Disentangling of Stellar Spectra

Abstract

The method of disentangling of stellar spectra and its applications is briefly reviewed and its complementarity with the interpretation of other observational methods like photometry and interferometry is discussed. Special attention is given to the comparison with Doppler tomography of interacting binaries and to the Bayesian estimate of errors.

Appendix: Bayesian Estimation of Parameters Errors

The errors of parameters of a stellar system determined by disentangling of spectra can be estimated using Bayesian statistics. The errors are caused mainly by the noise in the observed spectra but they are also influenced by the phase distribution of the observations and by the sensitivity of the available data to a particular parameter. Usually we “know” in advance only that the values of parameters p can be expected in some “reasonable” regions (e.g., hundreds of km/s for orbital RVs, days or from hours to months for the orbital period, the line strengths and widths corresponding to the spectral types and rotational broadening, etc.). If previous studies of the same system exist their results can be taken as a more specific limitation of the possible range of p (and a first estimate of the new solution), which we want to verify or improve using a set of new data. If the old data are available they can be included together with the new ones into the new solution, otherwise the reliability of the old results must be estimated and they can be included into the new solution as constraints to the solution in the space of parameters p. Having the set of new observations, i.e. the spectra F(x), we can find the “posterior” probability P( p | F) of p agreeing with the new data according to Bayes’ theorem

$$\displaystyle{ P(\,p\vert F) = \frac{P(F\vert p)P(\,p)} {P(F)} \;, }$$

(7.15)

where P( p) is the “prior” probability of p (i.e. either a smooth characteristic of the wide range of “reasonable” values or the constraint resulting from the old data), P(F | p) is the “liability” that for a chosen value of p the particular shape of F(x) will be detected and P(F) is a general probability do detect the signal F(x). The best new solution p _max can be then defined as the maximum of P( p | F) in the p-space or we can find a mean value

$$\displaystyle{ \langle p\rangle =\int pP(\,p\vert F)dp }$$

(7.16)

(if P( p | F) is normalised to unity, i.e., ∫ P( p | F)dp = 1). In both cases the result depends on our choice of measures in the p space. The errors of resulting parameters p can be estimated from the behaviour of P( p | F) around the found solution. Because p is generally multidimensional (and different dimensions can be of different nature), it is not sufficient to attribute some error bar to each component of p separately as it is a common habit. If P( p | F) is sufficiently smooth around its maximum, it can be expanded into a Taylor series up to quadratic terms in variation of δ p and the non-diagonal components of the corresponding quadratic form determine the correlation of the parameters. Generally, however, there may be several local maxima in the p-space, which may be treated as different solutions.¹ The errors of these different solutions must be then determined from a local behaviour of P( p | F). Generally, the information about the studied system is more completely characterised by mapping of P( p | F) than by a simple list of found values of p and their errors.

7.1.1 Errors of Line Strengths and Radial Velocities

We shall illustrate the calculation of P( p | F) first on a toy model of measurements of one spectrum of a single star. Let us assume that the observed spectrum F(x) is given by

$$\displaystyle{ F(x) = f(x,p) +\delta F(x)\;, }$$

(7.17)

where δ F(x) is a random observational noise. We fit F(x) by a model f(x, p) dependent on a parameter (or a set of parameters) p by minimising the residual error

$$\displaystyle{ S(\,p) =\int _{D}(F(x) - f(x,p))^{2}dx }$$

(7.18)

integrated over the whole observed region D of x. The equation(s) for p thus reads

$$\displaystyle{ 0 = \frac{\partial S(\,p)} {\partial p} = 2\int \frac{\partial f(x,p)} {\partial p} (\,f(x,p) - F(x))dx\;. }$$

(7.19)

The random fluctuations δ F in the observed signal F(x) blur this condition and result in a deviation δ p of the solution from its correct value. These variations are related by the condition

$$\displaystyle{ \delta p \frac{\partial } {\partial p}\int \frac{\partial f(x,p)} {\partial p} (\,f(x,p) - F(x))dx =\int \frac{\partial f(x,p)} {\partial p} \delta F(x)dx\;. }$$

