# Computing radial-velocities from Cross Correlation data

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How can I better fit a Gaussian curve to a CCF so that I get the most precise RV value? The image below shows the fitting where I compared the fitting by weighting by the uncertainties and not. There is not a big difference between them because the errors are nearly the same for all data points.

The RV is the $$mu$$ and the uncertainty was gotten from the covariance matrix first time in the diagonal. Performing a similar procedure but for summing several orders (such as fig. below) gives me the RV time-series which has too much spread. I need to find a way to reduce the noise as much as possible. I am not showing the errors in RV time-series because I thought it to be coming from the covariance matrix but it looks unrealistic (too big).

Why not fit the orders you are interested in separately and then use the standard error of the mean (possibly weighted by the signal-to-noise in each CCF) as the precision in the final, averaged RV.

In terms of what to fit, I don't see why a Gaussian is so bad? You probably need to limit the fit to the inner $$sim pm 1 sigma$$ to avoid noise outside the peak pulling the fit one way or another. Other options are to just use the numerical centroid (but this may be affected by asymmetry) or you could use a sinc function to the central region (sometimes a better model of the peak of a CCF).

Edit: You haven't followed my advice, which was to limit the Gaussian fit to $$pm 1$$ sigma from the peak (do it iteratively). At the moment, the "wings" of the Gaussian are just adding noise.

As a rule of thumb though, you are not going to do much better than $$2.2sigma$$ divided by the signal-to-noise ratio. It looks like your $$sigma sim 2$$ km/s and your signal-to-noise ratio is about 30, so I don't see how you can have gotten a scatter as small as you have from data like this? And the graph below contains no points with the value $$-10.76$$. There is still something missing from your question.

## Title: Radial velocities of southern visual multiple stars

High-resolution spectra of visual multiple stars were taken in 2008–2009 to detect or confirm spectroscopic subsystems and to determine their orbits. Radial velocities of 93 late-type stars belonging to visual multiple systems were measured by numerical cross-correlation. We provide the individual velocities, the width, and the amplitude of the Gaussians that approximate the correlations. The new information on the multiple systems resulting from these data is discussed. We discovered double-lined binaries in HD 41742B, HD 56593C, and HD 122613AB, confirmed several other known subsystems, and constrained the existence of subsystems in some visual binaries where both components turned out to have similar velocities. The orbits of double-lined subsystems with periods of 148 and 13 days are computed for HD 104471 Aa,Ab and HD 210349 Aa,Ab, respectively. We estimate individual magnitudes and masses of the components in these triple systems and update the outer orbit of HD 104471 AB.

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## 1. Introduction

Binary stars are ubiquitous throughout the galaxy and an important source of astrophysical parameters. Photometric and spectroscopic studies of eclipsing binaries, in particular, reveal fundamental stellar parameters such as masses and radii that inform our understanding of stars and constrain stellar evolutionary models. The frequency of binaries and their properties also serve as testbeds for star-formation theories, essential for constraining current theoretical models of stellar and planetary formation alike. Known statistics of the field population for solar-type stars indicate that at least 40% are binaries, with

12% in higher order multiples (Raghavan et al. 2010). Observational evidence, however, has shown that many such binaries are in fact triples (Eggleton et al. 2007 Tokovinin 2014a, 2014b), especially short-period binaries with separations comparable to the stellar radii (Tokovinin et al. 2006). This prevalence of tertiary companions orbiting close binaries has strong implications for star-formation mechanisms because the protostellar radii would be too large to fit inside their present-day orbits (Rappaport et al. 2013). Theoretical studies, however, suggest that the presence of a third star can cause large eccentricity excitations to the inner orbit causing tidal forces to shrink and circularize the inner orbit via the eccentric Kozai–Lidov mechanism (Fabrycky & Tremaine 2007 Naoz 2016). This mechanism has also been proposed for planet migration, specifically to explain the presence of hot Jupiters with eccentric and misaligned orbits, as distant stellar or planetary companions with highly inclined orbits can perturb the planetary orbit and cause it to decay (Naoz et al. 2012).

