Precise determination of the orbital elements of binary stars using ...

Astrophys Space Sci DOI 10.1007/s10509-011-0715-9

O R I G I N A L A RT I C L E

Precise determination of the orbital elements of binary stars using Differential Evolution Abdel-Fattah Attia

Received: 13 February 2011 / Accepted: 19 April 2011 © Springer Science+Business Media B.V. 2011

Abstract This paper presents an ‘adaptive probability of crossover’ technique, as a variation of the differential evolution algorithm (ACDE), for optimal parameter estimation in the general curve-fitting problem. The technique is applied to the determination of orbital elements of a spectroscopic binary system (eta Bootis). In the ACDE, Varying the crossover probability rate (Cr) provides faster convergence than keeping it constant. The Cr is determined for each trial parameter vector (‘individual’) as a function of fit goodness. The adaptation automatically updates control parameter to an appropriate value, without requiring prior knowledge of the relationship between particular parameter settings and a given problem optimization characteristics. The presented analysis of eta Bootis derives best-fitting Keplerian and phasing curves. Error estimation of the optimal parameters is also included. Comparison of the results with previously published values suggests that the ACDE technique has a useful applicability to astrophysical data analysis. Keywords Methods: numerical · Stars: orbital elements

1 Introduction A considerable number of published orbits are not sufficiently well determined to allow the use of orbital elements. A.-F. Attia () Deanship of Scientific Research, King Abdulaziz University, P.O. Box 80230, Jeddah 21589, Saudi Arabia e-mail: [email protected] A.-F. Attia National Research Institute of Astronomy and Geophysics, Helwan, Cairo, Egypt

Some of the main problems encountered when numerical simulations are used for determining orbital parameters are that the parameter space which needs to be searched is often very large, and that the results of each simulation must be compared with the observational data. While methods for automatic comparison of data between observations and simulations were used in some cases (e.g. Engström and Athanassoula 1991), very little were achieved to find an efficient method for reducing the amount of search necessary to find the orbit in a general case. In addition, researchers applied the varieties of the optimization algorithms in order to solve such a problem. The basic idea is to solve an optimization problem by evolving the best solution from an initial set of completely random guesses. An optimization scheme based on differential evolution (DE) can avoid the problems inherent in the traditional approach. Restrictions on the range of the parameter space are imposed only by observational constraints and by the physics of the model. Although the parameter space is often quite large, the DE provides a relatively efficient means of searching globally for the best-fit model. DE used in many branches of science, for instance several optimization algorithms in Sherpa were used by Siemiginowska et al. (2009). They applied these methods to a variety of astronomical data (X-ray spectra, images, timing, optical data, etc.). The global optimization methods such as genetic algorithms and differential evolution recently are used in astrophysics. For examples, solar coronal modeling (Gibson and Charbonneau 1996), pulsar planet searching (Lazio 1997), eclipsing binary stars (Hakala 1995), gamma-ray astronomy (Lang 1995), determining orbital parameters for pairs of interacting galaxies (Wahde 1998) and determining the age and relative contribution of different stellar populations of the barred spiral galaxy NGC 3384 (Attia et al. 2005). For an excellent review of GAs in astronomy and astrophysics,