(7.20)

For example, let the spectrum be rectified, i.e. normalised to the continuum, which means f = 1 −φ, where φ(x) corresponds to the spectral lines. We shall assume first that the only unknown free parameter p is the strength of lines, i.e., the model has the form

$$\displaystyle{ f(x,p) = 1 - p\varphi _{1}(x)\;, }$$

(7.21)

where φ ₁ is a pattern of the line-profile(s) imprinted into the observed spectrum F(x) with an unknown amplitude (e.g., due to uncertainty in element abundances, due to a contamination of the signal by light of another star or due to instrumental error in subtracting dark signal). Equation (7.19) is then a simple linear equation with solution

$$\displaystyle{ p = \frac{\int (1 - F(x))\varphi _{1}(x)dx} {\int \varphi _{1}^{2}(x)dx} }$$

(7.22)

and in agreement with Eq. (7.20) its variations are given by

$$\displaystyle{ \delta p = -\frac{\int \varphi _{1}(x)\delta F(x)dx} {\int \varphi _{1}^{2}(x)dx} \;. }$$

(7.23)

The integrals in these equations are actually summations over K individual pixels (each one of size Δ _x  = D∕K) in real observations,

$$\displaystyle\begin{array}{rcl} \int \varphi _{1}(x)\delta F(x)dx& =& \varDelta _{x}\sum _{i=1}^{K}\varphi _{ 1}(x_{i})\delta F(x_{i})\;,{}\end{array}$$

(7.24)

$$\displaystyle\begin{array}{rcl} \int \varphi _{1}^{2}(x)dx& =& \varDelta _{ x}\sum _{i=1}^{K}\varphi _{ 1}^{2}(x_{ i})\;.{}\end{array}$$

(7.25)

Let us assume that the probability distribution of the noise δ F of the signal F(x) in each pixel can be approximated as a Gaussian with standard deviation σ, i.e., the probability of its value δ F in one pixel x _i is

$$\displaystyle{ P(\delta F(x_{i})) \simeq \exp (-\frac{\delta F(x_{i})^{2}} {2\sigma ^{2}} ) }$$

(7.26)

and, hence, the statistical mean value is 〈δ F ²〉 = σ ². In addition, the noise in different pixels is supposed to be statistically independent, i.e.,

$$\displaystyle{ \langle \delta F(x_{i})\delta F(x_{j})\rangle =\sigma ^{2}\delta _{ ij}\;. }$$

(7.27)

The probability distribution of error δ p is then also Gaussian² with

$$\displaystyle{ \langle \delta p^{2}\rangle = \frac{\varDelta _{x}^{2}} {\left [\int \varphi _{1}^{2}(x)dx\right ]^{2}}\langle [\sum _{i=1}^{K}\varphi _{ 1}(x_{i})\delta F(x_{i})]^{2}\rangle = \frac{\varDelta _{x}\sigma ^{2}} {\int \varphi _{1}^{2}(x)dx}\;. }$$

(7.28)

The squared error of p thus decreases inversely proportionally with the number of pixels covering the profile φ(x). It can be seen from Eqs. (7.22) and (7.23) that the strength p and its uncertainty δ p are predominantly determined by the parts of the spectrum where the lines φ ₁ of the model are deep.

