# Using the 'exoplanet' module to fit a radial velocity curve for a binary star system

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As part of a project, I am trying to fit a radial velocity curve using the below tutorial for the binary star system (EBLM J0608-59): https://docs.exoplanet.codes/en/stable/tutorials/rv/

I have the code below (similar to the tutorial) but as it is only one body orbiting the main star, I have not used the shape arguments in the 'Model' section. I have the time measurements and corresponding radial velocities below.

In my code:

bjd_arr = [2458549.66666822, 2458488.75868917, 2458721.8842678, 2458216.52944096, 2458777.86519712, 2458730.87419918, 2458503.73189952, 2458740.81585069, 2458717.86935648, 2458566.60445361, 2458708.90058602, 2458698.93553665, 2458556.64600867, 2458537.693232, 2458553.66510611, 2458436.82194322, 2458726.88080012, 2458516.71230084] RVarr = [40.4432860896165, 23.5780860488492, 19.1436760180712, 50.786661126373, 7.57399767050893, 33.7977531116774, 26.2274095792521, 46.7356539726255, 14.8623891652762, 50.1464417200217, 30.9220153319253, 50.7551660448898, 20.5739440645183, 50.6114884906977, 49.0286951865467, 48.9542544284423, 49.1472667204292, 14.7099410033466] RVerr_arr = [0.00746727240668619, 0.0071426759147196405, 0.007709525952827119, 0.00749855434080898, 0.00811465321639329, 0.00802464048008298, 0.00773122855735923, 0.00789124901574783, 0.007773933655270579, 0.00753401931897191, 0.00822360876229237, 0.00741660549431721, 0.0079125927446526, 0.007308327482575159, 0.00868241919549683, 0.00794049877685647, 0.00781335663783952, 0.0076679825576695] periods = 14.608810 import numpy as np import matplotlib.pyplot as plt import pymc3 as pm import exoplanet as xo import theano.tensor as tt x = np.array(bjd_arr) y = np.array(RVarr) yerr = np.array(RVerr_arr) period_errs = 0.011184 t0s = (min(bjd_arr)+max(bjd_arr))/2 # transit time (rough estimate) t0_errs = 0.011184 Ks = xo.estimate_semi_amplitude(periods, x, y, yerr, t0s=t0s) print(Ks, "km/s") print(type(Ks)) print(float(Ks)) x_ref = 0.5 * (x.min() + x.max()) t = np.linspace(x.min() - 5, x.max() + 5, 1000)"Making RV model"with pm.Model() as model: # Gaussian priors based on transit data (from Petigura et al.) t0 = pm.Normal("t0", mu=(t0s), sd=(t0_errs)) P = pm.Bound(pm.Normal, lower=0)( "P", mu=(periods), sd=(period_errs), testval=(periods), ) # Wide log-normal prior for semi-amplitude logK = pm.Bound(pm.Normal, lower=0)( "logK", mu=float(np.log(Ks)), sd=10.0, testval=float(np.log(Ks)) ) # Eccentricity & argument of periasteron ecc = xo.distributions.UnitUniform( "ecc", testval=0.1 ) omega = xo.distributions.Angle("omega") # Jitter & a quadratic RV trend logs = pm.Normal("logs", mu=np.log(np.median(yerr)), sd=5.0) trend = pm.Normal("trend", mu=0, sd=10.0 ** -np.arange(2)[::-1], shape=2) # Then we define the orbit orbit = xo.orbits.KeplerianOrbit(period=P, t0=t0, ecc=ecc, omega=omega) # And a function for computing the full RV model def get_rv_model(t, name=""): # First the RVs induced by the planets vrad = orbit.get_radial_velocity(t, K=tt.exp(logK)) pm.Deterministic("vrad" + name, vrad) # Define the background model A = np.vander(t - x_ref, 2) bkg = pm.Deterministic("bkg" + name, tt.dot(A, trend)) # Sum over planets and add the background to get the full model return pm.Deterministic("rv_model" + name, tt.sum(vrad, axis=-1) + bkg) # Define the RVs at the observed times rv_model = get_rv_model(x) # Also define the model on a fine grid as computed above (for plotting) rv_model_pred = get_rv_model(t, name="_pred") # Finally add in the observation model. This next line adds a new contribution # to the log probability of the PyMC3 model err = tt.sqrt(yerr ** 2 + tt.exp(2 * logs)) pm.Normal("obs", mu=rv_model, sd=err, observed=y)"Initial model"plt.figure(figsize=(10,10)) plt.errorbar(x, y, yerr=yerr, fmt=".k") with model: plt.plot(t, xo.eval_in_model(model.vrad_pred), "--k", alpha=0.5,label='star') plt.plot(t, xo.eval_in_model(model.bkg_pred), ":k", alpha=0.5,label='bkg') plt.plot(t, xo.eval_in_model(model.rv_model_pred), label="model") plt.legend(fontsize=10) plt.xlim(t.min(), t.max()) plt.xlabel("time [days]") plt.ylabel("radial velocity [km/s]") _ = plt.title("initial model")