Astrophys Space Sci

see Charbonneau (1995), and Charbonneau (2002). Attia et al. (2009) determined the orbital elements of the binary star using the modified genetic algorithm AGAPOP. Shahbaz et al. (2002) used differential evolution (DE) the same phase binned outburst and decay light curves of XTE of J2123058. The standard differential evolution was introduced by Storn and Price (1995). Nowadays the DE has become one of the most frequently used evolutionary algorithms solving the global optimization problems. There many research papers published about the adaptation of control parameters in DE. The crossover rate Cr in Omran et al. (2005) is generated for each individual from a normal distribution N (0.5, 0.15), this approach called SDE. Abbass (2002) introduced a self-adaptive Pareto Differential Evolution (SPDE). SPDE is an adaptation of his Pareto differential evolution (PDE) algorithm described in Abbass et al. (2001). Both methods self adapt the crossover and mutation rates. Wenli and Feng (2005) used the idea of Srinivas and Patnaik (1994) for adapting scaling factor and crossover rate in differential evolution based on the fitness function. Qin and Suganthan (2005) proposed self-adaptive differential evolution algorithm (SaDE), where the choice of learning strategy and the two control parameters scaling factor (F ) and crossover rate (Cr) are not required to be pre-defined. During evolution, the suitable learning strategy and parameter settings are gradually self-adapted according to the learning experience. In this paper, adaptive differential evolution (ACDE) will be used for searching the space of possible orbital elements of binary stars. Section 2 provides a brief overview of the DE family of algorithms and describes a recent state-of-the-art version of DE algorithm. The proposed differential evolution (ACDE) is discussed and presented in Sect. 3. Section 4 explains the observations and data reduction and how the radial velocity is determined by knowing the orbital elements. Section 5 contains a description of the problem. Section 6 explains the implementation details of ACDE for determining the orbital elements of binary stars. The simulation results and discussions are discussed in Sect. 8. The conclusion and some references for further readings are presented.

1 , . . . , x D }, i = 1, . . . , NP towards the global opti{xi,G i,G mum. The initial population should better cover the entire search space as much as possible by uniformly randomizing individuals within the search space constrained by the prescribed minimum and maximum parameter bounds 1 , . . . , x D } and X 1 D Xmin = {xmin max = {xmax , . . . , xmax }. For min th example, the initial value of the j parameter in the i th individual at the generation G = 0 is generated by:   j j j j xi,0 = xmin + rand (0, 1) . xmax − xmin ,

j = 1, 2, . . . , D

Where rand (0, 1) represents a uniformly distributed random variable within the range [0, 1]. 2.1 Mutation operation The mutation operator produce mutant vector with respect to each individual Xi,G , so-called target vector. For each target vector at the generation G, its associated mutant vector 1 , . . . , v D } can be generated via certain mutaVi,G = {vi,G i,G tion strategy1 are listed as follows (Qin et al. 2009): • DE/rand/1

  Vi,G = Xr i ,G + F. Xr i ,G − Xr i ,G 1

DE shares a common terminology of selection, crossover, and mutation operators with GA. However it is the application of these operators that make DE different from GA. Whereas in GA crossover plays a significant role, it is the mutation operator which affects the working of DE (Karaboga and Koyuncu 2005). DE Algorithm working as follows (Brest et al. 2006): DE Algorithm aim at evolving a population of NP for a D-dimensional parameter vectors, so-called individuals which encode the candidate solutions, i.e., Xi,G =

2

3

(2)

• DE/best/1

  Vi,G = Xbest,G + F. Xr i ,G − Xr i ,G 2

3

(3)

• DE/rand-to-best/1

  Vi,G = Xi,G + F. Xbest,G − Xi,G   + F. Xr i ,G − Xr i ,G 2

3

(4)

• DE/best/2

  Vi,G = Xbest,G + F. Xr i ,G − Xr i ,G 1 2   + F. Xr i ,G − Xr i ,G 3

• DE/rand/2 2 Differential evolution algorithm

(1)

Vi,G = X i

r1 ,G

4

(5)

  + F. Xr i ,G − Xr i ,G 2



+ F. Xr i ,G − Xr i ,G 4

5



3

(6)

The five indices r1i , r2i , r3i , r4i , r5i are mutually randomly generated within the range [1, NP], which are different from the index i. The scaling factor F is a positive control parameter from interval [0, 2]. A good initial guess is the interval [0.5, 1], e.g. 0.8 (Storn and Price 1997). 1 Publicly available online at http://www.icsi.berkeley.edu/~storn/code.

html.

