56

# Gravitational lensing

Lecture of the course "General Relativity With Astrophysical Applications", taught by Prof. Rodrigo Nemmen (USP).

• Einstein ring and angle
• Derivation scheme for Einstein angle
• Einstein angle inside Our Galaxy
• Einstein angle for galaxy clusters
• Lensing and the search for MACHOs

Credit for the slides/figures belongs to Rodrigo Nemmen, unless otherwise stated. ## Rodrigo Nemmen

November 13, 2018

## Transcript

1. AGA0319
Rodrigo Nemmen
Gravitational lensing
Astrophysical Applications of GR II

2. alterations are being made. The date for the reopen-
ing of the building has not yet been set, but it will
probably be early in the coming year.
north of San Francisco. Fort Ross, the chief Russian
port and settlement on the Californian coast, is about
sixty miles north of San Francisco.
DISCUSSION
LENS-LIKE ACTION OF A STAR BY THE
DEVIATION OF LIGHT IN THE
GRAVITATIONAL FIELD
SOME time ago, R. W. Mandl paid me a visit and
asked me to publish the results of a little calculation,
with his wish.
The light coming from a star A traverses the gravi-
tational field of another star B, whose radius is BR.
Let there be an observer at a distance D from B and
at a distance x, small compared with D, from the ex-
tended central line AB. According to the general
theory of relativity, let a., be the deviation of the light
ray passing the star B at a distance BR from its center.
For the sake of simplicity, let us assume that AB
is large, compared with the distance D of the observer
from the deviating star B. We also neglect the eclipse
(geometrical obscuration) by the star B, which indeed
is negligible in all practically important cases. To
permit this, D has to be very large compared to the
radius B, of the deviating star.
It follows from the law of deviation that an observer
situated exactly on the extension of the central line
AB will perceive, instead of a point-like star A, a
luminius circle of the angular radius , around the
center of B, where
Bio-
It su bo D
t
It should be noted that this angular diameter ,B does
not decrease like 1/D, but like 1/VD, as the distance
D increases.
Of course, there is no hope of observing this phe-
nomenon directly. First, we shall scarcely ever ap-
proach closely enough to such a central line. Second,
the angle ,B will defy the resolving power of our
instruments. For, axO being of the order of magnitude
of one second of arc, the angle RO/D, under which the
deviating star B is seen, is much smaller. Therefore,
the light coming from the luminous circle can not be
distinguished by an observer as geometrically different
from that coming from the star B, but simply will
manifest itself as increased apparent brightness of B.
The same will happen, if the observer is situated at
a small distance x from the extended central line AB.
But then the observer will see A as two point-like
light-sources, which are deviated from the true geo-
metrical position of A by the angle ,3, approximately.
The apparent brightness of A will be increased by
the lens-like action of the gravitational field of B in
the ratio q. This q will be considerably larger than
unity only if x is so small that the observed positions
of A and B coincide, within the resolving power of our
to the expression
X
21
Iq- 212
\$ Y
\1+ 1
where
I= VaoDBo.
on November 12, 2018
http://science.sciencemag.org/
Einstein 1936 Science
http://science.sciencemag.org/content/84/2188/506