Graph here

I thought this would work but I am getting the wrong model curve, where the model curve is just a straight line and was wondering if I was missing something obvious? I suspect there is something wrong in maybe thedef get_rv_modelfunction but I am not sure (I'm still a bit inexperienced with coding), so any guidance would be appreciated. This code works fine when I use it for K2-24 like they do in the example.

## Weekly literature review #1: Exoplanet detection using deep learning

This month on, I make a resolution to read through as much scientific literature as possible, and try my best to summarize it up, as a series of posts- one every week. Most of these will be centered on, but not restricted to astrophysics.

This week I had a chance to read through the paper on Exoplanet detection of NASA and UT Austin [0]. NASA held a conference for the same, claiming

Artificial Intelligence helps find an exoplanet.

In this post, I will try to b reakdown the original paper, remove the “journalism science”, and try to get to the bottom of “how much of AI” is actually into the work!

Prerequisites:

1. Basic knowledge of planetary motion (high school stuff!)
2. Basic neural networks (not really required, but recommended).

The original paper which I am reading can be found in arXiv: 1712.05044.

A couple of good references for Convolutional networks:

starry : Analytic occultation light curves for astronomy.

PyTransit : Fast and easy transit light curve modeling using Python and Fortran.

batman : Fast transit light curves models in Python.

robin : Robust exoplanet radii from ingress/egress durations

ktransit : A simple exoplanet transit modeling tool in python

planetplanet : A general photodynamical code for exoplanet light curves

ketu : Search for transiting planets in K2 data

ttvfast-python : Python interface to the TTVFast library

TTV2Fast2Furious : Construct and fit linear transit timing variation models

pysyzygy : A fast and general planet transit (syzygy) code written in C and in Python

wellfit : Turnkey transit modeling with starry and celerite

lcps : A tool for pre-selecting light curves with possible transit signatures

## Stellar Spectroscopy

### III.B Rotation

In addition to their radial velocities, orbital motion, pulsation, or complex atmospheric oscillation, all stars exhibit another motion: They rotate. The solar rotation can be detected easily by tracking the positions of sunspots, but the spectrum also shows this rotation. The equator at the east limb of the sun is approaching us at 2 km s −1 , which corresponds to a rotation period of 27 days. As a result, the spectral lines from the eastern solar hemisphere are systematically blueshifted and the lines from the western hemisphere are all redshifted. In fact, the first strong confirmation of the Doppler effect for light came in the 1880s, when the of solar rotation rates as a function of latitude from the sunspot and spectroscopic methods were shown to be in agreement.

The same systematic velocity shifts are of course present in other stars, but, with few exceptions, we see only the light integrated over the whole disk and we observe spectral line profiles that are broadened compared with the nonrotating case. It is necessary to use a model to derive the stellar rotation speed. The model of stellar rotation must take several factors into account. First, the maximum speed occurs at the equator because the matter must travel the greatest distance during the rotation period moving away from the equator the rotational speed drops, reaching zero at the poles. Second, we only observe part of the rotational speed of any point on the stellar disk, the component that is moving toward or away from us that can produce a Doppler shift the material at the limbs, whose motion is most closely aligned with our line-of-sight, produces the largest shifts. This reduces to the fact that any chord on the apparent stellar disk that is parallel to the central meridian is a line of constant radial velocity . The wavelength shift is

for a given equatorial rotation velocity, υeq and inclination angle i, and μ = cosθ, where θ is the angle of the chord from the center meridian: 0° at the center of the disk and 90° at the limb.

