Spectroscopic Binary

The pair you cannot see

Most binary stars in the Galaxy are too close together — in real space, or just in projection on the sky — for a telescope to resolve into two separate points of light. A typical solar-type pair at 30 parsecs separated by 0.1 astronomical units subtends about 30 microarcseconds, four orders of magnitude below the diffraction limit of a 10-metre telescope at optical wavelengths. They blend into a single image. Yet the spectrum tells a different story. Stars have sharp absorption lines whose laboratory rest wavelengths are known to fractions of a picometre. If two stars are orbiting one another, each line in each star moves periodically toward the blue when the star approaches us and toward the red when it recedes. The composite spectrum oscillates, and from that oscillation alone one can recover the orbit, the mass ratio, and — with luck — the individual masses.

That is a spectroscopic binary. The pair is unresolved as photons, but resolved as wavelengths.

SB1 versus SB2 — one set of lines or two

If the secondary is much fainter than the primary, only the primary's spectrum is visible. As the primary swings around the centre of mass, its lines slide back and forth by an amount Δλ/λ = v_r/c, with v_r the line-of-sight velocity. You see one set of lines doing the sliding. The system is single-lined — SB1. Without a second curve you can only extract a single quantity that depends on both the unseen secondary's mass and the unknown inclination: the mass function.

If both components contribute lines of comparable strength, you see two superimposed sets that alternately separate and merge each orbital cycle. Near conjunction the lines blend; near quadrature they reach maximum split. The two amplitudes K₁ and K₂ relate the radii of the two orbits around the barycenter:

M₁ K₁ = M₂ K₂      →      q ≡ M₂ / M₁ = K₁ / K₂

The mass ratio drops out instantly. The system is double-lined — SB2 — and is by far the more informative configuration. The Pourbaix et al. 2004 Ninth Catalogue of Spectroscopic Binary Orbits (SB9) lists more than 2500 systems with published orbital elements; a comfortable majority are SB1 because the second star is fainter, evolved, or a compact remnant.

The radial-velocity curve

For a circular orbit, the line-of-sight velocity of one component is

v_r(t) = γ + K sin( 2π(t − t₀)/P )

where γ is the systemic velocity of the barycenter, K is the semi-amplitude, P is the orbital period, and t₀ is the time of ascending node. Eccentric orbits deform the sinusoid into a characteristic asymmetric shape parameterised by eccentricity e and argument of periastron ω; the classical Lehmann-Filhés method (1894) recovers all five elements (P, K, γ, e, ω) plus t₀ from the fit. The semi-amplitude is geometrically

K = (2π a sin i) / (P √(1 − e²))

so K, P, and e determine a sin i — the projected semi-major axis of the star's path around the barycenter. Inclination i remains a free parameter until extra information arrives.

The mass function and the sin³i degeneracy

Substituting K into Kepler's third law and rearranging gives the canonical mass function:

f(M) ≡ (K₁³ P (1 − e²)^(3/2)) / (2π G)
     = M₂³ sin³i / (M₁ + M₂)²

For an SB1, the left-hand side is a direct observable; the right-hand side mixes the companion mass with the inclination. The two cannot be separated. Conventionally one reports f(M) itself, or a minimum companion mass under the assumption sin i = 1 (edge-on). Doubling the mass function gives a useful lower bound on M₂ when M₁ is independently known.

For an SB2 you measure K₁ and K₂. The mass ratio q = K₁/K₂ is exact (and inclination-independent), and

M₁ sin³i = (P (1 − e²)^(3/2) (K₁ + K₂)² K₂) / (2π G)
M₂ sin³i = (P (1 − e²)^(3/2) (K₁ + K₂)² K₁) / (2π G)

Both individual masses are known up to the same sin³i factor. Statistical inversion across an ensemble of orbits — assuming random orientations — was already in use by 1924 (Aitken) and remains the standard cluster-binary technique today. For any one system, only an eclipse or astrometric orbit pins i down.

When the orbit also eclipses

An eclipse implies i is near 90°. Geometrically, a transit happens when cos i ≲ (R₁ + R₂)/a, so a star whose secondary blocks any portion of its disk has an inclination of at most a few degrees from edge-on. The eclipse duration and depth additionally constrain i, R₁/a, R₂/a, and the surface brightness ratio.

