Question

Use the simulator to investigate the relationship between RV amplitude and stellarmass. Report your findings and conclusions.

Answer 1

RV jitter from rotational modulation:-

In this section we will try to observe how decreased detection efficiency at the rotation period and it's two harmonic scale with stellar mass. we will also try to establish typical time amplitude RV variations due to stellar roatation as a function of stellar mass.

Amplitude:-

we will calculate typical amplitude of rotational RV variation due starports. The physical mechanism behind these thing is that as cool spots moves across the limb of the star they introduce an asymmetry in the rotational velocity profile's of the star disc and therefore forms an asymmetry in the spectral line. It can also be caused palgues and facule the effect of these active groups are more difficult to quantify with a light curve and are only dominant cause of RV variations for slowly roatating stars. Aigrain et al. give an analytic model for for a spot induced an RV time series with a flux time series which we simplify in the form to estimte peak-peak amplitudes

RV_PP$\simeq$ F_PP * $\nu$ sin(i)

where,

R_vv= Peak-Peak RV variation caused by the starports

F_PP= Flux variation in the passband

$\nu$sin(i)= Projected rotational velocity of star

We will see the flux variation over the same bandpass as radial vellocities and will focus on optical flu variations by keppler telescope which are likely lower due to due to lessened flux contrast with cool starports.

We will calculate for peak- peak variation in optical flux from the result of Mc quillen et. al. who measured the periodic photometric amplitude variations of keppler targets. they fdeined the periodic amplitude as 5th-95th % of median divided unity, and subtarcted and 10 h-box car smoothed keppler light curves. After that dominant source of photometric variation is a starport modulation

We will use the result of Mc quillan et. al. to establish relation between rotational modulation and stellar mass. we divided the data into bins and take and take 16th and 84th data roughlu 1$\sigma$ of each mass bin as ;lowwer as upper bounds for typical rotational and modulation amplitude. This sample target very few stars with less then 0.3M_$\odot$thius we extrapolate our relation to small dawrfs

We measure the value of $\nu$ sin(i) by combining our estimates of rotation period for main sequence star of stellar radii from a 5Gyr Dartmouth isochrone, and the average value of sin(i) over all possible spin axis orientation using equation:-  

  $\nu$sin(i) $\simeq$ 2$\pi$R_$\star$/ P_rot * (sin(i)

& sin(i)$\simeq$ 0.79

R_$\star$ = Stellar radius

& P_rot = Stellar rotation period