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Spectroscopic Binaries & Mass Determination Cheat Sheet
A printable reference covering spectroscopic binaries, Doppler shifts, radial velocity curves, Kepler’s laws, mass functions, and inclination effects for grades 11-12.
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Spectroscopic binaries are star systems that look like one point of light but reveal two orbiting stars through their spectra. This cheat sheet helps students connect observed wavelength shifts to orbital motion and stellar mass. It is useful because many binary stars cannot be resolved in a telescope, so spectroscopy becomes the main evidence. Understanding these systems shows how astronomers measure masses beyond the solar system. The core idea is that Doppler shifts give radial velocity, which is the speed toward or away from Earth. A radial velocity curve gives the orbital period, velocity amplitudes, and evidence for one-lined or two-lined binaries. Kepler’s third law connects orbital size and period to total mass. Because inclination affects what we observe, spectroscopic masses are often lower limits unless the orbit angle is known.
Key Facts
- The Doppler shift formula for nonrelativistic speeds is v_r / c = Δλ / λ0, where v_r is radial velocity, c is the speed of light, Δλ is wavelength shift, and λ0 is rest wavelength.
- A positive radial velocity usually means the star is moving away from the observer, producing a redshift.
- A negative radial velocity usually means the star is moving toward the observer, producing a blueshift.
- For a circular two-lined spectroscopic binary, the mass ratio is M1 / M2 = K2 / K1, where K is the radial velocity amplitude.
- Kepler’s third law for a binary system is M1 + M2 = a^3 / P^2 when mass is in solar masses, separation a is in AU, and period P is in years.
- The observed radial velocity amplitude depends on inclination: K_observed = K_true sin i, so face-on systems have very small Doppler shifts.
- For a single-lined spectroscopic binary, the mass function is f(M) = (M2^3 sin^3 i) / (M1 + M2)^2 = P K1^3 / (2πG) for a circular orbit.
- Double spectral lines that move in opposite directions over time are strong evidence for a two-lined spectroscopic binary.
Vocabulary
- Spectroscopic binary
- A binary star system detected by periodic Doppler shifts in its spectral lines rather than by direct visual separation.
- Radial velocity
- The component of an object’s velocity along the observer’s line of sight.
- Doppler shift
- A change in observed wavelength caused by motion toward or away from the observer.
- Velocity amplitude
- The maximum radial speed measured from a star’s radial velocity curve, often written as K.
- Inclination
- The tilt of an orbit relative to the observer’s line of sight, with i = 90° meaning edge-on.
- Mass function
- A formula that uses the observed period and radial velocity to place a lower limit on the companion’s mass.
Common Mistakes to Avoid
- Confusing radial velocity with total orbital velocity is wrong because spectroscopy measures only motion along the line of sight.
- Ignoring inclination gives incorrect masses because the observed velocity is reduced by the factor sin i.
- Assuming one visible spectrum means one star is wrong because a faint companion can still affect the visible star’s motion.
- Using wavelength shift without the rest wavelength is wrong because Δλ must be compared to λ0 in the Doppler formula.
- Mixing units in Kepler’s third law leads to wrong masses because M1 + M2 = a^3 / P^2 works only with AU, years, and solar masses.
Practice Questions
- 1 A spectral line with rest wavelength 500.0 nm is observed at 500.2 nm. Using c = 3.00 x 10^5 km/s, what is the star’s radial velocity?
- 2 A two-lined spectroscopic binary has velocity amplitudes K1 = 80 km/s and K2 = 120 km/s. What is the mass ratio M1 / M2?
- 3 A binary has orbital separation a = 2.0 AU and period P = 1.0 year. What is the total mass M1 + M2 in solar masses?
- 4 Why can a spectroscopic binary with a face-on orbit be difficult to detect even if the stars are orbiting quickly?