visual binary
Definition: For a visual binary with a fully resolved relative orbit, Kepler’s third law (Lecture 08) gives the total mass; if each star’s individual displacement from the system’s center of mass is also measured, the mass ratio m1/m2 = a2/a1 (inversely proportional to each star’s distance from the barycenter) splits the total mass into individual masses.
Why it matters here: Binary stars provide the only direct, model-independent measurements of stellar mass in astrophysics, with visual, spectroscopic, and eclipsing binaries each contributing a different, complementary piece of the full orbital and physical picture.
Relationship: Explain how a visual binary’s astrometric orbit yields both the total mass (Kepler’s third law) and, with center-of-mass data, the individual masses.
Boundary or common confusion: Treating a single-lined spectroscopic binary’s mass function as a measured mass. Without an independently known inclination (or a visible second set of spectral lines), the mass function only gives a lower limit / degenerate combination of the companion mass and sin(i); this is exactly the ambiguity that made many early radial-velocity exoplanet "minimum masses" (m sin i) rather than true masses.
spectroscopic binary
Definition: For a spectroscopic binary, the observed radial-velocity semi-amplitude K depends on the true orbital speed times sin(i); the derived "mass function" therefore only constrains a combination of masses and sin(i), not the individual masses, unless the inclination is broken by an independent method (e.g., eclipses, or astrometry).
Why it matters here: Binary stars provide the only direct, model-independent measurements of stellar mass in astrophysics, with visual, spectroscopic, and eclipsing binaries each contributing a different, complementary piece of the full orbital and physical picture.
Relationship: For an eclipsing, double-lined spectroscopic binary (both stars’ lines visible and eclipses observed), the combination of radial-velocity curves (giving masses, with i fixed near 90° by the eclipses) and eclipse light-curve timing (giving radii) yields precise, nearly assumption-free masses and radii for both stars — the single most reliable source of fundamental stellar parameters in all of astrophysics.
Boundary or common confusion: Treating a single-lined spectroscopic binary’s mass function as a measured mass. Without an independently known inclination (or a visible second set of spectral lines), the mass function only gives a lower limit / degenerate combination of the companion mass and sin(i); this is exactly the ambiguity that made many early radial-velocity exoplanet "minimum masses" (m sin i) rather than true masses.
eclipsing binary
Definition: For an eclipsing, double-lined spectroscopic binary (both stars’ lines visible and eclipses observed), the combination of radial-velocity curves (giving masses, with i fixed near 90° by the eclipses) and eclipse light-curve timing (giving radii) yields precise, nearly assumption-free masses and radii for both stars — the single most reliable source of fundamental stellar parameters in all of astrophysics.
Why it matters here: Binary stars provide the only direct, model-independent measurements of stellar mass in astrophysics, with visual, spectroscopic, and eclipsing binaries each contributing a different, complementary piece of the full orbital and physical picture.
Relationship: For an eclipsing, double-lined spectroscopic binary with measured radial-velocity semi-amplitudes K1 and K2 and eclipse-confirmed i ≈ 90°, the individual masses follow directly (no sin³i ambiguity) from m1 sin³i and m2 sin³i ≈ m1 and m2 — the method underlying the great majority of precisely measured stellar masses used to calibrate the mass-luminosity relation in ASTR 230.
Boundary or common confusion: Treating a single-lined spectroscopic binary’s mass function as a measured mass. Without an independently known inclination (or a visible second set of spectral lines), the mass function only gives a lower limit / degenerate combination of the companion mass and sin(i); this is exactly the ambiguity that made many early radial-velocity exoplanet "minimum masses" (m sin i) rather than true masses.
radial velocity curve
Definition: Spectroscopic binaries (unresolved, but showing periodic Doppler shifts in one or both stars’ spectral lines) give a radial-velocity curve; because only the line-of-sight velocity component is measured, the derived masses carry an unavoidable sin(i) ambiguity unless the inclination i is independently known.
Why it matters here: Binary stars provide the only direct, model-independent measurements of stellar mass in astrophysics, with visual, spectroscopic, and eclipsing binaries each contributing a different, complementary piece of the full orbital and physical picture.
Relationship: For an eclipsing, double-lined spectroscopic binary (both stars’ lines visible and eclipses observed), the combination of radial-velocity curves (giving masses, with i fixed near 90° by the eclipses) and eclipse light-curve timing (giving radii) yields precise, nearly assumption-free masses and radii for both stars — the single most reliable source of fundamental stellar parameters in all of astrophysics.
Boundary or common confusion: Treating a single-lined spectroscopic binary’s mass function as a measured mass. Without an independently known inclination (or a visible second set of spectral lines), the mass function only gives a lower limit / degenerate combination of the companion mass and sin(i); this is exactly the ambiguity that made many early radial-velocity exoplanet "minimum masses" (m sin i) rather than true masses.
mass function
Definition: For a spectroscopic binary, the observed radial-velocity semi-amplitude K depends on the true orbital speed times sin(i); the derived "mass function" therefore only constrains a combination of masses and sin(i), not the individual masses, unless the inclination is broken by an independent method (e.g., eclipses, or astrometry).
Why it matters here: Binary stars provide the only direct, model-independent measurements of stellar mass in astrophysics, with visual, spectroscopic, and eclipsing binaries each contributing a different, complementary piece of the full orbital and physical picture.
Relationship: Without an independently known inclination (or a visible second set of spectral lines), the mass function only gives a lower limit / degenerate combination of the companion mass and sin(i); this is exactly the ambiguity that made many early radial-velocity exoplanet "minimum masses" (m sin i) rather than true masses.
Boundary or common confusion: Treating a single-lined spectroscopic binary’s mass function as a measured mass. Without an independently known inclination (or a visible second set of spectral lines), the mass function only gives a lower limit / degenerate combination of the companion mass and sin(i); this is exactly the ambiguity that made many early radial-velocity exoplanet "minimum masses" (m sin i) rather than true masses.
orbital inclination
Definition: For a spectroscopic binary, the observed radial-velocity semi-amplitude K depends on the true orbital speed times sin(i); the derived "mass function" therefore only constrains a combination of masses and sin(i), not the individual masses, unless the inclination is broken by an independent method (e.g., eclipses, or astrometry).
Why it matters here: Binary stars provide the only direct, model-independent measurements of stellar mass in astrophysics, with visual, spectroscopic, and eclipsing binaries each contributing a different, complementary piece of the full orbital and physical picture.
Relationship: Spectroscopic binaries (unresolved, but showing periodic Doppler shifts in one or both stars’ spectral lines) give a radial-velocity curve; because only the line-of-sight velocity component is measured, the derived masses carry an unavoidable sin(i) ambiguity unless the inclination i is independently known.
Boundary or common confusion: Treating a single-lined spectroscopic binary’s mass function as a measured mass. Without an independently known inclination (or a visible second set of spectral lines), the mass function only gives a lower limit / degenerate combination of the companion mass and sin(i); this is exactly the ambiguity that made many early radial-velocity exoplanet "minimum masses" (m sin i) rather than true masses.