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Gravitational lensing

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.

https://rodrigonemmen.com/teaching/relatividade-geral-e-aplicacoes-astrofisicas/

Rodrigo Nemmen

November 13, 2018
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  1. AGA0319
    Rodrigo Nemmen
    Gravitational lensing
    Astrophysical Applications of GR II

    View full-size slide

  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,
    which I had made at his request. This note complies
    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
    instruments. Simple geometric considerations lead
    to the expression
    X
    21
    Iq- 212
    $ Y
    \1+ 1
    where
    I= VaoDBo.
    on November 12, 2018
    http://science.sciencemag.org/
    nloaded from
    Einstein 1936 Science
    http://science.sciencemag.org/content/84/2188/506

    View full-size slide

  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,
    which I had made at his request. This note complies
    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
    instruments. Simple geometric considerations lead
    to the expression
    X
    21
    Iq- 212
    $ Y
    \1+ 1
    where
    I= VaoDBo.
    on November 12, 2018
    http://science.sciencemag.org/
    nloaded from
    Einstein 1936 Science
    http://science.sciencemag.org/content/84/2188/506

    View full-size slide

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

    View full-size slide

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

    View full-size slide

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

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

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

    View full-size slide

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

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

    View full-size slide

  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

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  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!

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  13. Einstein angle with a galaxy cluster as the lens

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  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

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

    View full-size slide

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

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  17. Magnification of light with gravitational
    lensing

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  18. Search for massive compact halo objects
    (MACHOs)

    View full-size slide

  19. tempo
    fluxo
    t
    Δt =

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

    View full-size slide

  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

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  21. Github
    Twitter
    Web
    E-mail
    Bitbucket
    Facebook
    Group
    figshare
    [email protected]
    rodrigonemmen.com
    @nemmen
    rsnemmen
    facebook.com/rodrigonemmen
    nemmen
    blackholegroup.org
    bit.ly/2fax2cT

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