Doppler Tomography: Tracing a Planet's Shadow Across the Stellar Line Profile

What Doppler Tomography Is and Why It Works

A rotating star does not emit a single sharp spectral line. Because one limb spins toward you and the other away, every point on the disc contributes a slightly Doppler-shifted copy of the line, and the observed profile is the rotationally broadened sum of all of them. Crucially, position on the disc maps to velocity: a patch at projected rotation velocity x along the equator contributes light at exactly that Doppler shift within the line.

When a planet transits, it blocks a small circular patch of photosphere. The light removed corresponds to one narrow velocity, so the planet subtracts a Gaussian-shaped notch of missing flux at that position in the line. In the residual (transit minus out-of-transit) spectrum this appears as a bright bump — the Doppler shadow. As the planet crosses the disc, the patch it covers changes its projected rotation velocity, so the bump marches across the line profile.

• Disc position ↔ line-profile velocity is the core mapping.
• The planet is a moving "pixel eraser" whose track is read off directly.

The Mechanism and Governing Relation

In practice one does not use a single line but the cross-correlation function (CCF) built from thousands of lines, which has the same rotation profile with far higher signal-to-noise. Collier Cameron and collaborators model the CCF as a limb-darkened rotation profile convolved with a Gaussian representing the local (thermal + instrumental) line width. The transiting planet removes a Gaussian of area set by the depth (Rp/R*)² at the local velocity of the patch it hides.

The velocity of the shadow at any phase depends on where the planet sits on the disc, (X, Y) in units of the stellar radius, and on the sky-projected obliquity λ:

• x = v sin i · (cos λ · X − sin λ · Y), where X runs along the sky-projected spin axis' perpendicular and Y along the transit chord.

Because the planet moves on a straight chord, the shadow traces a straight line in a phase-versus-velocity map. An aligned prograde orbit (λ ≈ 0°) gives a shadow that starts on the blueshifted side, crosses zero at mid-transit, and exits redshifted. A misaligned orbit tilts and offsets that line; a retrograde orbit reverses its sense.

Key Quantities and a Worked Example

The signal amplitude scales with the transit depth: for a hot Jupiter with Rp/R* ≈ 0.11, roughly 1.2% of the light is blocked, so the Doppler-shadow bump is about 1% of the line's central depth — small, hence the need for high resolution (R ≳ 50,000–100,000) and many exposures.

Worked example — WASP-33b. The host is an A5 star with effective temperature ≈ 7400 K and a huge projected rotation speed v sin i ≈ 86 km/s, so its line is broadened over ±86 km/s — far wider than the ~2 km/s Keplerian reflex the planet induces. Classical radial velocities could not confirm the planet. But the Doppler shadow, a bump ~1% deep sweeping ~170 km/s across the profile over the ~2.7-hour transit (period 1.22 days), was unmistakable.

• Projected obliquity: λ ≈ −108° — a strongly misaligned, essentially retrograde orbit.
• Shadow slope gives v sin i; its width gives the intrinsic local line width; its depth cross-checks Rp/R*.

How It Is Observed and Detected

An observing campaign captures a time series of high-resolution spectra across one full transit, plus baseline spectra before and after. Instruments such as HARPS, HARPS-N, HDS/Subaru, HIRES/Keck, ESPRESSO, and PEPSI/LBT provide the resolving power needed. Each spectrum is cross-correlated with a stellar template to produce a CCF; the out-of-transit average CCF is subtracted to reveal the residual bump.

Stacking the residual CCFs by transit phase produces the signature tomogram: a two-dimensional map (velocity on one axis, phase on the other) in which the planet's shadow appears as a bright diagonal streak. Reading its entry velocity, exit velocity, and tilt yields λ and v sin i simultaneously.

• Works best on hot, rapidly rotating A/F stars where lines are wide and the shadow is well resolved.
• Can even confirm planethood: the straight, coherent trail crossing the disc at the transit's ephemeris is a signature no stellar spot or pulsation mimics.
• Overlapping non-radial pulsations (as in the δ Scuti star WASP-33) must be modelled and removed.

Relation to the Rossiter–McLaughlin Effect and Cousins

Doppler tomography and the Rossiter–McLaughlin (RM) effect are two views of the same physics. The RM effect is the net radial-velocity distortion produced when the planet blocks blue- or red-shifted photosphere: the star appears to shift velocity anomalously during transit. Doppler tomography instead resolves the full line shape and watches the actual missing-light bump, rather than collapsing it to a single velocity.

• RM velocimetry excels for slow rotators (v sin i ≲ 10 km/s), where the line is nearly Gaussian and the anomaly is clean.
• Tomography excels for fast rotators (v sin i ≳ 15–20 km/s), where RM suffers degeneracies between λ, v sin i, and impact parameter.
• A newer relative, the reloaded RM technique, similarly isolates the local stellar CCF behind the planet and can even map stellar surface velocity fields, including differential rotation and center-to-limb convective effects.

All three trace their lineage to Doppler imaging of spotted stars and to accretion-disc Doppler tomography developed by Marsh and Horne in the late 1980s.

Significance, Famous Cases, and Open Questions

Spin–orbit angles are a fossil record of planet migration. A well-aligned orbit is consistent with smooth migration through a protoplanetary disc; large misalignments and retrograde orbits point to violent histories — planet–planet scattering, Kozai–Lidov cycles from a distant companion, or primordial tilting of the star itself.

• WASP-33b (λ ≈ −108°): first planet confirmed by Doppler tomography rather than radial velocity, and later used to detect nodal precession of its orbit.
• Kepler-13Ab: a misaligned, hot, prograde orbit mapped tomographically.
• KELT-9b: an ultra-hot Jupiter on a near-polar orbit around a ~10,000 K star, where fast rotation makes tomography the natural tool.

Open questions remain. λ is only the sky-projected angle; the true 3D obliquity ψ requires knowing the stellar inclination (from v sin i, rotation period, and radius, or from asteroseismology). Whether hot-Jupiter misalignments reflect the planet's dynamics or a tilted star, and why cool stars (below the ~6250 K Kraft break) tend to be more aligned, are debated — with tidal re-alignment a leading but contested explanation.

Doppler tomography versus the classical Rossiter–McLaughlin (RV) method for measuring spin–orbit alignment

Property	Doppler tomography	Rossiter–McLaughlin (RV anomaly)
What is measured	Bright bump moving through the resolved line profile / CCF	Net radial-velocity distortion (one number per spectrum)
Ideal host star	Fast rotator, v sin i ≳ 15 km/s	Slow rotator, v sin i ≲ 10 km/s
Planet's path in data	Straight line (trail) — extrapolatable from partial coverage	Anomaly curve degenerate with v sin i and impact parameter
Can confirm the planet?	Yes — WASP-33b confirmed this way when RV failed	No — needs an independent mass signal
Extra outputs	Rp/R*, intrinsic line width, non-radial pulsations	Mainly λ and v sin i
Failure mode	Needs high S/N, many spectra through transit	Degeneracies inflate errors for rapid rotators