A third factor is the fraction of the star's light that has a particular projected velocity. Matter at the equatorial limbs has the largest radial velocity, but this represents a tiny fraction of the stellar disk. On the other hand, matter along the rotation axis represents the largest portion of the disk, but it has no velocity shift because the material is moving perpendicular to our line of sight. This is accounted for by weighting the contribution from each chord by its length, approximately 2 μ. The residual flux of the line, the ratio of the flux at λ to the flux of the adjacent continuum,

where r0 is the residual flux depth of the line in the absence of rotation. A rapidly rotating star will have shallow line cores which appear to be washed out relative to the slowly rotating case. To incorporate limb darkening into this model for rotational line broadening, a rotation broadening function,

where β is the limb darkening coefficient, is convolved with the intrinsic residual profile, r′(x):

where x is measured in units of the equatorial velocity shift, Δ λ c / λ 0 υ eq . .

As indicated by the appearance of sin i above, the rotation axis of the star is not necessarily perpendicular to our line of sight. So in general we only see some unknown projection of the equatorial rotation speed. The projected rotation speed of the star is found by computing the broadened profile for various values of the equatorial rotation speed and interpolating to find the value giving the closest match to the observations. Figure 10 illustrates this process.

FIGURE 10 . A portion of an observed spectrum of the bright A-type star Deneb showing absorption lines of Fe I and Fe II, Ti II, and Cr II. Also shown is the computed spectrum for three different assumed values of the star's projected equatorial rotation velocity. Note the blended Ti II and Fe I lines near 3758 Å. Comparision of the blend with the computed spectra suggests that the projected rotational velocity of Deneb is close to 25 km s −1 . [Deneb spectrum kindly provided by A. Kaufer.]

The discussion of rotation to this point, as well as the description of the method by which rotation is calculated, has assumed that a star can be characterized by a single equatorial rotation speed. As noted above, the sun's surface rotates differentially, with the rotation period increasing from about 27 days at the equator to more than 35 days near the poles. In addition, the sun's rotation appears to vary with time and to exhibit a north–south asymmetry. These facts all show that the solar rotation is a much more complex phenomenon than might be apparent at first. Studies of the differential rotation in other stars are just beginning and are closely linked to the study of stellar surface structure, which is described next.

## How to find exoplanets and ‘listen’ to their stars with TESS

Title: A HOT SATURN ORBITING AN OSCILLATING LATE SUBGIANT DISCOVERED BY TESS
Authors: Daniel Huber, William J Chaplin et al
First Author’s Institution: Institute for Astronomy, University of Hawai‘i, USA
Status: Submitted to AAS journals, closed access

NASA’s space mission TESS is currently hunting for new exoplanets in the southern hemisphere sky. But while its primary aim is to find 50 small planets (with radii less than 4 Earth radii) with measurable mass, there is a lot of other interesting science to do. Today’s paper presents the discovery of a new exoplanet that is quite precisely characterised thanks to the complementary technique of asteroseismology used on the same data.

### Meet TESS

TESS will survey stars over the entire sky, studying 26 strips for 27 days each. Data for selected bright stars is downloaded to give data points every 2 minutes, then processed through a pipeline to produce lightcurves. Another pipeline detects transit-like signals in these lightcurves. One of the planet candidates it identified was TOI-197.01 (see Figure 1a).

### Is it an exoplanet?

The authors used high resolution imaging by the NIRC2 camera on the Keck telescope to rule out companion stars that could produce a similar lightcurve. An intense spectral monitoring campaign of 111 spectra from 5 different instruments in a seven week period let them search for periodic Doppler shifts in the stellar spectrum caused by the mass of another object tugging on the star. The mass they calculated from these radial velocities (seen in Figure 3) confirmed TOI-197.01 as an exoplanet.

### Stellar pulsations

Photometry from space is not only useful for finding exoplanets: Kepler could detect the periodic changes in stellar brightness caused by stellar pulsations or ‘star quakes’ (listen to them here). Asteroseismology, the study of these pulsations, allows astronomers to investigate the inner structure of bright stars and calculate their key properties, including radius and mean density, very precisely. Astronomers expected they could also study stellar pulsations using TESS data.

After removing the transit signal from the TESS light curves (giving Figure 1b), the light curve is fourier transformed from time (days) into frequency (microHz), giving the power spectrum seen in Figure 1c. Modelling the stellar pulsations along with the stellar granulation and white noise (see Figure 1c), the authors then ‘smoothed’ the power spectrum to identify the tallest peak, i.e. the frequency of maximum power at 430 microHz and its height, or power.