The combined data products are spectacular. A double-lined eclipsing binary delivers, with no astrophysical assumptions beyond Kepler's law:

• Both stellar masses M₁ and M₂ to ≲1% precision.
• Both radii R₁ and R₂ from eclipse duration / ingress.
• Surface gravities g = GM/R² (the most precisely known stellar parameter of all).
• Effective temperature ratio from depth of primary versus secondary eclipse.

The DEBCat catalogue maintained by John Southworth lists ~250 detached double-lined eclipsing binaries with masses and radii better than 2%. These systems anchor the empirical mass-luminosity and mass-radius relations and provide the calibration data for stellar isochrones, asteroseismic scaling, and gyrochronology.

From naked-eye spectroscopy to one metre per second

Pickering's 1889 detection of Mizar relied on visually inspecting photographic plates and noticing the calcium K-line had split. The precision was on the order of tens of km/s — adequate for a 70 km/s binary. Modern echelle spectrographs reach 1 m/s and below, an improvement of more than four orders of magnitude. Two innovations made this possible:

• The iodine-cell technique (Marcy & Butler 1992). A heated glass cell of I₂ gas is placed in the beam ahead of the spectrograph. The starlight then passes through and acquires a forest of thousands of narrow molecular absorption lines from the iodine, superimposed on the stellar spectrum. Because the iodine lines and the stellar lines traverse the same optics on the same exposure, instrumental drift cancels to first order. The HIRES spectrograph at Keck and the HARPS spectrograph at La Silla (which uses ThAr instead) reach ~0.5–1 m/s.
• The thorium-argon (ThAr) hollow-cathode lamp. A separate calibration spectrum from a Th/Ar lamp is recorded with each exposure, providing a dense reference grid tied to laboratory atomic standards. With careful drift modelling HARPS achieves 0.6 m/s.
• Laser frequency combs. Stabilised mode-locked lasers produce an evenly spaced, absolutely calibrated grid of reference lines; ESPRESSO at the VLT now reaches ~10 cm/s on bright stars.

The improvement in precision shifted the discovery space from stellar binaries with K of tens of km/s to planetary companions with K of a few m/s — and finally to small rocky planets with K of cm/s, the regime where Earth-twin detection becomes possible.

Modern surveys and the binary-star census

The 1980s SB9 catalogue compiled orbits one paper at a time. Today, ensemble RV surveys deliver them by the thousand.

Survey	Spectral range	Targets	Output
Gaia DR3 RVS	847–874 nm (Ca II triplet)	~33 million stars with RV	10⁵+ RV-variable candidates, ~250k orbital solutions
APOGEE (SDSS)	1.51–1.70 μm (H-band)	700k+ stars, mostly red giants	Multi-epoch RVs cutting through Galactic dust
LAMOST	370–900 nm	10⁷ low-res spectra	RV variability flags; SB2 detection
TESS	Photometry, 600–1000 nm	Full-sky transit photometry	Eclipsing-binary identification for follow-up
HARPS/HARPS-N/ESPRESSO	378–691 nm	Bright FGK stars	1 m/s and sub-m/s precision for exoplanets
HIRES/HARPS-N planet-search	Optical	~5000 bright stars	Confirmed exoplanets, hot/dormant compact-companion finds

APOGEE's near-infrared coverage is critical for studying dust-obscured regions like the Galactic plane and the bulge, where optical RV is blocked. Gaia's RV instrument is comparatively low-resolution and low-precision (R ≈ 11500, ~1 km/s), but the sheer N is transformative; one expects the SB1+SB2 catalogue to grow by two orders of magnitude in the next decade.

From stellar binaries to 51 Pegasi b

The Doppler technique scales seamlessly between stellar and planetary mass companions; only the amplitude shrinks. A Jupiter at 0.05 AU produces K ≈ 50 m/s on a Sun-like star; an Earth at 1 AU produces 9 cm/s. In late 1995 Michel Mayor and Didier Queloz, using the ELODIE spectrograph at Haute-Provence, reported a 51.6 m/s sinusoidal RV variation of 51 Pegasi at a 4.231-day period — too brief to be a brown dwarf or low-mass star, too smooth to be activity, exactly what a 0.47 M_Jup planet at 0.05 AU would produce. The discovery, announced at the Florence IAU symposium on 6 October 1995, opened the exoplanet era and earned the 2019 Nobel Prize.

Every subsequent radial-velocity exoplanet discovery — Geoff Marcy and Paul Butler's tally that ran into the hundreds, the HARPS Mediterranean catalogue, the M-dwarf programs of CARMENES and MAROON-X — has used the same machinery a 19th-century astronomer would have recognised. Lines shift; you measure how much; you fit a sinusoid.

