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Draft Model of Neutron Star (M=1.5*M_Sun) Hannu Poropudas 14.5.2024

SI-units used here.

Schwarzschild metrics

ds^2=(1-2MG/(rc^2))c^2dt^2-dr^2/(1-2MG/(rc^2))-r^2dtheta^2- r^2sin(theta)^2*dphi^2

Schwarzschild metrics modified used here

ds^2=(1-2m(r)G/(rc^2))c^2dt^2-dr^2/(1-2m(r)G/(rc^2))-r^2dtheta^2- r^2sin(theta)^2*dphi^2

Einstein field equations

-(2*(rc^2-2Gm(r))Gdm(r)/dr)/(r^3c^2)=8Pirho(r)

(2Gdm(r)/dt)/(r*(rc^2-2Gm(r)))=-(8Pi*G/c^4)*p1(r)

Gd^2m(r)/dr^2r/c^2=-(8PiG/c^4)*p2(r)

-Gd^2m(r)/dr^2r*(-1+cos(theta)^2)/c^2=-(8PiG/c^4)*p3(r)

Solutions:

rho(r)=-(-c^2rsqrt(1-2MG/(rc^2))-rc^2+MG)/(8PiGr^3)

p1(r)=(3MG+2MGsqrt(1-2MG/(rc^2)))-rc^2-rc^2sqrt(1- 2MG/(rc^2))c^4/(8PiGr^2*(-rc^2+2M*G))

p1(r)<0, range 2MG/c^2 -> approx 5906

p1(r)>0, range approx 5907 →

p2(r)= (rc^2-2MG)Mc^2exp(-(2sqrt(1-2MG/(rc^2))(rc^2- 2MG)(3MG+rc^2))/(15r^2M^2G^2))/(8PiMGr^3sqrt(1-2MG/(r*c^2)))

p3(r)=p2(r), due isotropy theta = Pi/2

2MG/c^2 = 4429.729416

Plots results rho(r):

(kgm^(-3)) rho(r), plot starts approx 1.4710^18, plot max approx 1.92*10^18

Plot results log10(p1(r)): plot starts positive values approx 10^31.7 = 5.01187233610^31 at approx r = 5906 p1(5900)=-1.00100753910^33 p1(5906)=-4.82739179410^31 (Nm^(-2)) p1(5907)= 1.094350802*10^32

plot max approx 10^34.9 = 7.94328234710^34 Plot results log10(abs(p1(r))): Negative pressures plot starts approx -10^38.2 = -1.58489319210^38 at r = approx 5906 approx -10^31.85 = -7.07945784410^31 p1(4429.729416) = infinity Positive pressures (Nm^(-2)) Plot results log10(p1(r)) from 4429.729416 to 20* 4429.729416 plot starts approx 10^33.32 = 2.08929613110^33 max approx 10^34.88 = 7.58577575010^34

Plot results p2(r) and p3(r):

(Nm^(-2)) plot stats approx 10^33.96 = 9.12010839410^31 max approx 10^34.84 = 6.918309709*10^34

1Nm^(-2) = 10dyncm^(-2)

Refrence:

Adler R. Bazin M. Schiffer M. 1965. Introduction to General Relativity. McGraw-Hill Book Company. Printed in the United States of America. 651 pages, pp. 276-295.

P.S. I. Equation of state can be plotted with Maple 9: (positive pressure p1(r) range) plot([log10(rho(r)),log10(p1(r)),r=4429.729416..24429.729416]) (positive pressure p1(r) range and negative pressure p1(r) range with abs due negative log10 is not possible for plot) II. Equation of state can be plotted with Maple 9: (positive pressure p2(r)=p3(r)) plot([log10(rho(r)),log10(p2(r)),r=4429.729416..24429.729416]) plot([log10(rho(r),log10(abs(p1(r)))),r=4429.729416..2*4429.729416])

Used constants in SI-units: G=6.6738810^(-11), c=2.9979248810^8, M_Sun=1.9884710^30, M=1.5M_Sun=2.98270510^30, 2M*G/c^2=4429.729416.

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  • $\begingroup$ What is your question here? Please use mathjax here to illustrate the math much more legibly and include the actual plots you talk about. $\endgroup$ May 15 at 10:51
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    $\begingroup$ This appears to be borderline spam. The questioner is asking (in fact, not asking; there is no question here) about their own personal theory. $\endgroup$ May 15 at 12:28

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