The Mass of Stars and Binary Stars (Part 7)

Determining Stellar Masses

  • Stellar mass determination cannot occur directly by examining isolated stars.

  • Mass can be inferred via gravitational effects on other bodies by applying Newton’s Law of Gravity.

Newton's Law of Gravity and Stellar Mass

  • Newton's exploration of Kepler's third law leads to the conclusion that the orbital period of two objects depends on the sum of their masses.

  • For example, in a star system:

    • If a planet orbits the Sun, we can neglect the planet's mass.

    • However, for binary star systems, we use Newton’s more complex equations taking into account both masses.

Binary Star Systems

  • More than half of the stars near Earth form binary systems where two stars orbit each other, though many appear as isolated in the night sky.

  • Observations using telescopes allow for measurements of orbital periods and distances, facilitating mass calculations.

Types of Double Stars

  1. Optical Double (Apparent Binaries):

    • Appearing close together in the sky but may not be physically near each other (optical doubles).

    • Example: \deltaHerculis (an optical double).

  2. True Binary Stars:

    • Pairs of stars that are physically orbiting a common center of mass.

    • They are Visual Binaries if both of the stars can be seen and distinguished from earth (using a telescope if necessary).

Kepler's Third Law and Mass Calculation

  • Newton rewrote Kepler’s third law as a relation between the masses of the stars (in solar masses).

  • (M_1+M_2)=a^3/P^2  where

    • M_1 and M_2 are the two masses expressed in solar masses.

    • a is the length of the semimajor axis of the ellipse in astronomical units which is also the average separation between the two bodies.

    • P is the orbital period in years.

  • Example: For a star with an orbit radius of 4 au and period 2.5 years, the total mass is calculated.

  • To find individual stellar masses, knowledge of their distances from the center of mass is necessary.

Using Kepler’s Third Law to Identify an Individual Star’s Mass

  • To determine the individual masses of stars in a binary system, knowledge of their distances from the common center of mass is essential.

    • Both stars orbit this center of mass; the more massive star is closer (similar to a balanced seesaw).

  • For Visual Binaries:

    • The center of mass can be located as the common focus of their elliptical orbits, observed against background stars.

    • The main challenge is determining the plane of orbit.

  • For Eclipsing Binary Systems:

    • Stars periodically block each other, implying their orbital plane is nearly parallel to our line of sight.

    • This allows for the comparison of their orbit sizes around the center of mass.

    • This comparison yields the mass ratio (M1/M2).

  • Calculating Individual Masses:

    • Combine the mass ratio (M1/M2) with the total mass (M1+M2) (derived from Newton's rewritten Kepler’s third law).

    • This system of two equations with two unknowns allows for the calculation of the individual masses of both stars.

Characteristics of Binary Stars and Eclipses

Eclipsing Binary Detection and Analysis

  • Eclipsing binaries can be detected even if their individual stars cannot be visually resolved in a telescope.

  • Their detection relies on changes in apparent magnitude:

    • The image dims each time one star partially or fully blocks the other.

    • Astronomers measure light intensity over time to create light curves.

  • Interpreting Light Curves:

    • V-shaped trough: Indicates a partial eclipse.

    • Flat-bottomed trough: Indicates a total eclipse.

    • Light curves provide information on how close the orbital plane is to being perpendicular to our line of sight.

Utilizing Light Curves to Study Stellar Atmospheres

  • Light curves can reveal details about stellar atmospheres.

  • Example: If a tiny white dwarf eclipses a much larger giant star.

    • By observing the gradual cutoff of the white dwarf’s light as it passes behind the giant, astronomers infer the pressure and density in the giant’s upper atmosphere.

    • This data is crucial for validating stellar structure models.

Effects of Binary Star Separation

  • Wide Binary Systems:

    • Stars are separated by several astronomical units or more.

    • They behave largely as isolated stars, meaning models for individual stellar evolution generally apply.

  • Close Binary Systems:

    • Stars are only a few stellar diameters apart.

    • The mutual gravitational pull profoundly affects each other's appearance and evolutionary path.

    • Mass Transfer: If one star is a giant, gas from its outer layers can be stripped and transferred to its more compact companion.

Multiple Star Systems

  • Beyond binary systems, some gravitationally bound systems consist of three or more stars.

  • Example: Polaris, the North Star, is part of a triple system where two stars orbit closely, and a third is much farther away.

Mass-Luminosity Relationship of Main-sequence Stars

  • There is a positive correlation between a star’s mass and luminosity on the main sequence.

  • The more luminous a star is within this sequence, the more massive it generally is, confirming a direct relationship between mass, luminosity, and energy production.

  • Average circulation around the H-R diagram positions stars by their temperature, mass, and luminosity characteristics.

  • The importance of mass is evident in stellar evolution and energy generation mechanisms.

Spectroscopic Binary Detection

  • Spectroscopic binaries are identified when individual stars cannot be visually resolved, but their binary nature is revealed through spectral analysis.

  • Initial Detection Clues:

    • Incongruous spectral lines (e.g., both hot (hydrogen) and cool (titanium oxide) absorption lines) from what appears to be a single star indicate a binary system.

  • Doppler Shift Application:

    • The movement of stars orbiting each other causes Doppler shifts in their spectral lines.

    • Approaching sources have shorter (blueshifted) wavelengths; receding sources have longer (redshifted) wavelengths.

    • The magnitude of the shift is proportional to the star's speed.

  • Types of Spectroscopic Binaries:

    • Double-line spectroscopic binary: Both stars' spectral lines are visible and show periodic shifts, allowing for more comprehensive information.

    • Single-line spectroscopic binary: Only one star's spectral lines are detectable, shifting regularly back and forth while the companion remains unseen or too dim.

  • Interpreting Spectral Shifts (Example from Figure 12-16 conceptually):

    • When star A approaches Earth (blueshifted) and star B recedes (redshifted), two offset sets of spectral lines appear.

    • When neither star moves towards or away from Earth, spectral lines return to their normal positions (no Doppler shift).

    • The pattern of blueshifts and redshifts periodically reverses as the stars complete their orbits.

  • Radial-Velocity Curve:

    • Observations yield a radial-velocity curve, which graphs radial velocity over time, demonstrating velocity changes related to the orbital period.

    • This curve can also show the overall motion of the binary system relative to Earth.

  • Combined Data for Comprehensive Analysis:

    • For rare cases where an eclipsing binary is also a double-line spectroscopic binary, astronomers can combine light curves and radial-velocity curves.

    • This allows for the calculation of individual masses, diameters, relative brightnesses, speeds, and stellar separation of the stars.

    • Most spectroscopic binaries do not eclipse from Earth's perspective due to orbital plane tilt.