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

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

I found one Figure 32.1. (a) Schwarzschild coordinates on page 848 in the book of
(Misner C. W. Thorne K. S. Wheeler J. A., 1973), which is agreement of my analytic
second solution orbit of S2 star, which orbit was spiralling from outside event horizon
to inside event horizon and finally to the singularity (at origin) of SgrA* black hole
(sign changes (from + to -) of velocity of S2-star when crossing event horizon was described here
in this posting chain of mine):

The free-fall collapse of a star if initial radius R_i = 10 M as depicted [...] in
(a) Schwarzschild coordinates. [...] (Misner C. W. Thorne K. S. Wheeler J. A., 1973).

"This shows the surface of a collapsing star (the boundary of the gray region)
graphed in [...] Schwarzschild coordinates (which has the weird property
that inside the horizon, the collapsing surface is actually moving
backwards in time relative to the time coordinate) [...]."

This copy "[...]" was taken from the First Answer from Physics stack exchange
of the question:
"Formation of the event horizon seems impossible with singularity inside
seems impossible [duplicate]"
Asked 8 years, 1 month ago.
3 Answers.

the First Answer Apr 1, 2015 at 18:42
Hypnosift
https://physics.stackexchange.com/users/59406/hypnosifl
(reputation score=6130, gold badges=2, silver badges=23, bronze badges=38)

Reference:
Misner C. W., Thorne K. S., Wheeler J. A., 1973.
Gravitation.
W.H. Freeman and Company San Francisco.
Printed in the United States of America.
1279 pages. (Figure 32.1 (a) Schwarzschild coordinates on page 848.)

Best Regards,
Hannu Poropudas
Kolamäentie 9E,
90900 Kiiminki / Oulu,
Finland.

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o Re: Analytic GR solutions of S2-star orbits and prec

By: Hannu Poropudas on Wed, 16 Nov 2022

2Hannu Poropudas
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