=Paper=
{{Paper
|id=Vol-1746/paper-19
|storemode=property
|title=L2 Point vs. Geosynchronous Orbit for Parallax Effect by Simulations
|pdfUrl=https://ceur-ws.org/Vol-1746/paper-19.pdf
|volume=Vol-1746
|authors=Lindita Hamolli,Mimoza Hafizi
|dblpUrl=https://dblp.org/rec/conf/rtacsit/HamolliH16
}}
==L2 Point vs. Geosynchronous Orbit for Parallax Effect by Simulations==
https://ceur-ws.org/Vol-1746/paper-19.pdf
L2 point vs. geosynchronous orbit for parallax effect by simulations
Lindita Hamolli Mimoza Hafizi
Physics Dept. Physics Dept.
Faculty of Natyral Science Faculty of Natyral Science
lindita.hamolli@fshn.edu.al mimoza.hafizi@fshn.edu.al
radius, RE and u0 is the minimum separation
obtained at the moment of the peak
Abstract magnification t0 [Pac86]. The light curve
obtained by above equation is symmetric
The isolated dark low-mass objects in around the peak and by it can be defined: u0,
our Galaxy, such as free-floating t0 and TE. By them, only TE carries
planets (FFP), can be detected by information about the lens. So, the detection
microlensing observations. By the of the dark objects by gravitational
light curve can be defined three microlensing is usually limited by a well-
parameters, but only the Einstein time, known degeneracy in TE. In principle, a way
TE involves the mass, the distance and to break partially this degeneracy is the
the transverse velocity of the lens. To parallax effect, which can be detected when
break this degeneracy, have to be (i) the event can be observed during the
detected the perturbations in the light Earth’s orbital motion creating a shift relative
curve due to the relative accelerations to the simple straight motion between the
among the observer, the lens and the source and lens or (ii) two observers at
source. Recently, toward Galactic different locations looking contemporarily
bulge are planned space-based towards the same event can compare their
microlensing observations by observations. The first way (i) is possible in
WFIRST, which can be located in L2 the long microlensing events or in short
point or geosynchronous orbit (GSO). microlensing events caused by FFPs when
Using the simulations Monte Carlo in they are observed by the space. These objects
C++ we investigate that the better are discussed more in the scientific
position for the parallax effect community, recently. The formation
detection in microlensing events mechanisms of FFPs remain an open
caused by FFPs is L2 point than GSO. theoretical question in astrophysics. Sumi et
al. [Sum11] examined the data of two years
1. Introduction
microlensing survey from the Microlensing
In the simplest case, when both the lens and Observations in Astrophysics (MOA)
the source can be considered as point-like collaboration and found an excess of short
objects and their relative motion with respect timescale events with duration less than 2
to the observer is assumed to be linear, the days, which is expected to be caused by FFPs
amplification of the source star follows the with mass range 10−5 ÷ 10−2 M ⊙ . By ground-
Paczynski profile, based observations, the deviations in the light
As = [u 2 (t ) + 2] / [u (t ) u 2 (t ) + 4] curve with short timescale, are small and
where u(t) is the separation between the lens generally undetectable. But, the era of space-
and the line of sight in units of Einstein based microlensing observations has already
started with Spitzer and Kepler telescopes, additional observing constraints in GSO due
which will help to detect the short to the Earth and the Moon. The galactic
microlensing events caused by FFPs. Actually coordinates of the WFIRST line of sight are
two other space-based missions are planned b = −1.6 , l = 1.1 , the cadency is 15min and the
for the future as WIRST and Euclid. In this threshold amplification Ath=1.001.
