceleration at aphelion
> > > > sqrt(((J^2/y^3)*(1-2*MG/y)^2-MG/y^2+MG*( (1-2*MG/y)^2-(E+J^2/y^2)*(1-2*MG/y)^3 )^2/(y^2*(1-2*MG/y))-x*((J/y^2)*(1-2*MG/y))^2)^2+(y*((-2/(y*(1-2*MG/y)))*((J/y^2)*(1-2*MG/y))*((1-2*MG/y)^2-(E+J^2/y^2)*(1-2*MG/y)^3))-2*sqrt((1-2*MG/y)^2-(E+J^2/y^2)*(1-2*MG/y)^3)*((J/y^2)*(1-2*MG/y)))^2);
> > >
> > > > # 0.6642424264e-21
> > > > # cm/s^2
> > > > (2.99792458*10^8*10^2)^2*0.6642424264e-21;
> > >
> > > > # 0.5969913206
> > > > #####################################################
> > > > ####
> > > > K := 2*MG;
> > >
> > > > # K := 0.1258418194e13
> > > > # K = 2*MG = 0.1258418194e13 cm EVENT HORIZON of SgrA* Black Hole
> > >
> > > > # total coordinate acceleration at event horizon K=2*MG(+ sign selected for both)
> > > > sqrt(((J^2/K^3)*(1-2*MG/K)^2-MG/K^2+MG*( (1-2*MG/K)^2-(E+J^2/K^2)*(1-2*MG/K)^3 )^2/(K^2*(1-2*MG/K))-K*((J/K^2)*(1-2*MG/K))^2)^2+(K*((-2/(K*(1-2*MG/K)))*((J/K^2)*(1-2*MG/K))*((1-2*MG/K)^2-(E+J^2/K^2)*(1-2*MG/K)^3))-2*sqrt((1-2*MG/K)^2-(E+J^2/K^2)*(1-2*MG/K)^3)*((J/K^2)*(1-2*MG/K)))^2);
> > >
> > > > # 0.3973241981e-12
> > > > # cm/s^2
> > > > (2.99792458*10^8*10^2)^2*0.3973241981e-12;
> > >
> > > > # 357097180.7
> > > > ### total proper acceleration at event horizon K = 2*MG(+ sign selected for both)
> > > > # ONE NOTICE: pure imaginary number acceleration for example at 3*MG, this is due 3*MG is over definition area of the second analytic solution..
> > > > # Different selections of +, - signs (- + sign d^2Phi/dta^2)(+ - sign dr/da) are not treated here, + sign selections is made for both cases.
> > > > ####
> > > > # K = 2*MG = 0.1258418194e13 EVENT HORIZON of SgrA* Black Hole
> > > > K := 2*MG;
> > >
> > > > # K := 0.1258418194e13
> > > > ### total proper acceleration
> > > > sqrt(((J^2/(sqrt(E)*K^3))*(1-2*MG/K)-(MG/(sqrt(E)*K^2*(1-2*MG/K)))+(MG/(sqrt(E)*K^2))*(1-(E+J^2/K^2)*(1-2*MG/K))-K*(J/K^2)^2)^2+(K*((2*J/(K^3))*sqrt(1-(E+J^2/K^2)*(1-2*MG/K)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/K^2)*(1-2*MG/K)))*(J/K^2))^2);
> > >
> > > > # 0.9932912900e-3
> > > > # cm/s^2
> > > > (2.99792458*10^8*10^2)^2*0.9932912900e-3;
> > >
> > > > # 0.8927256908e18
> > > > ### total coordinate velocity
> > > > sqrt((1-2*MG/K)^2*(1-(E+J^2/K^2)*(1-2*MG/K))+(J^2/K^2)*(1-2*MG/K)^2);
> > >
> > > > # 0.1460945827e-7
> > > > # cm/s
> > > > 2.99792458*10^8*10^2*0.1460945827e-7;
> > >
> > > > # 437.9805405
> > > > ### total proper velocity
> > > > sqrt((1/E)*(1-(E+J^2/K^2)*(1-2*MG/K))+J^2/K^2);
> > >
> > > > # 36.52364512
> > > > # cm/s
> > > > 2.99792458*10^8*10^2*36.52364512;
> > >
> > > > # 0.1094951335e13
> > > > ####
> > > > ##########################################################
> > > > ###### Inside SgrA* Black Hole if S2 follows somehow the second analytic solution
> > > > z := MG;
> > >
> > > > # z := 0.6292090968e12
> > > > ### total coordinate velocity
> > > > sqrt((1-2*MG/z)^2*(1-(E+J^2/z^2)*(1-2*MG/z))+(J^2/z^2)*(1-2*MG/z)^2);
> > >
> > > > # 103.2754259
> > > > # cm/s
> > > > 2.99792458*10^8*10^2*103.2754259;
> > >
> > > > # 0.3096119378e13
> > > > ### total proper velocity
> > > > sqrt((1/E)*(1-(E+J^2/z^2)*(1-2*MG/z))+J^2/z^2);
> > >
> > > > # 103.2743723
> > > > # cm/s
> > > > 2.99792458*10^8*10^2*103.2743723;
> > >
> > > > # 0.3096087792e13
> > > > ### total proper acceleration
> > > > sqrt(((J^2/(sqrt(E)*z^3))*(1-2*MG/z)-(MG/(sqrt(E)*z^2*(1-2*MG/z)))+(MG/(sqrt(E)*z^2))*(1-(E+J^2/z^2)*(1-2*MG/z))-z*(J/z^2)^2)^2+(z*((2*J/(z^3))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))*(J/z^2))^2);
> > >
> > > > # 0.3494379988e-7
> > > > # cm/s^2
> > > > (2.99792458*10^8*10^2)^2*0.3494379988e-7;
> > >
> > > > # 0.3140592111e14
> > > > ### total coordinate acceleration
> > > > sqrt(((J^2/(sqrt(E)*z^3))*(1-2*MG/z)-(MG/(sqrt(E)*z^2*(1-2*MG/z)))+(MG/(sqrt(E)*z^2))*(1-(E+J^2/z^2)*(1-2*MG/z))-K*(J/z^2)^2)^2+(z*((2*J/(z^3))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))*(J/z^2))^2);
> > >
> > > > # 0.3790004584e-7
> > > > # cm/s^2
> > > > (2.99792458*10^8*10^2)^2*0.3790004584e-7;
> > >
> > > > # 0.3406286247e14
> > > Best Regards,
> > > Hannu Poropudas
> > > Finland
> > IMPORTANT REMARK:
> >
> > I have recalculated proper time acceleration formulae and coordinate time
> > acceleration formulae component by component.
> >
> > My calculation work is not ready at the moment,
> >
> > and
> >
> > I have some difficulties with these new acceleration formulae now.
> >
> > I don't recommend to use these above old
> > acceleration formulae now.
> >
> > Best Regards, Hannu Poropudas
> It seems to me that acceleration formulae in polar coordinates
> are not suitable in these general relativistic calculations although
> these formulae contains correctly components by component
> Christoffel symbols of second kind ,
> but these Christoffel symbols of second kind are correct
> for polar coordinates and these are
> NOT correct for Schwarzschild metrics.
>
> The reference is below which I have used in my calculations.
>
> Reference
>
> Weinberg Steven, 1972.
> Gravitation and Cosmology: Principles and Applications of the
> General Theory of Relativity.
> John Wiley & Sons, Inc.
> Printed in the United States of America.
> 657 pages, pp. 185-188.
>
> Best Regards, Hannu Poropudas


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