If the unknown free parameter p is the Doppler shift of the spectrum, the model has the form

$$\displaystyle{ f(x) = 1 -\varphi _{0}(x - p)\;, }$$

(7.29)

where φ ₀(x) is a model line-profile in laboratory wavelength scale. The residual noise

$$\displaystyle{ S(\,p) =\int (\varphi _{0}(x - p) -\varphi (x) +\delta F(x))^{2}dx }$$

(7.30)

is then no more a simple quadratic function of p and Eq. (7.19), which reads now

$$\displaystyle{ 0 = \frac{\partial S(\,p)} {\partial p} = -2\int \frac{d\varphi _{0}(x - p)} {dx} (\varphi _{0}(x - p) -\varphi (x) +\delta F(x))dx, }$$

(7.31)

may have several solutions for p corresponding to coincidences of some improperly identified observed lines with wrong lines in the model. The uncertainty of p in each of these solutions can be estimated from the depth and width of the local minimum of the residual noise. Equation (7.20) reads now

$$\displaystyle{ \delta p\int \left ( \frac{d\varphi _{0}} {dx}\right )^{2}dx =\int \frac{d\varphi _{0}} {dx}\delta F(x)dx\;, }$$

(7.32)

where we have skipped the term $\frac{d^{2}\varphi _{ 0}} {dx^{2}} (\varphi _{0}-\varphi )$ which should vanish at the correct value p = p ₀ where φ = φ ₀ (this need not be true at a false minimum). This equation shows that the error δ p of RV is dominated by the noise δ F in the steepest parts of the line profile. Analogously to Eq. (7.28), the mean squared value of this error is now

$$\displaystyle{ \langle \delta p^{2}\rangle = \frac{\varDelta _{x}\sigma ^{2}} {\int (\frac{d\varphi _{0}} {dx})^{2}dx}\;. }$$

(7.33)

The difference between Eqs. (7.28) and (7.33) is due to the fact that the fit of line strengths is most sensitive to the part of the spectrum where the line is deepest, while the Doppler shift is most sensitive to the wings where the line profile is steepest.

7.1.2 Errors in Multidimensional Space of Parameters

We can optimise the fit of the observed spectrum simultaneously with respect to the line-strength, Doppler shift and some additional parameters. Generally, if the model spectrum φ ₀(x, p) is a function of several parameters p _i  |  _i = 1 ^m, we get for them a set of m generally non-linear equations for p

$$\displaystyle{ 0 =\int \frac{\partial \varphi _{0}} {\partial p_{i}}(1 - F(x) -\varphi _{0}(x,p))dx\;. }$$

(7.34)

Linearising, we arrive at the set of equations

$$\displaystyle{ M_{ij}\delta p_{j} =\delta R_{i} }$$

(7.35)

for variations δ p _j caused by variations δ F(x) of the observed spectrum. The matrix M _ij is given here by

$$\displaystyle{ M_{ij} \equiv \int \left ( \frac{\partial \varphi _{0}} {\partial p_{i}} \frac{\partial \varphi _{0}} {\partial p_{j}} - \frac{\partial ^{2}\varphi _{0}} {\partial p_{i}\partial p_{j}}(1 - F(x) -\varphi _{0}(x,p))\right )dx }$$

(7.36)

and the right-hand side by

$$\displaystyle{ \delta R_{i} \equiv -\int \frac{\partial \varphi _{0}} {\partial p_{i}}\delta F(x)dx =\varDelta _{x}\sum _{k=1}^{K}\frac{\partial \varphi _{0}(x_{k})} {\partial p_{i}} \delta F(x_{k})\;. }$$

(7.37)

Depending on the form of the model φ ₀, some variation δ F can be compensated by a change of different p _i , so that their variations may be correlated

$$\displaystyle{ \langle \delta p_{j}\delta p_{k}\rangle = M_{ji}^{-1}M_{ kl}^{-1}\langle \delta R_{ i}\delta R_{l}\rangle \;. }$$

(7.38)

Assuming that the noise δ F(x) is statistically independent at different pixels x _k , x _l , we have

$$\displaystyle\begin{array}{rcl} \langle \delta R_{i}\delta R_{l}\rangle & =& \varDelta _{x}^{2}\sum _{ k,l=1}^{K}\frac{\partial \varphi _{0}(x_{k})} {\partial p_{i}} \frac{\partial \varphi _{0}(x_{l})} {\partial p_{j}} \langle \delta F(x_{k})\delta F(x_{l})\rangle = \\ & =& \varDelta _{x}\int \frac{\partial \varphi _{0}} {\partial p_{i}} \frac{\partial \varphi _{0}} {\partial p_{j}}\langle \delta F(x)^{2}\rangle dx\;. {}\end{array}$$