One of several methods to find such triple-star systems involves long-term monitoring of binary eclipses for periodic perturbations caused by the presence of a third star. The nearly continuous photometry of over 150,000 stars and more than 2000 eclipsing binaries (Kirk et al. 2016) collected by NASA's Kepler mission (Borucki et al. 2010) created an ideal data set for identifying eclipse timing variations, and has resulted in the discovery of hundreds of triple star candidates (Gies et al. 2012 Rappaport et al. 2013 Conroy et al. 2014 Gies et al. 2015 Borkovits et al. 2016).

In Gies et al. (2012, 2015), we reported eclipse timing variations for a subset of 41 eclipsing binaries chosen to optimize the chances of discovery of a third body in the system and enable follow-up ground-based spectroscopy. In total, we identified seven probable triple systems and seven additional systems that may be triples with orbits longer than the Kepler baseline (Gies et al. 2015). Subsequently, we have completed a large set of spectroscopic observations of this sample in order to determine spectroscopic orbits, estimate stellar properties, compare with evolutionary codes (Matson et al. 2016), and explore pulsational properties (Guo et al. 2016, 2017a, 2017b) of the component stars.

Of the 41 eclipsing binaries selected for eclipse timing analysis via Kepler, approximately two-thirds were reported only recently to be eclipsing based on automated variability surveys such as the Hungarian-made Automated Telescope Network (HATnet), whose goal is to detect transiting extrasolar planets using small-aperture robotic telescopes (Hartman et al. 2004), and the All Sky Automated Survey (ASAS) which monitors V-band variability among stars brighter than 14th magnitude (Pigulski et al. 2009). Most of the remaining binaries have been known since prior epochs, but typically have little more than times of eclipse minima and orbital ephemerides published.

To characterize this set of eclipsing binaries further and derive spectroscopic orbital elements, we collected an average of 11 ground-based optical spectra per binary. Ideally, when measuring radial velocities, high-resolution spectra and complete phase coverage of the orbit are desired. However, moderate resolution ( ) optical spectra in the wavelength range of 3930–4600 Å provided a high density of astrophysically important atomic lines and molecular bands (traditionally used for stellar classification) that allowed us to derive accurate radial velocities of intermediate-mass (

1–5 ) stars. In addition, the ephemerides determined in the eclipse timing analysis (Gies et al. 2015) enabled us to concentrate our observations during velocity extrema to best constrain the spectroscopic orbits with a modest number of spectra.

We discuss our observations in Section 2, followed by the determination of radial velocities and orbital parameters in Section 3. Discussion of the radial velocity results, mass ratio trends, and suspected triple systems is given in Section 4. Finally, a brief summary of our results is given in Section 5.

## Title: Finding proto-spectroscopic binaries: Precise multi-epoch radial velocities of 7 protostars in rho-Ophiuchus

60000) multi-epoch radial velocity survey of 7 YSO in the star forming region (SFR) rho-Ophiuchus. The radial velocities of each source were derived using a two-dimensional cross-correlation function, using the zero-point established by the Earth's atmosphere as reference. More than 14 spectral lines in the CO (0-2) bandhead window were used in the cross-correlation against LTE atmospheric models to compute the final results. We found that the spectra of the protostars in our sample agree well with the predicted stellar photospheric profiles, indicating that the radial velocities derived are indeed of stellar nature. Three of the targets analyzed exhibit large radial velocity variations during the three observation epochs. These objects - pending further confirmation and orbital characteristics - may become the first evidence for proto-spectroscopic binaries, and will provide important constraints on their formation. Our preliminary binary fraction (BF) of

71% (when merging our results with those of previous studies) is in line with the notion that multiplicity is very high at young ages and therefore a byproduct of star formation

## 36-Inch Telescope

The telescope was built in 1951-55 by the now-defunct firm of Sir Howard Grubb, Parsons & Co. at Newcastle-upon Tyne. It replaced a much older telescope of the same aperture, which was brought to Cambridge from South Kensington when the Solar Physics Observatory moved here in 1913. That telescope was returned to its owners (The Science Museum) before the new one was installed the Director of the Observatories at the time (Professor R.O. Redman), who in his youth had made substantial use of the old telescope, always averred that it should never have left the Museum!