Astrophys Space Sci Fig. 1 Illustration of the crossover process for D = 6 parameters

2.2 Crossover operation

3 Adaptive probability of crossover in differential evolution (ACDE)

In order to increase the diversity of the perturbed parameter vectors, crossover is introduced (Zhang and Sanderson 1 , . . . , v D }and the tar2007). The mutant vector Vi,G = {vi,G i,G get vector Xi,G are used in recombination to build the trail vector Ui,G = {u1i,G , . . . , uD i,G }:

Srinivas and Patnaik (1994) and (1996) modified probability of crossover, mutation in Genetic Algorithm optimization method to prevent sticking at a local maximum based on the various fitness of the population. They determined probability of crossover, Pc in every generation for each chromosome as a function of its fitness; where Pc has a value that corresponds to half population size (NP). Probability of crossover is adapted in proportion to the fitness maximum and fitness mean. This idea will be further enhanced and modified to control crossover rate Cr of differential evolution as follows:  k1 (fmax − f )/(fmax − f¯), fmax ≥ f¯ and Cr ≤ 1.0 Cr = fmax < f¯ k2 ,

 j Ui,G

=

j

if (randj ≤ Cr) or (j = jrand )

j

otherwise

vi,G , xi,G ,

j = 1, 2, . . . , D

, (7)

Cr is the third control parameter of DE that decides in a comparison with a random number randj ∈ [0, 1] which components are copied from V i,G or Xi,G , respectively. At least one component is taken from V i,G by randomly choosing jrand ∈ {0, D} for each individual, so Ui,G = Xi,G is ensured. Figure 1 gives an example of the crossover mechanism for 6-dimensional vectors. 2.3 Selection operation Selection is the step to choose the vector between the target vector and the trail vector with the aim of creating an individual for the next generation. In single objective optimization the comparison is based on the objective function values, and the vector yielding the smaller result is kept for the next generation (for minimization problems) (Tvrdk 2002).

(8) Where: f is the current fitness and is equal to the inverse of objective function (f = 1/χ 2 in the application problem), fmax is the maximum of the fitness values of the solutions in every population, and f¯ is the mean of fitness. The values of k1 , and k2 lie in the range [0, 1]. The recommended value for k1 is 0.5, and for k2 is 0.8. The adaptation policy in ACDE is different from all the approaches described in Qin and Suganthan (2005), Zhang and Sanderson (2008) and Tvrdk et al. (2002). Cr is determined for each individual as a function of its fitness. One of the goals of ACDE approach is to prevent DE from getting stuck at a local optimum. To achieve this goal, solutions with sub-average fitness is employed to the region containing global optimum in the search space.

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4 Observations and reduction of data The relatively frequent occurrence of spectroscopic binaries and their interesting apparent distribution with spectral type, demonstrate their importance in studies of stellar origin and evolution and of the origin of double stars. Assignment of variable velocity depends on a number of factors: the number and quality of lines measured on the spectrograph, the number and quality of plates available, the stability of spectrograph, and the observed velocity range. This paper uses the radial velocity variation observed in the spectroscopically visible component of the spectroscopic binary star η Bootis, MK type G0 IV, where the variability of its radial velocity was discovered by Moore (1905). A combination of astrometry information with the spectroscopic results for η Bootis explains the invisibility of the companion. Data from (Bertiau 1957); including observations from both Lick observatories in the interval (1897–1947) and Ottawa observatory in the interval (1907–1910), are used to derive accurate spectroscopic elements for the system. Systematic corrections are tabulated to be applied to radial velocities from different observatories. These corrections are derived to reduce all velocities to the system of the Mills three-prism observations made at Lick observatory. The choice of the Lick system as a standard due to its reliability established by the large number of radial velocities obtained directly with the Mills spectrograph, and also because of the care made by its staff in defining and maintaining the velocity system (Basic Astronomical Data 1963). A systematic correction of +1.43 km/sec was used to make the Ottawa velocities consistent with the Lick observations, while some observations are totally rejected due to poor plate quality (Bertiau 1957). η Bootis data are considered to offer a moderately difficult global optimization problem. These elements represent the parameters to be determined using the proposed ACDE algorithm as follows: p, the orbital period, τ , the time of perihelion passage (zero point of the orbit),  , the longitude of perihelion, e, the orbital eccentricity, K, the orbital velocity amplitude and Vo the system’s radial velocity. By knowing the values of the above parameters the expected radial velocity variations can be computed and compared to observations. Determining the radial velocity variations associated with the motion of a binary component in an arbitrarily positioned elliptical orbit about the common center of mass is carried out by using the following equations (9) to (12): V (t) = Vo + K(cos(ν(t) +  ) + e cos ω)