work we are interested for the parallax effect
in microlensing events caused by FFPs. Since 3. The parallax effect
WFIRST telescope is going to observe toward The parallax effect is an anomaly in the
the Galactic bulge and its orbit is still under standard light curve caused due to the
discussion, the main purpose of the paper is to acceleration of the observer around the Sun as
define which location is more suitable for the L2 point or around the Earth center as the
parallax effect, the L2 point or the GSO GSO, which can be observable. Since the
[Spe13]. The plan of the paper is as follows: parameters of these systems and the
in Sec. 2 we present the review of the coordinates of the source star are known, we
WFIRST satellite. In Sec. 3 we show the can calculate precisely the microlensing event
parallax effect induced in light curves. In light curves caused by a FFP as observed by
Sec.4 are shown the main results for the satellite at GSO or L2 point. Following
parallax effect detected by WFIRST located Dominik [Dom98], the trajectory of the
in L2 point and GSO and in Sec. 4 we observer in the L2 point and in GSO can be
summarize our main conclusions. projected onto the lens plane. The coordinates
of the L2 point on the lens plane are:
2. Overview of WFIRST mission
x1 (t ) = ρ{− sin χ cos φ (cos ξ (t ) − ε ) − sin χ sin φ 1 − ε 2 sin ξ (t )}
Here we provide an overview of the WFIRST
x2 (t ) = ρ{− sin φ (cos ξ (t ) − ε ) + cos φ 1 − ε 2 sin ξ (t )}
telescope that is potentially relevant for
monitoring microlensing events and where ρ = 1.01a⊕ (1 − x) / RE is the length of the
combined with the results from Kepler, L2 point orbit semi-major axis projected onto
WFIRST will produce the first statistically the lens plane and measured in RE. Here, a⊕
complete census of exoplanets. WFIRST is is the semi-major axis of the Earth orbit
planned to launch in the mid-2020s into a around the Sun, ε = 0.0167 is the eccentricity
Solar orbit at Sun-Earth L2 or in to a GSO. and ξ is a parameter which is related to the
WFIRST will conduct a ~ 432-day time by t = a⊕3 (ξ − ε sin ξ ) / GM ⊙ [Ham13].
microlensing survey towards the Galactic
bulge divided equally between six 72-day The φ , χ characterize the position of the
seasons. These seasons will be split between source, which give the longitude measured in
the beginning and the end of the mission to the ecliptic plane from the perihelion towards
maximize the ability of WFIRST to measure the Earth motion and the latitude measured
the parallax effect for the detected events from the ecliptic plane towards the northern
[Spe13]. In observations towards the Galactic point of the ecliptic. We find the following
bulge, the Sun angle constraints are the same values for parameters: φ = 167.9 and
at GSO and at L2, but WFIRST suffers from χ = −5.5 . The coordinates on the lens plane
0.2
of GSO satellite, orbiting in the equatorial
M = M⊕
plane around the Earth at the distance 6R⊕ 0.1 T E ≈ 0.25 day
and targeting the Galaxy bulge are:
0.0
x1 (t ) = ρ '{− sin δ cos α cos(ωt + ϕ ) − sin δ sin α cos(ωt + ϕ )}
Residuals
x2 (t ) = ρ '{− sin α cos(ω t + ϕ ) + cos α sin(ωt + ϕ )} -0.1
where ρ ' = 6 R⊕ (1 − x) / RE is the distance of -0.2
the GSO satellite from center of the Earth
-0.3 L2
projected onto the lens plane and measured GSO in heliocentric system
in RE, ω = 2π / P is its angular velocity and ϕ -0.4
GSO
-0.4 -0.2 0.0 0.2 0.4
is the satellite orbital phase with respect to the 0.2
t(day)
vernal equinox. The parameters α , δ are the M = MJ
right ascension and the declination of the 0.1 T E ≈ 4.4 day
WFIRST line of sight. In fact, the GSO 0.0
satellite together with the Earth will orbit
Residuals
around the Sun. So, we consider this case -0.1
calling GSO in heliocentric system (HS). We
-0.2
calculate theoretically the amplification Ap (t )
L2 point
of the light curve with parallax effect and then -0.3
GSO in heliocentric system
defined the residuals between the two curves. -0.4
GSO
-8 -6 -4 -2 0 2 4 6 8
The parallax effect is detectable on a light t(day)
curve when there are at least eight 0.2
consecutive points with residuals Re s > 0.001 . -2
M = 10 M Sun
0.1
T E ≈ 14.4 day
4. Results
0.0
The parallax effect depends on the observer
Residuals
orbit, position of it in orbit in conjunction -0.1
with line of sight [Ham13’] and u0. Firstly, we -0.2
calculate the parallax effect in microlensing
events caused by a FFP with mass: Earth -0.3 L2
GSO in heliocentric system
mass ( M ⊕ ), Jupiter mass ( M J ) and -0.4
GSO
102 M ⊙ when WFIRST satellite is in L2 point, -30 -20 -10 0
t(day)
10 20 30
GSO and GSO in HS. In fig. 1 we have
Fig. 1. The residuals between the light curves
shown the residuals between the light curve detectable and standard curves for an microlensing
with parallax effect and standard curve for event caused by a lens with mass M = M ⊕ (top), M = M J
different mass of the lens at the distance (middle) and M = 10−2 M ⊙ (bottom). The curve are
DL=4kpc, transit velocity vT =50 km/s, the calculated when the WFIRST is at the L2 point (black
source distance DS = 8.5 kpc and u0=0.05. The lines), in the GSO (dashed lines) and when the
period and radius of GSO are considered WFIRST is at the GSO in the HS (gray lines).