Figure 1: The TESS lightcurve of TOI-197. a) Raw TESS lightcurve showing two transits marked by grey triangles. b) Corrected TESS lightcurve with transits and instrumental effects removed. c) Power spectrum of the corrected lightcurve, where dashed red lines show the granulation and white noise. The solid red line is a fit to these as well as the stellar pulsations. Figure 1 from today’s paper.

The authors converted the maximum power into amplitude and plotted this against the frequency of maximum power. Comparison against 1500 stars from the Kepler mission confirmed it had solar-like oscillations. Another important value is the large frequency separation, found by identifying the difference in frequency between the radial mode peaks. Figure 2 shows the radial mode peaks in blue and that the large frequency separation is 29.84 microHz.

Figure 2: a) Power spectrum of TOI-197.01 in frequency space showing the oscillations. Vertical lines mark identified individual frequencies, with blue showing the radial modes. b) Frequencies of individual peaks are plotted against a frequency difference of 28.94 microHz. The radial modes (blue circles) line up at this frequency difference, so 28.94 microHz is the large frequency separation. Figure 2 from today’s paper.

### Modelling stellar properties

The authors then used stellar evolution and oscillation codes to model the stellar properties. The luminosity for the models was calculated by combining the Gaia parallax with photometry from many different catalogues. They also input properties from spectral modelling – temperature, surface gravity (log g) and metallicity – and the individual frequencies and large frequency separation from asteroseismology. This resulted in two preferred models: i) a lower mass, older star (1.15 Msol,

6Gyr) or ii) a higher mass, younger star (1.3Msol,

4Gyr). An independent constraint on surface gravity from an autocorrelation analysis of the lightcurve favours a higher mass model. Thanks to asteroseismology, the final estimates of stellar parameters have small uncertainties: radius (2%), mass (6%), mean density (1%) and age (22%).

### Characterising the planet

Using the mean stellar density from asteroseismology, the authors jointly fit the photometric and radial velocity data to obtain the planet properties, including period, radius and mass. Figure 3 shows photometric data (top) and radial velocity data (bottom), both phase folded on the best period of 14.3 days. The mass ratio is calculated from the maximum amplitude of the radial velocity data. Combining this with the modelled stellar mass gives a minimum planet mass 35% lighter than Saturn. The transit depth gives the radius ratio, which combined with the modelled star radius gives a planet radius the same as Saturn.

Figure 3: Data for TOI-197 phase folded on the best period of 14.3 days. Top: The TESS lightcurve of TOI-197. Bottom: radial velocity curve. Figure 5 from today’s paper.

### A Hot Saturn and a Bright Future!

The result: TOI-197.01 is a hot Saturn orbiting a late subgiant/early red giant star. The combination of spectra and the large frequency separation from asteroseismology shows the star has just started ascending the red giant branch. TOI-197.01 represents the starting point before gas giants reinflate due to the strong flux from their evolved stars. This is significant as it is TESS’s first detection of a transiting planet orbiting a late subgiant/early red giant with detected oscillations, and only the sixth ever detected.

This is an exciting result as it shows that even with 27 days of data, TESS should allow us to study the oscillations of thousands of bright stars which have 2 minute cadence data. TOI-197.01 is also one of the most precisely characterised Saturn sized planets, with density constrained to 15%. This demonstrates what we can gain by ‘listening’ to exoplanet host stars.

## Study unveils detailed properties of the eclipsing binary KOI-3890

The power spectrum of KOI-3890 is shown in black alongside the fit to the background (excluding the Gaussian component describing the power excess) in red. Credit: Kuszlewicz et al., 2019.

By combining transit photometry, radial velocity observations, and asteroseismology, astronomers have gathered important information about the properties of a highly eccentric, eclipsing binary system known as KOI-3890. The new findings are presented in a paper published April 30 on arXiv.org.

Eclipsing binaries with red giants exhibiting solar-like oscillations are a rare find. When searching for new systems of this type, astronomers are especially interested in finding ones that showcase ellipsoidal variations in their light curve caused by large tidal distortions of the surface of the star during periastron. Due to these variations, such binaries were dubbed the "heartbeat" systems.

Heartbeat systems with solar-like oscillations can be characterized in great detail using radial velocity data and by conducting astroseismologic studies. For instance, astroseismology enables determining the obliquity (the angle between the stellar rotation axis and the angle normal to the orbital plane) of these systems and stellar parameters of their oscillating red-giant primary components.