(7.39)

If the noise 〈δ F(x)²〉 = σ ² is statistically the same in all pixels and if the second term in the integral in Eq. (7.36) is negligible compared to the first one (which is the case if the fit is good and F(x) ≃ 1 −φ ₀(x, p)), then

$$\displaystyle{ \langle \delta R_{i}\delta R_{l}\rangle =\varDelta _{x}\sigma ^{2}M_{ il} }$$

(7.40)

and

$$\displaystyle{ \langle \delta p_{j}\delta p_{k}\rangle =\varDelta _{x}\sigma ^{2}M_{ jk}^{-1}\;. }$$

(7.41)

As an example, let us investigate a fit by a simple Gaussian profile

$$\displaystyle{ f(x,p) = 1 -\varphi _{0}(x,p) = 1 - p_{1}\exp \left (-\frac{(x - p_{3})^{2}} {p_{2}^{2}} \right )\;. }$$

(7.42)

Equation (7.34) then reads

$$\displaystyle{ 0 =\int (x - p_{3})^{k}\varphi _{ 0}(1 - F -\varphi _{0})dx\;, }$$

(7.43)

where k = 0, 2, 1 in conditions for p ₁, p ₂, p ₃, resp. The variation of φ ₀ reads

$$\displaystyle{ \delta \varphi _{0} = \left ( \frac{1} {p_{1}}\delta p_{1} + 2\frac{(x - p_{3})^{2}} {p_{2}^{3}} \delta p_{2} + 2\frac{x - p_{3}} {p_{2}^{2}} \delta p_{3}\right )\varphi _{0}\;, }$$

(7.44)

so that neglecting the second term in Eq. (7.36), the matrix M has the form

$$\displaystyle{ M_{ij} \simeq \int \frac{\partial \varphi _{0}} {\partial p_{i}} \frac{\partial \varphi _{0}} {\partial p_{j}}dx = \sqrt{ \frac{\pi } {2}}\left (\begin{array}{ccc} p_{2} & \frac{p_{1}} {2} & 0 \\ \frac{p_{1}} {2} & \frac{3p_{1}^{2}} {4p_{2}} & 0 \\ 0 & 0 &\frac{p_{1}^{2}} {p_{2}} \end{array} \right ) }$$

(7.45)

and hence according to Eq. (7.41) the correlation matrix of parameter errors reads

$$\displaystyle{ \langle \delta p_{j}\delta p_{k}\rangle \simeq \sqrt{\frac{2} {\pi }} \varDelta _{x}\sigma ^{2}\left (\begin{array}{ccc} \frac{3} {2p_{2}} & - \frac{1} {p_{1}} & 0 \\ - \frac{1} {p_{1}} & \frac{2p_{2}} {p_{1}^{2}} & 0 \\ 0 & 0 & \frac{p_{2}} {p_{1}^{2}} \end{array} \right )\;. }$$

(7.46)

It means that the errors of the depth p ₁ and the width p ₂ of the line-profile given by Eq. (7.42), which are due to the part of the perturbation δ F symmetric with respect to the centre of the line, are anti-correlated, while the error in the position p ₃ of the line centre is not correlated with them and its squared value is half of that of the width.