Hunting compact remnants in plain sight

One of the powerful uses of single-lined spectroscopy is finding stellar-mass black holes and neutron stars that emit no light themselves. The recipe: identify a bright star with anomalously large RV amplitude and no second set of lines, derive f(M), and check whether the implied minimum companion mass exceeds the maximum neutron-star mass (~2.3 M☉). If so, the companion must be a black hole.

• Cygnus X-1. The blue supergiant HD 226868 shows a 70 km/s RV amplitude on a 5.6-day orbit; combined with X-ray constraints and inclination from polarimetry, M₂ ≈ 21 M☉ — the canonical Galactic stellar black hole.
• LB-1 (controversial). Initially proposed as a 70 M☉ black hole binary (Liu et al. 2019); subsequent analysis attributed the apparent SB1 to a stripped helium star.
• Gaia BH1. El-Badry et al. (2023) used Gaia DR3 astrometry + ground-based RV follow-up to identify a 9.6 M☉ black hole in a 185.6-day orbit with a Sun-like star, at 480 pc — the closest known.
• Gaia BH2, BH3. 2023–2024 follow-ups extended the catalogue to wider orbits.

The challenge for these searches is rejecting false positives: stripped stars, post-mass-transfer systems, and active stars all mimic large RV with no visible companion. Detailed spectroscopic diagnostics — line broadening, He I/He II strength, evolutionary status — are what distinguish a true dark companion from an exotic but luminous one.

Worked example: extracting masses from an SB2 + eclipse

Suppose you observe an eclipsing SB2 with the following measurements:

• Orbital period P = 4.000 days = 3.456 × 10⁵ s.
• Eccentricity e = 0 (circularised).
• Primary RV semi-amplitude K₁ = 75 km/s.
• Secondary RV semi-amplitude K₂ = 125 km/s.
• Eclipses present (so sin i ≈ 1).

Mass ratio: q = M₂/M₁ = K₁/K₂ = 75/125 = 0.60. (Primary is the more massive star, so q < 1.)

Total semi-major axis: a sin i = (K₁ + K₂) P / (2π) = (2 × 10⁵ m/s)(3.456 × 10⁵ s) / (2π) = 1.10 × 10¹⁰ m ≈ 0.0734 AU.

Total mass from Kepler: M₁ + M₂ = (4π² a³)/(G P²) with a in metres, P in seconds. Plugging in a = 1.10 × 10¹⁰ m yields M₁ + M₂ ≈ 4.06 × 10³⁰ kg = 2.04 M☉.

Individual masses: M₁ = M_total / (1 + q) = 2.04 / 1.60 = 1.28 M☉; M₂ = q M₁ = 0.77 M☉. A G dwarf with a K dwarf companion — entirely typical numbers.

Add the eclipse light curve and you obtain the radii R₁ and R₂ directly. Done. You have characterised a binary star to better precision than is achievable for any single star in the Galaxy except the Sun.

Common pitfalls and subtleties

• Confusing M sin i with M. For an SB1 you never recover M alone. Reporting M sin i as if it were the true mass is a category error; the convention is to print it explicitly with the sin i factor.
• Stellar activity masquerading as a planet. Starspots and faculae rotating across the disk produce sinusoidal RV signals at the rotation period. The 2014 case of GJ 581d, originally claimed as a habitable-zone planet, evaporated when activity was modelled out. Robust planet detection requires multi-band photometry and bisector analysis to separate activity from genuine Keplerian signals.
• Eccentric orbits and orbital aliasing. Sparse sampling can confuse a moderately eccentric orbit with a higher-eccentricity-but-shorter-period alternative. Bayesian fitting and proper period-search tools (Lomb-Scargle, Kepler-likelihood) are essential.
• Triples masquerading as binaries. Hierarchical triples are common; an unresolved third component can bias derived masses or introduce long-period RV modulation that looks like eccentricity drift. The Kozai-Lidov mechanism makes such systems astrophysically interesting in their own right.
• Line blending at low inclination or low resolution. Near orbital conjunction the two sets of lines overlap; if the spectrograph resolving power R is too low to separate them, the fitted velocities are biased. Two-dimensional cross-correlation (TODCOR; Zucker & Mazeh 1994) is the standard fix.
• γ velocity is not the same as systemic gravitational redshift. Different lines have slightly different formation depths and convective blueshifts; comparing γ-velocities across components requires care. For relativistic systems (compact-object binaries) this matters at the 1 km/s level.