KOI-3890 (aka KIC 8564976 and 2MASS J19350531+4438185) has been initially classified as a potential planet-hosting star. However, follow-up observations show that it is a heartbeat system—in particular, a highly eccentric 153-day-period eclipsing binary consisting of a red giant showing solar-like oscillations and an M-dwarf companion.

A team of astronomers led by James Kuszlewicz of Max Planck Institute for Solar System Research in Germany decided to conduct a comprehensive study of KOI-3890 using various methods to reveal detailed properties of the system.

"In this work, we aim to derive the properties of the components of an eclipsing binary system through the use of asteroseismology, eclipse fitting and radial velocity analysis. In addition, we aim to constrain the geometry of the system by inferring the inclination angle of the red giant primary to then obtain the obliquity of the system, which gives information as to whether the system is aligned," the researchers wrote in the paper.

The study found that red giant KOI-3890 A is almost six times larger than the sun (5.8 solar radii), but its mass is comparable to that of the sun (1.04 solar masses). The M-dwarf, named KOI-3890 B, was found to be approximately four times smaller and less massive than our sun (0.26 solar radii and 0.23 solar masses).

Astroseismic studies of KOI-3890 allowed the team to calculate the system's age. They found that the binary is about 9.1 billion years old. Moreover, astroseismic analysis revealed that the inclination angle of the rotation axis of KOI-3890 A is approximately 87.6 degrees. When it comes to the obliquity of the system, it was estimated to be at a level of approximately 4.2 degrees, which means that the plane of the orbit of KOI-3890 B is perpendicular to the stellar rotation axis of the primary star.

In addition, the astronomers estimated that the orbital eccentricity of KOI-3890 is about 0.61 and that the stars are separated from each other by around 0.19 AU. They also drew some conclusions regarding further evolution of the system.

"As the primary continues to evolve the M-dwarf may become embedded in the expanding envelope, leading to mass transfer between the stars in a common envelope phase. Additionally the strong drag forces on the secondary in such a configuration may lead to the ejection of the common envelope, and a significant decrease in the orbital period," the researchers concluded.

## Exoplanet system Kepler-2 with comparisons to Kepler-1 and 13

We have carried out an intensive study of photometric (Kepler Mission) and spectroscopic data on the system Kepler-2 (HAT-P-7A) using the dedicated software WinFitter 6.4 . The mean individual data-point error of the normalized flux values for this system is 0.00015, leading to the model’s specification for the mean reference flux to an accuracy of ∼0.5 ppm. This testifies to the remarkably high accuracy of the binned data-set, derived from over 1.8 million individual observations. Spectroscopic data are reported with the similarly high-accuracy radial velocity amplitude measure of ∼2 m s −1 . The analysis includes discussion of the fitting quality and model adequacy.

Our derived absolute parameters for Kepler-2 are as follows: (M_

) (Jupiter) 1.80 ± 0.13 (R_) 1.46 (pm 0.08 imes 10^<6>) km (R_

) 1.15 (pm 0.07 imes 10^<5>) km. These values imply somewhat larger and less condensed bodies than previously catalogued, but within reasonable error estimates of such literature parameters.

We find also tidal, reflection and Doppler effect parameters, showing that the optimal model specification differs slightly from a ‘cleaned’ model that reduces the standard deviation of the ∼3600 binned light curve points to less than 0.9 ppm. We consider these slight differences, making comparisons with the hot-Jupiter systems Kepler-1 (TrES-2) and 13.

We confirm that the star’s rotation axis must be shifted towards the line of sight, though how closely depends on what rotation velocity is adopted for the star. From joint analysis of the spectroscopic and photometric data we find an equatorial rotation speed of 11 ± 3 km s −1 .

A slightly brighter region of the photosphere that distorts the transit shape can be interpreted as an indication of the gravity effect at the rotation pole however we note that the geometry for this does not match the spectroscopic result. We discuss this difference, rejecting the possibility that a real shift in the position of the rotation axis in the few years between the spectroscopic and photometric data-collection times. Alternative explanations are considered, but we conclude that renewed detailed observations are required to help settle these questions.

