STELLAR MASSES and the H-R DIAGRAM

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BINARY STARS

Over 60% of all stars are in BINARY systems, i.e., one of two
stars orbiting each other, held together by mutual gravitational attraction.

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CLASSIFICATION OF BINARIES

FALSE BINARIES or optical doubles -- these are close together in the sky, but
signficantly separated in distance and not bound by gravity.

TRUE BINARIES

  • VISUAL binaries -- relatively widely separated stars that can be
    seen by eye (a few) or telescope (lots); they appear to be close together, and ARE so.
  • Speckle binaries -- only seen as two stars using speckle interferometry.
  • Composite spectrum binaries -- even through a telescope they appear
    to be just one star, but the spectrum shows two humps, implying two photospheres.
  • SPECTROSCOPIC binaries -- these are very useful close binaries and
    come in two classes:
  • Single-line spectroscopic -- the lines of only one star are seen, but they
    shift back and forth in wavelength, indicating orbital motion about a much fainter companion
    (usually a star, occasionally a planet);
  • Double-line spectroscopic -- orbital motion is detected in the spectra of
    BOTH stars; while one approaches, the other recedes.
  • ECLIPSING binaries -- close enough and with orbital plane near the line of sight so
    stars alternately (at least partially) cover each other up;
    therefore the LIGHT CURVE undergoes dips.
  • ASTROMETRIC binaries -- wiggly motion ACROSS the sky (determined similarly
    to parallax and proper motion) indicates gravitational tug of an unseen companion --
    either another star or even a planet.

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    DETERMINATION OF MASSES

  • Remember, mass is the most important property of a star,
    but it is very tough to determine.

    (THIS SUBSECTION WAS COVERED IN THE LAST LECTURE TOO)

    DOUBLE LINE SPECTROSCOPIC BINARIES ALLOW THE RATIO OF THEIR MASSES
    TO BE DETERMINED.

  • Stars are actually orbiting their center of mass (CM)
  • This CM is closer to the more massive star (think of solar system,
    where CM is inside the sun, but not exactly at its center).
  • The more massive star moves slower, the less massive, faster.
  • If the semi-major axes are a_1 and a_2 and their velocities are
    v_1 and v_2 and their masses are M_1 and M_2 we have
  • (a_1 / a_2 ) = (v_1 / v_2 ) = M_2 /M_1
  • Example: v_1 = 8 km/s, v_2 = 32 km/s, so star 1 has an orbit
    only 1/4 as large as star 2's; therefore a mass 4 times larger: M_2/M_1 = 1/4.
  • Recall these velocities come from the Doppler shifts of lines.

    ECLIPSING OR VISUAL BINARIES CAN ALLOW THE DETERMINATION OF
    THE SUM OF STELLAR MASSES.

    This comes from Newton's generalization of Kepler's 3rd Law:

  • P^2 ~ a^3
  • This comes from Newton's 2nd law of motion, F = ma
  • and his law of gravity: F_grav = G m_1 m_2 / r^2
  • together these imply
  • M_1 + M_2 ~ a^3/P^2

    THE BEST CASE: double-line spectroscopic binary that is eclipsing;
    if you know M_1/M_2 and M_1 + M_2 you trivially can find M_1 and M_2 separately.

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    Good things come from ECLIPSING BINARIES

  • The depth of the secondary eclipse relative to the total intensity
    gives the ratio of the luminosities of the stars.
  • The depth of the primary eclipse then gives a good estimate of
    the temperature ratio of the stars.
  • Together, these can give decent estimates for the radii of the stars.
  • The relative length of the eclipses, compared to the total binary period
    gives the ratio of the star's sizes to their separation.

    Putting all these together, astronomers get decent values for the semi-major
    axis of the orbit, a.

  • But, P, the period is very simply determined as the time from the
    center of one primary eclipse to the next.
  • And now the sum of the masses comes from:
  • M_1 + M_2 = a^3 / P^2
    (where M's are in M_sun units, a is in AU and P is in years).

    Example: For a binary a = 5 AU and P = 10 yr so
    M_1 + M_2 = a^3/P^2 = 5^3/10^2 = 125/100 = 1.25 M_sun

    In addition, detailed measurements of eclipsing binary light curves
    allow the determination of:

  • eccentricity of the orbit (seen in asymmetry of time of secondary
    eclipse with respect to primaries -- a circular orbit means the secondary
    is half-way between the primaries, but an elliptical orbit leads to the
    secondary closer to one primary than another.
  • tilt of the orbital plane:
    flat bottomed eclipses indicate one star passes completely behind the
    other and their orbital plane nearly includes the line-of-sight;
    round bottomed eclipses indicate their orbit is tilted to this plane;
    (of course if the tilt is too big, no eclipses will be seen at all!).

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    STELLAR PROPERTIES -- SPECTRAL CLASSES

    Recall that different photospheric temperatures mean that different
    photospheric absorption lines are strongest. For example:

  • Very hot stars have He II (once ionized Helium) lines;
  • Hot stars have very strong neutral H I and He I lines;
  • Stars like the Sun have very strong ionized atomic lines, of ionized "metals"
    such as Ca II and Fe II (reminder: metals are anything other than H and He!);
  • Cooler stars have neutral "metal lines";
  • The coolest stars have strong molecular lines (like TiO).