3. alterations are being made. The date for the reopen-
ing of the building has not yet been set, but it will
probably be early in the coming year.
north of San Francisco. Fort Ross, the chief Russian
port and settlement on the Californian coast, is about
sixty miles north of San Francisco.
DISCUSSION
LENS-LIKE ACTION OF A STAR BY THE
DEVIATION OF LIGHT IN THE
GRAVITATIONAL FIELD
SOME time ago, R. W. Mandl paid me a visit and
asked me to publish the results of a little calculation,
with his wish.
The light coming from a star A traverses the gravi-
tational field of another star B, whose radius is BR.
Let there be an observer at a distance D from B and
at a distance x, small compared with D, from the ex-
tended central line AB. According to the general
theory of relativity, let a., be the deviation of the light
ray passing the star B at a distance BR from its center.
For the sake of simplicity, let us assume that AB
is large, compared with the distance D of the observer
from the deviating star B. We also neglect the eclipse
(geometrical obscuration) by the star B, which indeed
is negligible in all practically important cases. To
permit this, D has to be very large compared to the
radius B, of the deviating star.
It follows from the law of deviation that an observer
situated exactly on the extension of the central line
AB will perceive, instead of a point-like star A, a
luminius circle of the angular radius , around the
center of B, where
Bio-
It su bo D
t
It should be noted that this angular diameter ,B does
not decrease like 1/D, but like 1/VD, as the distance
D increases.
Of course, there is no hope of observing this phe-
nomenon directly. First, we shall scarcely ever ap-
proach closely enough to such a central line. Second,
the angle ,B will defy the resolving power of our
instruments. For, axO being of the order of magnitude
of one second of arc, the angle RO/D, under which the
deviating star B is seen, is much smaller. Therefore,
the light coming from the luminous circle can not be
distinguished by an observer as geometrically different
from that coming from the star B, but simply will
manifest itself as increased apparent brightness of B.
The same will happen, if the observer is situated at
a small distance x from the extended central line AB.
But then the observer will see A as two point-like
light-sources, which are deviated from the true geo-
metrical position of A by the angle ,3, approximately.
The apparent brightness of A will be increased by
the lens-like action of the gravitational field of B in
the ratio q. This q will be considerably larger than
unity only if x is so small that the observed positions
of A and B coincide, within the resolving power of our
to the expression
X
21
Iq- 212
\$ Y
\1+ 1
where
I= VaoDBo.
on November 12, 2018
http://science.sciencemag.org/
Einstein 1936 Science
http://science.sciencemag.org/content/84/2188/506

4. https://oneminuteastronomer.com/9237/gravitational-lens/

5. https://oneminuteastronomer.com/9237/gravitational-lens/

6. Observer would see
ring of light
“Einstein ring”
image of background object
θE

7. Derivation of Einstein angle
lens
source
observer
dS
dLS
dL
M

8. Derivation of Einstein angle
lens
source
observer
dS
dLS
dL
fake apparent
position
perfect Einstein ring

9. Derivation of Einstein angle
lens
source
observer
Δϕdef
=
4GM
c2 b
dS
dLS
dL

10. Derivation of Einstein angle
lens
source
observer
θE
Δϕdef
=
4GM
c2 b
dS
dLS
dL

11. Derivation of Einstein angle
lens
source
observer
θE
dS
dLS
dL
θ
E
≡ 2R
S
(
d
LS
dS
dL )
1/2
Einstein angle
fake apparent
position

12. Einstein angle within the galaxy
M = MSun
dS = 10 kpc = 1017 km
θ
E
≈ 10−3 arcsec
(
M
1M⊙ )
1/2
(
d
50kpc )
−1/2
lens
source
observer
TOO
SMALL!

13. Einstein angle with a galaxy cluster as the lens

14. d
L = 500 Mpc
d
S = 1000 Mpc
Einstein angle with a galaxy cluster as the lens
θ
E
≈ 0.5 arcmin
(
M
1014M⊙ )
1/2
(
d
1000Mpc )
−1/2
CAN
RESOLVE
WITH
TELESCOPES

15. Abell 2218
z=0.18
d = 770 Mpc
http://hubblesite.org/newscenter/archive/releases/2001/32/image/b/

16. http://apod.nasa.gov/apod/ap111221.html
Einstein ring with a galaxy as the lens

17. Magniﬁcation of light with gravitational
lensing

18. Search for massive compact halo objects
(MACHOs)

19. tempo
ﬂuxo
t
Δt =

E
2v
≈ 90days
(
M
1M⊙ )
1/2
(
v
200kms−1 )
−1

20. Conclusões de busca por MACHOs
Não há muitos objetos com M < 0.08
Msolar (anãs marrons), poucos eventos com
Δt curto)
<20% da massa do halo pode estar na forma de
MACHOs
tipicamente MMACHO > 0.15 Msolar (Δt > 35 dias)
∴ MACHOs não explicam ME, pois a
massa predominante do Halo deve ser na forma
de uma componente distribuída uniformemente
No halo escuro da nossa Galáxia

21. Github
Web
E-mail
Bitbucket