Vittorio De Falco
Graduated in Mathematics at the University of Napoli
PhD student supervised by Prof. Maurizio Falanga
APPROXIMATION OF RELEVANT ELLIPTICAL
INTEGRALS IN THE SCHWARZSCHILD METRIC AND
SOME ASTROPHYSICAL APPLICATIONS
Collaborators:
Maurizio Falanga
Luigi Stella
22-25 September 2015
Monte Porzio Catone
Osservatorio Astronomico di Roma
ASTROPHYSICAL MOTIVATION Discovery of X-­‐ray emission coming from the accreCon disks around the black holes Study how the ma1er around the black holes appears to an observer at infinity connect first simulaCons Luminet 1979 RAY-­‐TRACING METHOD Image of a black hole 28/09/15 CNOC IX -­‐ Vi1orio De Falco 2 1
CHAPTER
INTRODUCTION Numerical simula2on of an accre2on disk placed around a black hole 28/09/15 CNOC IX -­‐ Vi1orio De Falco 3 1.1 Geometrical structure TARGETS b Ÿ LIGHT BENDING (ellip2c integral) z zobs i Ÿ TRAVEL TIME (ellip2c integral) θ BH y=yobs Ÿ GRAVITATIONAL LENSING (ellip2c integral) ϕ u
R x E Ÿ GRAVITATIONAL REDSHIFT (exact) α Ÿ FLUX (derived) Keplerian orbit xobs photon quadrivelocity Uα BH photon emission null geodesics equaCon in the Schwarschild metric kα observer quadrivelocity Eobs
(V ↵ k↵ )obs
g = (1 + z) =
=
Eem
(U ↵ k↵ )em
Vα F =
photon propagaCon observer 28/09/15 CNOC IX -­‐ Vi1orio De Falco Z
⌫o
Z Z
R
I⌫o d⌫o d⌦
'
4 1.2 SimulaCons of the gravitaConal effects Numerical simulaCons showing the gravitaConal effects menConed before: i=30° i=60° i=89° The relaCvisCc effects become stronger and stronger 28/09/15 CNOC IX -­‐ Vi1orio De Falco 5 2
CHAPTER
ELLIPTIC INTEGRALS Kandisky -­‐ Composi2on VIII (1923) 28/09/15 CNOC IX -­‐ Vi1orio De Falco 6 2.1 Light bending (*) Misner, Thorne & Wheeler -­‐ Gravita2on ✓=
Z

1
b2
r2
b
1
r2
R
✓
1
2M
r
◆
Beloborodov (2002) 1
2
dr
Subs2tu2on ad hoc cos ↵
cos ✓
z=1
in r sin ↵
b= q
1 2M
r
(*) considering α small Mathema2cal algebra cos ↵
Beloborodov (2002) (1
cos ✓)(1
28/09/15 u) = 1
cos ↵
where u =
rs
r
CNOC IX -­‐ Vi1orio De Falco 7 2.2 Time delay (*) Misner, Thorne & Wheeler -­‐ Gravita2on t=
Z
1
R
dr
1
2M
r
(
1
b2
r2
✓
1
2M
r
◆
1
2
Poutanen & Beloborodov (2006) )
1
z=1
t/(R/c)
Subs2tu2on ad hoc cos ↵
in r sin ↵
b= q
1 2M
r
(*) considering α small Mathema2cal algebra cos ✓
Poutanen & Beloborodov (2006) 
t
uy uy 2
=y 1+
+
R
8
24
28/09/15 rs
u2 y 2
where u =
r
112
CNOC IX -­‐ Vi1orio De Falco y=
(1 cos ↵)
(1 u)
8 2.3 Solid angle (*) Misner, Thorne & Wheeler -­‐ Gravita2on par2
par1
cos i
d⌦ = 2 2
D sin ✓ 1
2
2M
R
sin ↵
cos ↵
(Z
1
R