(9)

where K is the velocity amplitude and its magnitude is obtained by the following equation: na sin i K=√ 1 − e2

(10)

and n = 2π p , i is the inclination angle of the orbital plan with respect to the plan of the sky. a is the major axis, e is the eccentricity. The true anomaly ν is related to the eccentric anomaly E via the relation given by the following equation:  E 1+e ν tan (11) tan = 2 1−e 2 The eccentric anomaly is related to time via Kepler’s equation by the following equation E − e sin(E) =

2π (t − τ ) p

(12)

Going over the expressions (9) to (12) the six orbital elements to relate the radial velocity curve can be identified. Equation (12) is a transcendental in E and is solved by the bisection method for nonlinear root derived in Attia et al. (2009).

5 Problem description The orbital elements of the spectroscopic binary star η Bootis are grouped in one vector x = (p, τ, , e, K, Vo ) that minimizes the χ 2 value. Given N data points Vjobs ≡ V (tj ) with associated error estimates σj . the χ 2 is the objective function value that will be calculated for both Julian Date and Phase time as follows: 2 χphase (x) =

N −1 obs 1  Vj (phase) − V (phase; x) 2 N −6 σj (phase) j =0

(13) χJ2ulian Date (x) =

N −1 obs 1  Vj (Julian Date) − V (Julian Date; x) 2 N −6 σj (Julian Date) j =0

(14) 1 χ 2 (x) = (χJ2ulian Date (x) + χJ2ulian P hase (x)) 2

(15)

The aim is to minimize the objective function in (15) to give the best fit Phase and keplerian curves. The search domains of the parameters to be determined are mainly the boundaries of the orbital elements of the spectroscopic binary star η Bootis (see Table 1). The objective function of ACDE is introduced in (15). The position vector of each individual in ACDE consists of a vector of control variables such as p, the orbital period, τ , the time of perihelion passage (zero point of the orbit),  , the longitude of perihelion, e, the orbital eccentricity, K, the orbital velocity amplitude and Vo the system’s radial

Astrophys Space Sci Table 1

Vector x

Parameter

Lower boundary

Upper boundary

Dimension

x(1)

P

200

800

JD

x(2)

τ

to

to + P

JD

x(3)



0

2π

Radian

x(4)

e

0

1

Dimensionless

x(5)

K

0

max(Vjobs ) − min(Vjobs )

km/s

Vo

min(Vjobs )

max(Vjobs )

km/s

x(6)

velocity. The orbital elements of the spectroscopic binary star η Bootis are grouped in one vector: x = (p, τ, , e, K, Vo )

(16)

6 Implementation of ACDE to determine the orbital elements of the star η Bootis The optimal orbital elements of the spectroscopic binary star η Bootis based on ACDE, can be described in the following steps: Step 1: Read the observation data, reduction data and parameters of ACDE. Step 2: Initialize the positions of individuals (vectors) in the search space randomly and uniformly. Set the generation counter G = 0. Step 3: Calculate the fitness value of the initial individual through (9–16). The initial position of each individual is evaluated by the objective function. The vector x in (16) is used to generate an orbit to fit the given data set. The initial best individual among the population is achieved. Step 4: Let G = G + 1. Step 5: Update each individual position with the updating strategy: For every vector in the population, find difference vector and mutation vector as explained previously in mutation operation. Form crossover vector based on crossover operation and ACDE modification as in (8) and achieve trail vector based on ACDE approach. Step 6: Calculate evaluation values of the new individuals by χ 2 and objective function in (15). Step 7: Update the best individual of the last generation with current best individual. Step 8: Go to step 4 until a criterion is met, usually a sufficiently good evaluation value or a maximum number of generations. Figure 2 shows the flowchart of ACDE algorithm for determining the optimal orbital elements of binary star η Bootis of MK type G0 IV.

