__Write-up__

*May 28-June 17*

My research so far has focused on calculating the velocities of single M dwarfs and M dwarfs in binary pairs.

*This summer I am researching the velocities of close binary pairs. From SDSS, we have spectral data on these binaries for multiple exposures. Because the binary pairs that we’re examining are so close together and moving so quickly, in each exposure taken (which can be over a couple of hours), the binaries may have moved causing its spectrum to be redshifted.Using these SDSS exposures and previously compiled templates for each spectral type, we can actually calculate the velocity of these binary pairs. At this point in my research, I have calculated the velocities of over 4,000 M dwarfs that are in M dwarf and white dwarf binary pairs.*

__Getting Background Information__

For some background information, I read and took notes on Silverstri et al (2005), West et al (2008), West et al (2011), and Hilton et al (2010). Also read the chapter “Stellar Activity” in the Textbook __New Light on Dark Stars: Red Dwarfs, Low-Mass stars, Brown Dwarfs__ by Reid and Hawley. After doing this, I worked on some calculations on deriving velocity from wavelength change (Doppler shifts), and also the relationship between mass, separation, and velocity of binaries.

__Calculating the velocities of M dwarfs in DR7 Catalogue__

The first velocity calculation included calculating the velocities for the M dwarfs listed in AAW’s DR7 Catalogue. Using Doppler shifting techniques, we are able to calculate the velocities of M dwarfs from their spectra. Each binary is catalogued by plate, mjd, and fiber.Plate is the plug plate used, mjd is the mean julian time, which records the time that the exposure was taken, and fiber is the fiber number. In order to calculate a velocity of one of these dwarfs, we had to match spectral templates to the DR7 spectra. In order to normalize them, we zoomed into a flat area of the spectrum between 7300 and 8800 Angstrom and took the average over that region. We then divided the total flux by that average in order to normalize the spectrum. We did the same for the template spectra. Because there aren’t the same number of data points in the template and the DR7 spectra, we had to use the spline function in order to interpolate points in the DR7 spectra. Because we were going to apply this same process to the binary pairs, we zoomed in on the redder end of the spectrum, between 7300 and 8800 Angstroms. This range was determined to be the optimum range that focused on the redder range without giving up too much data accuracy in the spline process. Using the xcorl function, we were able to calculate the pixel shift between the template and the spectra.In order to convert pixel shift into velocity in units of km/s we multiplied the shift by 69.1.

__Calculating the velocities of M dwarfs in dM-WD binaries__

The same process was used with calculating the velocities of M dwarfs in M dwarf and white dwarf binary pairs. We first read in Dylan’s file, wddm_goods2.dat in order to get plate, mjd, fiber and template used (and therefore spectral type) for each of the M dwarfs in the binaries.We then matched the spectral types to each of the exposures. Depending on the binary, there were 3-9 exposures of each binary. Using the process described above, we calculated the velocities of the M dwarfs for each exposure. From these calculated velocities, we were able to calculate the expected spectrum Doppler shift; by plotting the image below, we can see how the expected shift (represented by dashed lines), corresponds with the movement in the corresponding exposure.

__Estimating the separation between the binaries__

The next step was to estimate the separation between the binaries. Using a two-body setup, we are able to calculate the separation, but we have to take into account the fact that the radial velocity of the entire system is also measured in our velocity calculation. To take this into consideration, we have to calculate what the radial velocity of the system is. See equations below:

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