Fig. 7.5

Gaussian line-profile (red line) with a simulated noise (black line) and its best fit (blue line; see text for a detailed description)

Results of a Monte Carlo simulation of this example can be seen in Fig. 7.5. The red line shows the profile (cf. Eq. (7.42)) for p ₁ = 0. 2, p ₂ = 0. 2 and p ₃ = 0. 0. The black line has added a randomly generated noise with σ = 0. 04 in 1024 pixels in the displayed interval of x ∈ (−1, +1), i.e., Δ _x  = 2⁻⁹ ≃ 0. 00195. The blue line gives the best fit to this particular simulated measurement, which was found for values of parameters p ₁ ≃ 0. 2002, p ₂ ≃ 0. 2085 and p ₃ = 0. 0029. The figure included in the bottom left corner shows the scatter of p ₁ and p ₂ around their true values for 1000 different choices of δ F (the drawn parts of coordinates correspond to ± 0. 01). Similarly the histogram in the right corner shows the distribution of p ₃ (the width of each histogram column is 0.001). If the noise σ is decreased then in agreement with Eq. (7.38) or its approximation (Eq. (7.41)) also the uncertainty 〈δ p _j δ p _k 〉 of the parameters p is reduced. This agrees with the Bayes theorem (Eq. (7.15)) according to which the probability P( p | F) of a larger difference of the true value of p from its best fit to particular data F(x) decreases with decreasing probability P(F | p) that the random noise will mimic a wrong spectrum.

For real observed spectra we do not know the exact value of σ (which can only be estimated from the level of signal integrated during the exposure), while we may have a rough estimate only of Δ _x and, generally, we do not know the explicit form of the matrix M _ij and its inversion. However, we can estimate these values from the value S ₀ of the residuum S( p) in its minimum and from variations of S(p) with respect to variations of p. If the spectrum is correctly fitted by a proper model, the residuum S ₀ should be given by the noise³ only, i.e.,

$$\displaystyle{ S_{0} \simeq \int \delta F^{2}(x)dx = D\sigma ^{2} = K\varDelta _{ x}\sigma ^{2}\;. }$$

(7.47)

The value of Δ _x and hence for known D also the number K of statistically independent pixels of noise can be estimated from auto-correlation of the residual noise. In our example, the numerical simulation results in S ₀∕Δ _x  ≃ 1. 73, giving σ ≃ 0. 0411 in good agreement with the value 0.04 for which the noise was generated. The residuum S( p ₃) is drawn by the green line in Fig. 7.5 as a function of p ₃ for p _1, 2 fixed to their best values (the zero level is shifted to the bottom of the panel for S( p ₃)). Its behaviour can be estimated substituting φ ₀ from Eqs. (7.42) into (7.30) also for φ,

$$\displaystyle\begin{array}{rcl} S(\,p_{3})& \simeq & \int (\varphi _{0}(x - p_{3}) -\varphi _{0}(x) +\delta F(x))^{2}dx \\ & \simeq & \int \delta F^{2}(x)dx +\int \varphi _{ 0}^{2}(x)dx +\int \varphi _{ 0}^{2}(x - p_{ 3})dx - 2\int \varphi _{0}(x - p_{3})\varphi _{0}(x)dx \\ & =& S_{0} + \sqrt{2\pi }p_{1}^{2}p_{ 2}\left (1 -\exp (-\frac{p_{3}^{2}} {2p_{2}^{2}})\right )\;. {}\end{array}$$

(7.48)

The residual S( p ₃) thus increases with the square of δ p ₃ around its minimum

$$\displaystyle{ S(\delta p_{3}) \simeq S_{0} + \sqrt{ \frac{\pi } {2}} \frac{p_{1}^{2}} {p_{2}} \delta p_{3}^{2}\;, }$$

(7.49)

but it approaches saturation at level $S_{\infty } = S_{0} + \sqrt{2\pi }p_{1}^{2}p_{2}$ for large δ p ₃ (in agreement with numerical results S _∞ ∕Δ _x  ≃ 12. 5 in our example). This saturation corresponds to the possibility that the line visible in the spectrum is actually a random fluctuation of the noise, while the real line is in the distant wavelength δ p ₃, but it is hidden in another noise fluctuation. The probability of this “hidden” solution is proportional to the width D of the wavelength interval. This may be in principle infinite, so that the probability of this solution may be high, even for a small value of σ. The limitation of D to the interval (−1, +1) in our example has a character of the prior P( p ₃) which is chosen equal to zero out of this interval.

