Microlensing effects and structure of gravitational lens systems
A study of gravitational microlensing of distant objects is presented. We performed simulations of light curves and trajectories of the image centroid of an extended source in the Chang–Refsdal lens with shear and continual dark matter. Various brightness distributions over the source (Gaussian, pow...
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| Zitieren: | Microlensing effects and structure of gravitational lens systems / E.V. Fedorova, A.N. Alexandrov, V.I. Zhdanov // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 276-278. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-796582015-04-04T03:01:54Z Microlensing effects and structure of gravitational lens systems Fedorova, E.V. Alexandrov, A.N. Zhdanov, V.I. MS3: Physics of Stars and Galaxies A study of gravitational microlensing of distant objects is presented. We performed simulations of light curves and trajectories of the image centroid of an extended source in the Chang–Refsdal lens with shear and continual dark matter. Various brightness distributions over the source (Gaussian, power-law, Shakura–Synyaev accretion disc) have been studied. We considered in detail approximate relations and corresponding algorithms used to fit observational data on high amplification events (HAE). The results are applied to interpretation of HAE observed by OGLE and GLITP groups. The source size and caustic crossing moment are estimated from these data, however, the determination of the brightness profile is statistically not reliable. 2005 Article Microlensing effects and structure of gravitational lens systems / E.V. Fedorova, A.N. Alexandrov, V.I. Zhdanov // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 276-278. — Бібліогр.: 5 назв. — англ. 0233-7665 http://dspace.nbuv.gov.ua/handle/123456789/79658 en Кинематика и физика небесных тел Головна астрономічна обсерваторія НАН України |
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MS3: Physics of Stars and Galaxies MS3: Physics of Stars and Galaxies Fedorova, E.V. Alexandrov, A.N. Zhdanov, V.I. Microlensing effects and structure of gravitational lens systems Кинематика и физика небесных тел |
| description |
A study of gravitational microlensing of distant objects is presented. We performed simulations of light curves and trajectories of the image centroid of an extended source in the Chang–Refsdal lens with shear and continual dark matter. Various brightness distributions over the source (Gaussian, power-law, Shakura–Synyaev accretion disc) have been studied. We considered in detail approximate relations and corresponding algorithms used to fit observational data on high amplification events (HAE). The results are applied to interpretation of HAE observed by OGLE and GLITP groups. The source size and caustic crossing moment are estimated from these data, however, the determination of the brightness profile is statistically not reliable. |
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Article |
| author |
Fedorova, E.V. Alexandrov, A.N. Zhdanov, V.I. |
| author_facet |
Fedorova, E.V. Alexandrov, A.N. Zhdanov, V.I. |
| author_sort |
Fedorova, E.V. |
| title |
Microlensing effects and structure of gravitational lens systems |
| title_short |
Microlensing effects and structure of gravitational lens systems |
| title_full |
Microlensing effects and structure of gravitational lens systems |
| title_fullStr |
Microlensing effects and structure of gravitational lens systems |
| title_full_unstemmed |
Microlensing effects and structure of gravitational lens systems |
| title_sort |
microlensing effects and structure of gravitational lens systems |
| publisher |
Головна астрономічна обсерваторія НАН України |
| publishDate |
2005 |
| topic_facet |
MS3: Physics of Stars and Galaxies |
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http://dspace.nbuv.gov.ua/handle/123456789/79658 |
| citation_txt |
Microlensing effects and structure of gravitational lens systems / E.V. Fedorova, A.N. Alexandrov, V.I. Zhdanov // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 276-278. — Бібліогр.: 5 назв. — англ. |
| series |
Кинематика и физика небесных тел |
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AT fedorovaev microlensingeffectsandstructureofgravitationallenssystems AT alexandrovan microlensingeffectsandstructureofgravitationallenssystems AT zhdanovvi microlensingeffectsandstructureofgravitationallenssystems |
| first_indexed |
2025-07-06T03:40:56Z |
| last_indexed |
2025-07-06T03:40:56Z |
| _version_ |
1836867394316795904 |
| fulltext |
MICROLENSING EFFECTS AND STRUCTURE
OF GRAVITATIONAL LENS SYSTEMS
E. V. Fedorova, A. N. Alexandrov, V. I. Zhdanov
Astronomical Observatory, National Taras Shevchenko University of Kyiv
3 Observatorna Str., 04053 Kyiv, Ukraine
e-mail: elena f@mail.univ.kiev.ua
A study of gravitational microlensing of distant objects is presented. We performed simulations of
light curves and trajectories of the image centroid of an extended source in the Chang–Refsdal lens
with shear and continual dark matter. Various brightness distributions over the source (Gaussian,
power-law, Shakura–Synyaev accretion disc) have been studied. We considered in detail approxi-
mate relations and corresponding algorithms used to fit observational data on high amplification
events (HAE). The results are applied to interpretation of HAE observed by OGLE and GLITP
groups. The source size and caustic crossing moment are estimated from these data, however,
the determination of the brightness profile is statistically not reliable.