The 36-inch, which is thought to be the largest telescope in the country, has three possible focal stations. There is a prime focus with a focal ratio of f/4.5 the primary mirror is a paraboloid, so no corrector is needed to obtain good images on the optical axis. In practice the prime focus has been little used: the telescope is large enough to make access to the focus difficult from the side of the tube. The other possible foci are coude, with a choice of two focal ratios, f/18 and f/30. The coude arrangement is unusual inasmuch as the light beam is directed UP the polar axis rather than downwards: that permits the shorter focal ratio to be exceptionally short for a coude, and results in a focus at a level near to that of the telescope, which is somewhat convenient for a lone observer who needs to operate both the telescope and whatever auxiliary equipment is placed at the focus. On the other hand, the arrangement lacks part of the advantage of a conventional coude focus, which is often in a basement that enjoys good passive thermal stability (and, from the point of view of the observer personally, protection from wind and extremes of cold!). Until recently the f/18 focus has been the favoured option, but new equipment that for the first time utilizes the f/30 arrangement has now been brought into use. The f/30 focus is just within the dome, high up to the north of the telescope, and its use involves a further reflection. In the present application, that reflection takes place close to the focus, and the beam is turned vertically downwards by successive internal reflections within two right-angle quartz prisms cemented together. The initial image is re-imaged at a focal ratio of f/14.5 at the position required for the auxiliary equipment. A simple plano-convex quartz field lens is cemented to the exit face of the quartz-prism assembly to image the telescope aperture upon the re-imaging lens.

In the early years of its operation, the telescope was used to send starlight into a spectrometer where the light intensities in several wavelength regions. which were accurately defined by masks in the focal plane of the spectrum, could be inter-compared. The intention (only partly realized, owing to the previously unrecognized individuality of the various stars) was to obtain astrophysically significant information about the chemical abundances and atmospheric characters of the stars surveyed. Three successive spectrometers, of progressively increasing size, resolution, and sophistication, were used in that effort.

The third spectrometer was further developed, some 30 years ago, to measure the doppler shift in the stellar spectra observed with it. It did that by means of a much more elaborate mask in the focal plane: instead of having just a few windows to isolate discrete bands of wavelength that were separately measured, it had a mask containing hundreds of narrow windows placed so as to match absorption lines in stellar spectra, the light from all of them being measured together by a single photomultiplier. The position of the spectrum could be sensed, and its doppler shift thereby accurately measured, by scanning the mask in the wavelength coordinate and looking for the more or less dramatic decrease in light transmission that occurs when every window is occupied by its corresponding absorption line. The plot of transmitted light against displacement of the mask is the cross-correlation function of the mask with the star spectrum, and has a pronounced minimum at the position of register.

That instrument, the orginal 'radial-velocity spectrometer', was the first application of cross-correlation to radial-velocity (or, indeed, any other astronomical) measurement. The method has now been adopted almost to the exclusion of the previous procedure involving the measurement of the positions of individual absorption lines, and has revolutionized the radial-velocity field, allowing observations to be made with enormously greater precision and sensitivity than was possible before. A few years before the instrument was brought into operation, a compilation of all known stellar radial velocities included only about 70 stars of 7.0 magnitude or fainter whose radial velocities were supposed to be known to an accuracy of 1 km/s more and fainter stars than that were sometimes observed to at least that accuracy on individual nights in Cambridge - a site that has not generally enjoyed much of a reputation for its excellence for observation. The original instrument remained in operation for 25 years, during which it provided most of the data for about 200 published scientific papers, and when it was de-commissioned it went straight to the Science Museum as an historic instrument. There were delays in commissioning its successor, which is however operating now and provides sensitivity, precision and convenience well beyond those of the pioneering instrument.