Regarding the assumption of the Gaussian noise (7.26), the liability P(δ F | p) of the noise δ F(x _i ) |  _i = 1 ^K in all pixels is

$$\displaystyle{ P(\delta F\vert p) \simeq \exp \left (-\frac{\sum _{i}\delta F(x_{i})^{2}} {2\sigma ^{2}} \right ) =\exp \left (-\frac{S_{0}} {2\varDelta _{x}\sigma ^{2}}\right )\;. }$$

(7.50)

Following Bayes’ theorem (Eq. (7.15)), the probability of p for known F (and a ’flat’ prior P( p)) reads

$$\displaystyle{ P(\,p\vert F) \simeq \exp \left (-\frac{S(\,p)} {2\varDelta _{x}\sigma ^{2}} \right ) \simeq \exp \left (-\frac{KS(\,p)} {2S_{0}} \right )\;. }$$

(7.51)

We thus obtain from Eq. (7.49) the probability distribution of radial velocity δ p ₃ of the “visible” line

$$\displaystyle{ P(\delta p_{3}\vert F) \simeq \exp \left (-\frac{KS(\delta p_{3})} {2S_{0}} \right ) \sim \exp \left (-\sqrt{ \frac{\pi } {2}} \frac{p_{1}^{2}} {2\varDelta _{x}\sigma ^{2}p_{2}}\right ), }$$

(7.52)

which yields, according to the definition (Eq. (7.16)), a mean value of squared velocity shift

$$\displaystyle{ \langle \delta p_{3}^{2}\rangle = \frac{\int \delta p_{3}^{2}P(\delta p_{ 3}\vert F)d\delta p_{3}} {\int P(\delta p_{3}\vert F)d\delta p_{3}} \simeq \sqrt{\frac{2} {\pi }} \frac{\varDelta _{x}\sigma ^{2}p_{2}} {p_{1}^{2}}, }$$

(7.53)

in agreement with component { j, k} = { 3, 3} of Eq. (7.46).

It is worth noting that the error $\sqrt{ \langle \delta p_{3}^{2}\rangle }$ on the radial velocity p ₃ is proportional to the ratio σ∕p ₁ of the noise to the line depth. Regarding the problem of desirable S/N-ratio mentioned in Sect. 7.2.4, we can see from this relation that there is no limitation. However, the higher the S/N-ratio is, the more precise is the determination of the disentangled parameters. From the observational and instrumental point of views, the S/N of spectra refers to the ratio of the noise to the overall spectral flux, i.e. the signal means the continuum. However, it is obvious that the signal which yields an information about RV and other spectral features is the modulation of the flux by spectral lines. It is thus desirable to optimise in the observations the ratio of line depths to the (photon) noise. The RV error is also proportional to the square root of the line width p ₂ which indicates that a higher precision can be achieved from sharper lines, although they are covered by smaller number of pixels and are thus more influenced by fluctuations in photon counts.

In the disentangling technique, we do not have explicitly given in Eq. (7.17) the template spectrum f(x, p), like in Eqs. (7.21) or (7.42), but a more complicated expression like the right-hand side of Eqs. (7.1) or (7.9). The component spectra F _j (x) whose imprints we want to find in the observations are now also unknown parameters that can be retrieved with some errors only. In analogy to Eq. (7.28), this error is proportional to the noise σ of the input spectra. However, usually the separation of the component spectra is overdetermined, as explained in Sect. 7.2.1 and hence, following the relations in the Footnote 2, the amplitude of the noise of the disentangled spectra decreases inversely proportionally to the square root of the number of observed spectra. The dependence of the model f(x, p) on nonlinear parameters p in the broadening functions Δ _j (x, t; p) is more complicated and generally cannot be expressed explicitly like in the above given toy models. However, it is possible to map the distribution of the residual noise numerically as a function of these parameters.