1. Gravitational microlensing may provide valuable information on masses of microlensing stars and structure
of distant sources. To obtain more detailed information about the source from microlensing processes a long-term
continuous monitoring is needed. Therefore, a considerable attention is paid to analysis of high amplification
events (HAE) that usually may be described by a small number of fitting parameters. This requires a detailed
investigation of the reliability of fitting procedures, which use some asymptotic relations for the source crossings
of a caustic of the gravitational lens system (GLS). In this connection, we have investigated the accuracy
of asymptotic relations used to treat caustic crossing microlensing events and model dependence of source
parameters that are obtained using existing observational data.
2. We performed simulations of magnifications and trajectories of the image centroid (IC) of an extended
source in the Chang–Refsdal lens with shear and convergence due to continuous dark matter. Various brightness
distributions over the source have been studied: Gaussian, power-law (PL)
FR(r) =
(p − 1)
πR2[1 + (r/R)2]p
,
and Shakura–Synyaev accretion disc (SSAD)
FR(r) =
3Rθ(r − R)
2πr3
(
1 −
√
R
r
)
,
where R is a characteristic source size. We show that astrometric microlensing effects may be helpful to reveal
important details of the GLS local structure, providing that astrometric accuracy of image centroid positioning
will reach the microarcsecond level. In particular, in case of a small source the occurrence of rapid jumps of
the image centroid in HAE seems to be an indicator of caustic crossing at the fold point. For more detailed
presentation see [2, 4].
3. We have studied in detail the approximate relations and corresponding algorithms used to fit observational
data on HAE in rather a general situation. The simplest approximation used to treat HAE is an approximation
of linear caustic, which is correct only in a very small area near the caustic. This may be used to obtain
the amplification coefficient of the extended source intensity only if its size is small as compared with the caustic
curvature radius. To modify the relation for this coefficient, we must expand the lens mapping (ξ, η) → (x, y)
in the vicinity of a critical point ξ = η = 0 up to the fourth order:
c© E. V. Fedorova, A. N. Alexandrov, V. I. Zhdanov, 2004
276
x = 2ξ + a(ξ2 − η2) + 2bξη + c(ξ3 − 3ξη2) − d(η3 − 3ξ2η) + e(η4 − 6ξ2η2 + ξ4) − 4f(ξη3 − ξ3η),
y = b(ξ2 − η2) − 2aξη + d(ξ3 − 3ξη2) + c(η3 − 3ξ2η) + f(η4 − 6ξ2η2 + η4) + 4e(ξη3 − ξ3η).
We restrict ourselves to an harmonicity condition, which means the absence of continuous matter on the line
of sight: ∂x/∂ξ+∂y/∂η = 2. Then, we suppose that the critical point is a fold; this means that b �=0. In this
case we derived the relation for the brightness amplification of the Gaussian source on the caustic as follows
K =
1
4
√
π|b|R
{
Γ(1/4) + Γ(3/4)
R
|b|
[
5(a2 + b2)
3a2 + 2b2
8b2
+
15c(c− 2a2)
8b2
+
(
3f
2b
− 15ad
4b
)]}
.
The expression in the square brackets defines the first non-zeroth correction to the linear caustic approxi-
mation. In case of, e.g., Q2237+0305 GLS, the source size is ∼ 0.1RE and applicability of the linear caustic
approximation seems to be marginal. Four additional parameters (as compared to linear caustic) must be de-
fined if we want to increase the accuracy and this is difficult because they are related with an effect of roughly
the same order as the measurement accuracy.
4. Taking into account simulations of microlensing light curves in a simple lens models we derived following
conclusions: (i) only a small area near the caustic (< 0.2RE) can be used to fit correct results; (ii) moment of
caustic crossing is defined rather well and it is almost independent on the brightness distribution over the source.
If the simulated model is known, then the fitting yields the source size with an accuracy of about 10–15%, in
case of unknown model, the accuracy is about 15–30%.
5. We have used our fitting routine to simulate observational light curves of the OGLE (Optical Gravita-
tional Lens Experiment) [5] and GLITP (Gravitational Lens International Time Project) [1] groups of HAEs in
the Q2237+0305 “Einstein Cross” GLS. We show that the existing data do not allow one to choose a concrete
brightness distribution for a source.