P=1day and R = 6 R⊕ . As one can see the microlensing events when the WFIRST
residuals is larger for the WFIRST location in telescope is positioned at L2 point and GSO
L2 point and when the mass of the lens is in HS using Monte Carlo simulations in C++
bigger the oscillations in the residual curves [Pre92]. For each event we extract, five
for GSO and GSO in HS are appeared. In parameters: 1) DL, based on the disk and
figure 2 we have shown the variation of the bulge FFP spatial distributions, [Gil89],
residuals by the mass of the lens when the [Pao01], [Haf04] along the WFIRST line of
WFIRST satellite is positioned at L2 point. sight; 2) M from the mass function
We have considered two values of lens mass: dN / dM ∼ M −α PL , with α PL in [0.9-1.6]
M = M ⊕ and M = M J . It is clear that the [Sum11]; 3) FFP relative transverse velocity
residuals roughly have the same magnitude by the Maxwellian distribution [Han95]; 4)
but the time extensions are different. So, to u0 is randomly extracted from a probability
detect the parallax effect in short events, the
cadency has to be smaller. In figure 3 we have 0 .0
shown the residuals between the light curves
for the lens mass equal to the Earth mass for -0 .1
Residuals
-0 .2
0.05 u 0 = 0 .0 5 L 2 p o in t
u 0 = 0 .5 M = M⊕
0.00 -0 .3
u 0= 1 T E ≈ 0 .2 5 d a y s
-0.05
-0 .4
-0.10 -0 .4 -0 .2 0 .0 0 .2 0 .4
Residuals
t(d a y)
-0.15
0 .1 5
-0.20
u 0 = 0 .0 5 GSO
M⊕ M = M⊕
-0.25 u 0 = 0 .5
MJ 0 .1 0
u 0= 1 T E ≈ 0 .2 5 d a y s
Residuals
-0.30
-0.35
-2 -1 0 1 2 0 .0 5
t(days)
Fig. 2. Simulated light curves for a microlensing event 0 .0 0
caused by a lens with mass M = M ⊕ (black line) and
-0 .4 -0 .2 0 .0 0 .2 0 .4
M = M J (gray line). The curve are calculated when the t(d a y )
WFIRST is in the L2 point. Fig. 3. The residuals between the light curves for a
microlensing event caused by a lens with mass M = M ⊕
three different values of the impact parameter, observed by L2 point (top) and GSO (bottom). The curve
u0 and the observations are taken by the L2 are calculated for different value of impact parameter
point and GSO. It is clear that the u0= 0.05 (black lines), u0 =0.5 (gray lines) and u0 =1
microlensing events with larger amplification (dashed lines).
(smaller u0) have bigger residuals and the distribution uniformly in the interval [0, 6.54]
residuals between curves observed by L2 because Ath = 1.001 ; 5) Ds from the Galactic
point is bigger than GSO. Also we calculate bulge spatial distributions of the stars
the efficiency of the parallax effect in [Haf04].We simulate 1000 microlensing
events for the populations of the FFPs in thin L2 point around the Sun we remark that the
disk, thick disk and bulge then calculate best period for microlensing observation is
theoretically the standard curve, the parallax near the solstice summer [Ham13’].
effect curve and the residuals between them.
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