Aggegrate Results

Quiz Talk

• I think many could have benefitted from a bit more studying, or perhaps just looking at the study materials a little more
• Quizzes can be a bit more punishing owing to limited points, plus this was the first one in this class, so perhaps you didn’t know what to expect
  • As such, I’ve added 1 point (5%) to your total score in the gradebook
• Grade Reports will be coming out as soon as I get more caught up in HW grading
• Focus on explaining the why in your computational essays. I promise it will help engrain the science concepts better in your mind

Recap

• The Lomb-Scargle Periodogram is an approach that works well for non-uniform data
  • Requires you to specify the range of angular frequencies / frequencies / periods that you want to computer the power over
  • Aliasing will generally be a factor you need to content with
• Planets and their parent star technically orbit their shared center of mass
• Causes the star to “wobble”, which can be measured!
  • Astrometric method looks for the shift in the star relative to background stars
  • Doppler method looks for the Doppler shift in light as the star move towards then away from us

Discussing Today

• Planet properties from wobbling methods
• Finding multiple signal components
  • Determining all component properties
  • Subtracting contributions
  • Rechecking the results

Animated Example

Beginning to extract planetary properties

• In general, to understand any exoplanet, you must first understand its parent star
• This is often considerably easier, since the star is big and bright
• Several parameters in particular are useful to know:
  • The mass of the star
  • The size of the star
• Both generally require knowing the distance to the star, but otherwise can be worked out from luminosities or location on the HR diagram

Planetary Period and Distance

• If you have the period of the star, then you have the period of the planet
  • Both move in lockstep about the center of mass
  • If you have multiple planets, you can separate the different components from the stars motion
• Extracting the distance to the planet / semimajor axis requires application of Kepler’s 3rd law along with the center of mass location: \[\text{Kepler's 3rd: } \frac{GM_{tot}}{4\pi^2} = \frac{a^3}{p^2} \qquad\qquad\text{Center of Mass: } M_1a_1 = M_2 a_2 \]

Planetary Mass (Part I)

• Combining \(a_1\) and \(a_2\): \[ a = a_1 + a_2 = a_1\left(1 + \frac{a_2}{a_1}\right) = a_1\left(1 + \frac{M_1}{M_2}\right) = \frac{a_1}{M_2}\left(M_2 + M_1\right) = \frac{a_1 M_{tot}}{M_2} \]
• Plugging into Kepler: \[ \frac{GM_{tot}}{4\pi^2} = \frac{1}{p^2} \left(\frac{a_1 M_{tot}}{M_2}\right)^3 \]
• If you know \(a_1\) by direct observation, then you are done, and can solve for \(M_2\)!
• Otherwise you need to write \(a_1\) in terms of a velocity
  • If you assume mostly circular orbits: \[v_1 = \frac{2\pi a_1}{p} \quad\Rightarrow\quad a_1 = \frac{v_1 p}{2\pi}\]

Planetary Mass (Part II)

• Plugging that back into Kepler: \[ \frac{GM_{tot}}{4\pi^2} = \frac{1}{p^2} \left(\frac{a_1^3 M^3_{tot}}{M^3_2}\right) \]

Some Caveats

• The true velocity is only what is measured at the peak of the Doppler curve if you are viewing the orbit perfectly edge-on
• In general: \[ v_{obs} = v_1\sin(i) \] where \(i\) is the orbital inclination (\(0^\circ\) if viewing face-on, or \(90^\circ\) if viewing edge-on)
  • If the inclination angle in unknown, then technically you are finding a minimum mass
• If the eccentricity is known, then: \[ M_2 = \left(\frac{M^2_{tot}}{G}\frac{v_1^3 p}{2\pi}(1- \epsilon^2)^{3/2}\right)^{1/3} \]

Hidden Signals

• Like with our FFT methods, a signal might be comprised of several different components
• This is especially true with Doppler methods, as each orbiting planet tugs on the star a bit differently
• Frequently then need to look for multiple significant peaks within the periodogram
• This is greatly complicated by aliases vs our FFT methods

Visualizing Aliases

• We saw last class that: \[f_{alias} = \left| f_{signal} \pm k \cdot f_{sampling} \right| \] where \(f_{sampling}\) you can get from the spectral window
  • Note that you might have several \(f_{sampling}\) values that all need to be accounted for
• Don’t just iterate over a few \(k\) values, go at least up to 4 or 5
  • Can be useful to compute a flattened outer product, or could just handle with looping

Flattened Outer Products

• Imagine an “outer product” as creating a grid of all the multiplied values of two arrays/vectors

• We don’t really care about the grid nature here, but would rather have just a list of all the possible combinations

• In Python:

  np.outer(array₁, array₂).ravel()

• In R:

  as.numeric(outer(array₁, array₂))

Proceeding Forward

• Once you have flagged all the potential aliases of a signal, you might be able to identify other potential component peaks
• A more robust way though is to subtract the found component wave from the signal and then repeat the process
• But how to compute the component wave?
  • We have the frequency/period, but what about the amplitude/phase angle?
  • No complex values to compute those from like with Fourier Transforms

Fitting the Wave Directly

• While we discussed early that non-linear fitting doesn’t work well by itself, it does work quite well if you have established and “locked” the period/frequency
• Define a way function but with your found peak frequency: \[ A \sin\left(2\pi f_{peak} t + \phi\right) + H \]
• Use your nonlinear fitting tool to find all the other unknown constants: \(A\), \(\phi\) and \(H\)
• Subtract the result from your original signal

Rinse, Wash, Repeat

• Now, with the new signal, compute a new Lomb-Scargle periodogram, and continue the process
  • Identify a candidate peak
  • Compute and visualize aliases to see if they make sense
  • Extract freqency, compute wave properties, and subtract
  • Repeat
• When does it end?
  • Subtracting will get rid of most of the signal, but not always quite all of it
  • If all you are left with in your Lomb-Scargle on peaks/aliases that you have already found, then you are done!

Practice

• The file here contains velocity information determined from the redshift/blueshift of a star with a single planet orbiting it.
• You can assume the planet is traveling in a mostly circular orbit.
• You know that the parent star has a mass of \(2\times10^{30}\) kg.
• Determine:
  • The period of the planet
  • The minimum mass of the planet