For size estimation, we used the rms radius Rrms and half-brightness radius R1/2 that are related more
directly to the observational quantities than the parameter R. It is supposed that
Rrms =
[∫
r2P (x, y)dxdy
/∫
P (x, y)dxdy
]1/2
and R1/2 = R∗/
√
ln 2,
where R∗ is defined by the condition
R∗∫
0
FR(r)rdr =
1
2
∞∫
0
FR(r)rdr.
In case of, e.g., the PL distribution Rrms = R/(p−2)1/2 (p > 2); R1/2 = R
√(
21/(p−1) − 1
)/
ln 2. The Gaussian
distribution is the limiting case of the PL one with a fixed Rrms and p → ∞; in this case R = Rrms = R1/2.
We have estimated the characteristic source size R, half-brightness radius R1/2 and rms radius for different
indexes p for the PL distribution, as well as for Gaussian and SSAD. However, within the least squares method
procedure, no preference can be given to any of these distributions. Some of the results are shown in Table 1
in terms of characteristic times T1/2 (T1/2 = R1/2/V , V is the normal velocity of the caustic with respect to
the source). The errors have been determined from a statistical simulation.
Figure 1 shows a model dependence for the fitting parameters derived from observational data. The data
were fitted using the PL brightness distribution with different p (TR = R/V ; Trms = Rrms/V ).
In summary, our numerical simulations show that for a correct determination of source parameters one must
use these relations only in a small area near the caustic (less than 0.1–0.2 of typical lens size RE). The source
size and caustic crossing moment can be estimated from these data, however, the determination of the brightness
profile appears to be statistically not reliable. Moreover, there is a considerable model dependence of the source
size derived from the above data. For more details see [3].
Acknowledgements. This work is supported by the Science and Technology Center in Ukraine (Project
NN43).
277
Table 1. Q2237+030 source crossing times estimated from HAE observed by OGLE and GLITP groups
Initial data, Image T1/2, R1/2 · v/300,
source model (band) day 10−3 pc km s−1
GLITP, Gaussian A (R) 38+44
−17 0.34+0.4
−0.15
GLITP, PL, p = 3/2 A (R) 80+36
−22 0.71+0.34
−0.19
GLITP, SSAD A (R) 58+6
−17 0.52+0.05
−0.16
GLITP, Gaussian A (V ) 29+25
−10 0.26+0.23
−0.09
GLITP, PL, p = 3/2 A (V ) 70+17
−39 0.63+0.16
−0.34
GLITP, SSAD A (V ) 52 ± 12 0.46 ± 0.11
OGLE, Gaussian A (V ) 40+17
−15 0.37+0.16
−0.14
OGLE, PL, p = 3/2 A (V ) 75 ± 51 0.67 ± 0.39
OGLE, SSAD A (V ) 58+12
−17 0.52+0.11
−0.16
OGLE, Gaussian C (V ) 42+21
−20 0.38+0.19
−0.18
OGLE, PL, p = 3/2 C (V ) 81+34
−24 0.73+0.30
−0.22
OGLE, SSAD C (V ) 70+17
−29 0.64+0.16
−0.26
Figure 1. Source sizes (days) derived from GLITP data for different values of the exponent p in the PL brightness
distribution. The black lines describe the V band, the grey ones – the R band
[1] Alcalde D., Mediavilla E., Moreau O., et al. QSO 2237+0305 VR Light Curves from Gravitational LensES Inter-
national Time Project Optical Monitoring // Astrophys. J.–2002.–572, N 2.–P. 729–734.
[2] Fedorova E. V., Zhdanov V. I., Alexandrov A. N. Motion of source image in Chang–Refsdal lens // J. Phys.
Studies.–2002.–6, N 4.–P. 465–468.
[3] Fedorova E. V., Alexandrov A. N., Zhdanov V. I. Source models in gravitational lens system Q2237+030 and high
amplification events.–Preprint of the Main Astronomical Observatory, GAO–04–1R.–2004.–22 p. (in Russian).
[4] Zhdanov V. I., Salata S. A., Fedorova E. V. Background-Field Effects in Astrometric Microlensing // Astron.
Lett.–2001.–27, N 9.–P. 562–567.
[5] Wozniak P. R., Udalski A., Szimansky M., et al. The Optical Gravitational Lensing Experiment: A Hunt for
Caustic Crossings in QSO 2237+0305 // Astrophys. J.–2000.–540, N 2.–P. L65–L67.
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