Fig. 2 Flowchart of ACDE algorithm for determining the optimal orbital elements

7 Simulation results and discussion The aim is to find the best Phase and keplerian curves to fit the given data in Bertiau (1957). The following subsections will show the results achieved by the proposed ACDE

Astrophys Space Sci Table 2 Comparison results (Bertiau 1957; Charbonneau 2002) and AGA-POP (Attia et al. 2009)

x

Bertiau (1957) LSE

Charbonneau (2002) SGA-PIKAIA

Attia et al. (2009) AGA

Attia et al. (2009) AGA-POP

Present work ACDE

P τ  e K Vo Generations POP Size End POP Time (min) χ 2 (Phase) χ 2 (Data) χ2

494.73 2428136.19 326.33 0.2575 8.42 1.01 – – – – 6.8667 9.2001 7.8640

494.2 2414299 326.86 0.2626 8.3836 1.0026 – – – – 6.8153 7.0358 6.9238

494.08 2414301.76 326.94 0.27175 8.4125 0.99554 3000 1000 1000 70.56 6.8005 7.1019 6.9479

494.2 2414298.52 326.81 0.26904 8.4132 1.06819 3000 1000 67 10.7 6.8254 7.0274 6.9249

494.153 2414300 326.95 0.25991 8.2914 1.0026 30 200 200 1 min & 6 sec 4.6598 4.8293 4.7446

Fig. 3 Phase plot: (a) Observed data and fitting curve; (b) (O–C) curve

approach. Whenever possible, comparisons carried out with various published techniques (Bertiau 1957; Charbonneau

2002; Attia et al. 2009). Table 2 shows a comparison between the values of different parameters of the x vector

Astrophys Space Sci Fig. 4 Keplerian plot (13000:19000 JD): (a) Observed data and fitting curve; (b) (O–C) curve

achieved from each technique, also different parameters of the technique itself are compared such as: maximum allowed number of generations (Generations), starting population size (POP Size), final population size (End POP), and time consumed by each technique (Time). 7.1 The phase plot The phase plot is shown in Fig. 3. Part (a) shows the given N data points used to plot the radial velocity versus the phase, while the continues curve is the one generated by the ACDE solution to best fit the N -point data. Substitutions in the previously indicated equations are done with the vector (P , τ, ω, e, K, Vo ) = ( 494.153, 2414300, 327.86, 0.2526, 8.1836, and 1.0026) which resulted from the ACDE solution. Figure 3, part (b) shows the (O–C) curve which plots the difference between the generated fitting curve and the discrete points to be fitted (error bars are also shown in Fig. 3). 7.2 The Keplerian plot Part (a) in both Figs. 4 and 5 shows the (N − 1) data points plotted with error bars. Radial velocity is plotted versus

Table 3 Results of velocity difference for different algorithms Algorithm

Velocity difference

Least Squares (LSE)

0.355036

SGA-PIKIA

0.04967

AGA

0.075701

AGA-POP

0.043038

ACDE

0.0328

the Julian Date. All data given is plotted except for only one datum at Julian Date = 2423175 for continues drawing purposes. It was used in different calculations. While the continues curve is the one generated by the ACDE solution to best fit the N data points, substitutions in the previously indicated equations are done with the vector [P , τ, ω, e, K, Vo ] = [494.153, 2414300, 327.86, 0.2526, 8.1836, and 1.0026] result from the ACDE solution. The asymmetrical curve of the orbit is obtained due to the nonzero eccentricity (e = 0.2526). As the value of the eccentricity decrease approaching zero, the non-symmetric shape fade away till a purely sinusoidal radial velocity variation is