dr
1
r2
2
b
r2
✓
1
2M
r
◆
3
2
)
1
zBH Ω zobs BH yBH xBH θ φ xobs 28/09/15 yobs ? no approximated equaCon found so far CNOC IX -­‐ Vi1orio De Falco 9 3
CHAPTER
MATHEMATICAL METHOD M. C. Escher -­‐ Rela2vity (1953) 28/09/15 CNOC IX -­‐ Vi1orio De Falco 10 3.1 Method Ÿ Approximate the ellipCc integrals with polynomial equaCons Z
f (x)dx
P (x)
polynomial ellip2c func2on Ÿ Get rid of the square root present in the integral general func2on sin ↵ = g(z)
z = z(↵)
Ÿ We assume that α is small, so we have: g(z) = sin ↵ ⇡ ↵ ⇡ 0
28/09/15 expand in Taylor series the integrand f(x) CNOC IX -­‐ Vi1orio De Falco 11 3.2 Light bending 28/09/15 CNOC IX -­‐ Vi1orio De Falco 12 Ÿ Having even powers of g(z) and researching a complete polynomial funcCon, we choose: g(z) =
28/09/15 p
Az 2 + Bz
CNOC IX -­‐ Vi1orio De Falco 13 Ÿ To find the approximaCon we compare the approximaCon with the original form in a parCcular compuCng convenience case. We choose u=0 and R=1, so we have: Ÿ We have to get rid of the square root with an even trigonometric funcCon, so we have: 28/09/15 CNOC IX -­‐ Vi1orio De Falco 14 where A =
✓
B
2
◆2
Ÿ TesCng in the parCcular case (u=0 and R=1) we have: 1
cos ↵ =
Bz
2
Ÿ Therefore choosing B = 2 and A = -­‐1, it implies that z = 1-­‐cosα. The final approximaCon is: (1
28/09/15 cos ✓)(1
u) = 1
CNOC IX -­‐ Vi1orio De Falco cos ↵
15 3.3 Time delay t=
Z
1
R
dr
1
2M
r
(
2
1
b
r2
✓
1
2M
r
◆
1
2
)
1
Subs2tu2on obtained rigorously with the mathema2cal method R = 4Rg
1
−0.5
28/09/15 R = 10Rg
0
Performing the same calculaCons made in the case of the light bending 
t
uy uy 2
=y 1+
+
R
8
24
R = 6Rg
2
cos ↵
∆t/(R/c)
z=1
De Falco & Falanga (2015) 0
0.5
cos
ψ
cos ✓
rs
u2 y 2
where u =
r
112
CNOC IX -­‐ Vi1orio De Falco 1
y=
−0.5
0
0.5
cos
ψ
cos ✓
1
−0.5
0
0.5
cos
ψ
cos ✓
(1 cos ↵)
(1 u)
16 1
3.4 Solid angle cos i
d⌦ = 2 2
D sin ✓ 1
2
sin ↵
cos ↵
2M
R
(Z
1
R

dr
1
r2
2
b
r2
1
2M
r
◆
3
2
)
1
De Falco & Falanga (2015) Subs2tu2on obtained rigorously with the mathema2cal method R = 4Rg
R = 10Rg
10
cos ↵
R = 6Rg
0
Performing the same calculaCons made in the case of the light bending 5
dΩ
z=1
✓
0
⇥
d⌦ = cons1 · R · 2z + (1
where 28/09/15 4 3u
C=
1 u
2
2C) z + 1
0.5
cos α
C + 2C
1
0
0.5
cos α
2
39u2 91u + 56
D=
56(1 u)2
CNOC IX -­‐ Vi1orio De Falco 2D z
3
1 −0.5
0
0.5
cos α
⇤
17 1
4
CHAPTER
APPLICATIONS Image of an accre2on disk’s forma2on in a binary system 28/09/15 CNOC IX -­‐ Vi1orio De Falco 18 4.1 Iron line profile De Falco & Falanga (2015) '
i = 30o
1 0
Ÿ Numerical code 28/09/15 Approximate: less than 1 min 0.5
i = 80o
0
Flux
40 millions of points Original: more than 60 min i = 60o
0.5
Flux
1 0
Flux
R
g 4 d⌦
1
F =
0.5
Ÿ Flux formula Z Z
CNOC IX -­‐ Vi1orio De Falco 0.8
0.9
1
E/E0
1.1
1.2
19 4.2 PolarizaCon Melia, Falanga & Goldwurm 2011 Schniaman & Krolik 2010 B
k
°°
b
θγ
z
z’
Bx
αsc
k
°
By
ϕγ
ksc
rsc
i
ψ
k
°
°
ϕ
°
n
y
α
° B
r
°
ksc
°°
x
28/09/15 ISSI -­‐ Vi1orio De Falco 20 5
CHAPTER
CONCLUSIONS ✓ PRESENT WORKS ý PUBLISH THIS WORK ý OPTIMIZE CODES TO CALCULATE THE FLUX AND THE POLARIZATION ✗ FUTURE PROJECTS ☐ EXTENSION TO THE KERR METRIC ☐ DEVELOP FAST NUMERICAL CODES IN THE KERR METRIC 28/09/15 CNOC IX -­‐ Vi1orio De Falco 21 THANK YOU FOR YOUR ATTENTION! 28/09/15 CNOC IX -­‐ Vi1orio De Falco 22