## Contents

For n = 1 , (2) gives the one-particle density which, for a crystal, is a periodic function with sharp maxima at the lattice sites. For a (homogeneous) liquid, it is independent of the position r 1 _<1>> and equal to the overall density of the system:

It is now time to introduce a correlation function g ( n ) > by

From (3) and (2) it follows that

### The potential of mean force Edit

It can be shown [2] that the radial distribution function is related to the two-particle potential of mean force w ( 2 ) ( r ) (r)> by:

In the dilute limit, the potential of mean force is the exact pair potential under which the equilibrium point configuration has a given g ( r ) .

### The pressure equation of state Edit

Developing the virial equation yields the pressure equation of state:

### Thermodynamic properties in 3D Edit

The radial distribution function is an important measure because several key thermodynamic properties, such as potential energy and pressure can be calculated from it.

For a 3-D system where particles interact via pairwise potentials, the potential energy of the system can be calculated as follows: [4]

The pressure of the system can also be calculated by relating the 2nd virial coefficient to g ( r ) . The pressure can be calculated as follows: [4]

This similarity is not accidental indeed, substituting (12) in the relations above for the thermodynamic parameters (Equations 7, 9 and 10) yields the corresponding virial expansions. [5] The auxiliary function y ( r ) is known as the cavity distribution function. [3] : Table 4.1 It has been shown that for classical fluids at a fixed density and a fixed positive temperature, the effective pair potential that generates a given g ( r ) under equilibrium is unique up to an additive constant, if it exists. [6]

In recent years, some attention has been given to develop Pair Correlation Functions for spatially-discrete data such as lattices or networks. [7]

One can determine g ( r ) indirectly (via its relation with the structure factor S ( q ) ) using neutron scattering or x-ray scattering data. The technique can be used at very short length scales (down to the atomic level [8] ) but involves significant space and time averaging (over the sample size and the acquisition time, respectively). In this way, the radial distribution function has been determined for a wide variety of systems, ranging from liquid metals [9] to charged colloids. [10] Going from the experimental S ( q ) to g ( r ) is not straightforward and the analysis can be quite involved. [11]

It is also possible to calculate g ( r ) directly by extracting particle positions from traditional or confocal microscopy. [12] This technique is limited to particles large enough for optical detection (in the micrometer range), but it has the advantage of being time-resolved so that, aside from the statical information, it also gives access to dynamical parameters (e.g. diffusion constants [13] ) and also space-resolved (to the level of the individual particle), allowing it to reveal the morphology and dynamics of local structures in colloidal crystals, [14] glasses, [15] [16] gels, [17] [18] and hydrodynamic interactions. [19]

Direct visualization of a full (distance-dependent and angle-dependent) pair correlation function was achieved by a scanning tunneling microscopy in the case of 2D molecular gases. [20]

It has been noted that radial distribution functions alone are insufficient to characterize structural information. Distinct point processes may possess identical or practically indistinguishable radial distribution functions, known as the degeneracy problem. [21] [22] In such cases, higher order correlation functions are needed to further describe the structure.

## Python

Reference –
Correlation coefficient – Wikipedia
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Borchers, H. and Schmidt, E.: 1964, Landolt-Börnstein (eds.),Numerical Data and Functional Relationships in Physics, Chemistry, Astronomy, Geophysics and Technology IV, 2b, Springer-Verlag, Berlin, p. 46.

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MacQueen, P. J.: 1986, Ph.D. Thesis, University of Canterbury.

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Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T.: 1988,Numerical Recipes, CUP, New York, Ch. 14.

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## Watch the video: Solved Problem: Cross Correlation (July 2022).

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