Relativistic Doppler vs. Emission Angle

Source

Collection: Wiki SR simulations (v1) — SuperSTEM special-relativity engine sim.
Concept: The relativistic Doppler factor f_obs/f_src = 1 / [ γ(1 − β cosθ) ] depends on the emission angle θ measured in the observer frame; the no-shift angle sits before transverse, and θ = 90° gives a pure 1/γ redshift.
Serves: Shows how frequency shift varies with viewing angle for a moving source, distinguishing relativistic Doppler (with its transverse 1/γ redshift) from the classical geometric shift.
Inquiry: At what emission angle does a moving source produce no frequency shift — exactly transverse (90°), tilted toward approach (<90°), or toward recession (>90°)?
Misconception: That "sideways = no Doppler," i.e. that θ = 90° is the no-shift angle. In fact the transverse direction is a genuine 1/γ redshift from time dilation, so the no-shift angle is always < 90°.

Pedagogy principles applied

ID	Principle	How it shows up here
H1	Misconception is structural, not a text card	The "θ = 90° is the no-shift angle" and "beyond 90°" errors are wrong prediction choices; the rebuttal appears only if the student picks one.
D6/D7	Guided inquiry reveals in stages	Sidebar step-cards run predict → observe → verify the no-shift angle → the transverse 1/γ redshift → resolve; the plot's θ₀ marker is revealed only after the student has watched the source emit.
A2	Fewest objects	Opens with one moving source, its wavefronts, and one observer at the slider angle; the f/θ curve carries a single marker bound to the θ slider.
C	Every number earns its place	γ, the Doppler factor at your θ, the no-shift angle θ₀, and the transverse 1/γ value are each read or compared by a step.
Physics honesty	No false animation of β	β and θ are static sliders the student sets; the time-animation is genuine wave propagation (expanding crests, blinking pulses), not a swept velocity.

Relativistic Doppler frequency shift vs. emission angle for a moving source

Speed

Moving source & observerobserver-frame geometry

f_obs / f_src vs θobserver frame · log scale

Guided Inquiry

1 · Predict the no-shift angle

A source moves at β = v/c and emits light. At what emission angle θ (measured in the observer frame) is there no frequency shift — f_obs = f_src?

2 · Watch the source emit

Press ▶ in the top bar. The source moves and emits wave crests: they bunch ahead (blueshift) and spread behind (redshift). The two pulse dots blink at the exact relativistic rates f_src and f_obs. Try dragging the θ slider through 90°.

3 · Where does the curve cross 1?

Read the No-shift angle θ₀ readout (revealed once the source has emitted) and the green crossing on the f/θ plot. Relative to transverse (90°), the no-shift angle θ₀ sits …

4 · What happens exactly at 90°?

Set θ = 90° with the slider. Classically, a source moving straight across your line of sight has zero Doppler shift. What does the relativistic factor read here?

5 · Angle is frame-dependent

The full curve f_obs/f_src = 1/[γ(1 − β cosθ)] runs from blueshift at θ = 0° through the no-shift angle θ₀ < 90°, the transverse 1/γ redshift at 90°, to deep redshift at 180°. The θ you set is the observer-frame angle; the same physical direction has a different angle in the source frame (aberration). Turn on ∑ Formal for the algebra, or sweep β and θ freely. Hit Finish to explore.

Controls

▾

Source speed β = v/c 0.60

0.050.95

Emission angle θ (observer frame) 60°

0°180°

Readouts

▾

Lorentz γ—

f_obs/f_src at your θ—

No-shift angle θ₀ (observer frame)—

Transverse factor (θ=90°) = 1/γ—

Relativistic Doppler factor (observer-frame θ)

θ is measured in the observer frame. θ = 0° → source approaching (blueshift); θ = 180° → receding (redshift); θ = 90° → transverse.

Transverse (θ = 90°)

The ratio reduces to 1/γ — a pure redshift from time dilation. Classically the transverse shift is zero; the 1/γ factor is a purely relativistic effect.

No-shift angle

Solving γ(1 − β cosθ) = 1 gives the angle where f_obs = f_src; it is always less than 90°.

Aberration (source-frame angle)

The same physical direction has a different angle in the source frame — angles are frame-dependent, just like lengths and times.