Astrophys Space Sci Fig. 5 Keplerian plot (27000:33000 JD): (a) Observed data and fitting curve; (b) (O–C) curve

Fig. 7 χ 2 decreases vs number of generations with best χ 2 = 4.6564 Fig. 6 Velocity difference of the ACDE solution

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Fig. 8 χ 2 isocontours in four hyperplanes of parameter space (χ 2 = 4.7446)

achieved for e = 0 (circular orbit). Part (b) shows the (O–C) curve which plots the difference between the generated fitting curve and the discrete points to be fitted. Error bars are also shown in Fig. 4b.

the Julian Date index V (JulianDatej ) at the same j instant. These two values are the same which is simply Vjobs . Therefore, for completely data-fitted phase and Keplerian curves we have:

7.3 The radial velocity Given N -row data set (tj , Vjobs and σj ) where j ∈ [1, N], the radial velocity V (tj ) at any instant j can be generated as a function of the phase index V (Phasej ) or as a function of

V (Phasej ) = V (JulianDatej )

or:

V (Phasej ) − V (JulianDatej ) = 0

(17)

Astrophys Space Sci

Fig. 9 χ 2 isocontours in four hyperplanes of parameter space (χP2 H ASE = 4.8293)

Equation (17) represents the ideal case and therefore, one may define the general case to be: Velocity Difference =

N −1 1  |V (Phasej ) − V (JulianDatej )| N

(18)

j =0

The pre-defined Velocity Difference is to be minimized. The values of the Velocity Difference obtained by ACDE approach are plotted in Fig. 6. Table 3 shows the numerical

comparison of the Velocity Difference values of different techniques. The worst (highest) value = (0.355036) corresponds to the least squares method (LSE), while the best (lowest) value = (0.0328) corresponds to the ACDE approach as shown in Fig. 6. Figure 7 shows convergence curve namely the χ 2 value for the best individual (orbital elements of the binary star) as a function of generation index. At the early generations, χ 2 started high where the population have diversity in chromosomes and accordingly in both χ 2 and operator probabili-

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2 Fig. 10 χ 2 isocontours in four hyperplanes of parameter space (χDATA = 4.6598)

ties values. At the final generations almost all chromosomes have converged to the best solution (optimal orbital elements reached) and, hence χ 2 stabilized at best value 4.6564. 7.4 Error analysis To calculate the error estimates, we perturb best fit solution vector x = (P , τ, ω, e, K, Vo ) and compute the χ 2 of this perturbed solution using ACDE objective function. In relatively low-dimensionality parameter spaces (such as for our

orbital fitting problem), it is often even simpler to just construct a hypercube centered about the optimal fit and directly compute χ 2 (x) at some preset spatial resolution across the cube (see Charbonneau 2002). This is shown in Fig. 8 which illustrates χ 2 isocontours, with the optimal solution indicated by the intersection of the dotted lines. Error estimation of the χ 2 isocontours for the orbital elements in case of Phase time and Julian date, are shown in Figs. 9 and 10; respectively.

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8 Conclusion An adaptive crossover rate in differential evolution ACDE was developed for precise determination of the orbital elements of binary stars. In adaptive differential evolution, the probability of crossover (Cr) is varied depending on the fitness values of the solutions. The method addresses the problem from the statistical point of view. The application of the ACDE promises to yield accurate results for determining optimal orbital elements of the star η Bootis of MK type G0 IV. Comparison with conventional and heuristic techniques verified the performance capability of differential evolutionary algorithm for determining the orbital elements. The error estimation for orbital elements is described. Simulation results of the ACDE approach depict a reliable and promising tool as a global optimization method. A good compromise between precision and speed of the algorithm was achieved. The global optimum in shorter computation time than all other reported approaches was reached, this means decreasing time complexity. Acknowledgements The author greatly thanks Prof. Al-Turki Y.A. and Dr. Abusorrah A. at King Abdulaziz University for their useful suggestions. Also to an anonymous referee for very useful comments that improved the presentation of the paper.

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