## Exoplanet system Kepler-2 with comparisons to Kepler-1 and 13

We have carried out an intensive study of photometric (Kepler Mission) and spectroscopic data on the system Kepler-2 (HAT-P-7A) using the dedicated software WinFitter 6.4 . The mean individual data-point error of the normalized flux values for this system is 0.00015, leading to the model’s specification for the mean reference flux to an accuracy of ∼0.5 ppm. This testifies to the remarkably high accuracy of the binned data-set, derived from over 1.8 million individual observations. Spectroscopic data are reported with the similarly high-accuracy radial velocity amplitude measure of ∼2 m s −1 . The analysis includes discussion of the fitting quality and model adequacy.

Our derived absolute parameters for Kepler-2 are as follows: (M_

) (Jupiter) 1.80 ± 0.13 (R_) 1.46 (pm 0.08 imes 10^<6>) km (R_

) 1.15 (pm 0.07 imes 10^<5>) km. These values imply somewhat larger and less condensed bodies than previously catalogued, but within reasonable error estimates of such literature parameters.

We find also tidal, reflection and Doppler effect parameters, showing that the optimal model specification differs slightly from a ‘cleaned’ model that reduces the standard deviation of the ∼3600 binned light curve points to less than 0.9 ppm. We consider these slight differences, making comparisons with the hot-Jupiter systems Kepler-1 (TrES-2) and 13.

We confirm that the star’s rotation axis must be shifted towards the line of sight, though how closely depends on what rotation velocity is adopted for the star. From joint analysis of the spectroscopic and photometric data we find an equatorial rotation speed of 11 ± 3 km s −1 .

A slightly brighter region of the photosphere that distorts the transit shape can be interpreted as an indication of the gravity effect at the rotation pole however we note that the geometry for this does not match the spectroscopic result. We discuss this difference, rejecting the possibility that a real shift in the position of the rotation axis in the few years between the spectroscopic and photometric data-collection times. Alternative explanations are considered, but we conclude that renewed detailed observations are required to help settle these questions.

## A New Magnetically Active Binary System Discovered in Yunnan-Hong Kong Wide Field Survey ⋆

We present a newly discovered magnetically active binary system detected by Yunnan-Hong Kong wide field survey, with an orbital period of 0.60286 days. Two color photometry for the system was performed using the 1 m Cassegrain telescope of Yunnan Observatories with its CCD (Charge-Coupled Device) camera. In the observed light curves, there are clearly different light maxima existed in the out-of-eclipse regions. We made spectroscopic observations for the binary system using the 2.4 m telescope and YFOSC (Yunnan Faint Object Spectrograph and Camera) of Lijiang station of Yunnan Observatories, China. The radial velocity curve was derived for primary star of the binary system. The primary star exhibited strong chromospheric activity, which confirms that the distortion of the light curves results from the starspot activity on the primary star. Through analyzing the light curves and RV (Radial Velocity) curve mentioned above by means of the W-D (Wilson-Devinney) code, orbital parameters and starspot configuration of the binary system are obtained. Finally, we have discussed the properties of the binary system, and given the prospects on the future work.

## Practice with the radial velocity simulator

There are a lot of concepts to understand with the Doppler technique. First, we use the spectrum of the star to measure wavelength shifts that correspond to changes in the velocity of the star. Second, we only see the projected radial component of the stellar velocity (this leads to a sinusoidal variation in the signal). Third, we don't know the inclination of the orbit - this tilt of the orbit means that the sinusoidal signal will be largest when viewed edge-on (90 ∘ inclination) and smaller as the inclination tilts toward face-on. This means that we only measure (m_sin i) with the Doppler method.

Let's walk through a series of exercises with the Nebraska Astronomy Applet Project (NAAP) simulator to help develop your intuition and understanding of the Doppler method. You will need to download the Simulations package (will install in your Applications directory, open Extrasolar Planets, Exoplanet Radial Velocity Simulator).

Begin by opening the simulator and setting up the inputs to match the values shown in the figure below. Simulated radial velocities are "phase-folded'' in the upper right hand plot. The radial velocities are obtained as a function of time, but if you repeat observations over more than one orbital period, the data can be folded so that all of the peaks and troughs of the curve line up. This is routinely done by astronomers to build evidence for repeating signals.

Figure (PageIndex<7>): Initial setup: show multiple windows, Inclination 76.8 degrees, longitude 45 degrees, 1 Msun star, planet mass 1 Mjup, semi-major axis 5.2 AU, eccentricity 0, show theoretical curve, show simulated measurements, noise = 2.0 m/s, number of observations = 30. The semi-amplitude of the radial velocities should be about 12 m/s (read this off the y-axis of the RV plot).