Radial Velocity Profiles of Outflows

• Radial velocity profiles of outflows define how ejected material's speed varies with radius, encoding key acceleration and deceleration mechanisms.
• Analytical formulations combine gravitational redshift with Doppler and transverse effects to accurately predict emission and absorption line morphologies.
• Understanding these profiles helps diagnose wind launching conditions, infer mass loss rates, and distinguish between decelerated and failed outflow models.

Radial velocity profiles of outflows describe how the bulk velocity of ejected material varies as a function of radius from the source—in contexts ranging from compact objects in strong gravity to star-forming galaxies and AGN winds. These profiles encode both the acceleration mechanisms and the interaction of the outflow with its local environment, and they are a central diagnostic in interpreting emission and absorption line spectra. The nature of these profiles is set by the wind launching conditions, local forces (including radiation, gravity, pressure gradients, and magnetic fields), and any subsequent deceleration due to ambient material or radiative losses.

1. Analytical Formulation of Velocity Profiles Near Compact Objects

In outflows from neutron stars and black holes, especially those launched within a few tens of Schwarzschild radii, strong gravity modifies the classical expectations for velocity and line profile morphology (Dorodnitsyn, 2011). The wind’s radial velocity profile can be parameterized as:

• For a purely decelerating wind:

$u(x) = \frac{U^\infty}{x^m}$

where $u(x)$ is the velocity normalized to the local thermal speed, $x = r/R_c$ with $R_c$ the launching radius, $U^\infty$ is the terminal velocity in units of thermal speed, and $m > 0$ controls the steepness of deceleration.

• For a "failed" wind (brief acceleration, then deceleration):

$$u(x) = U_0 \left\{ \frac{1}{ x^m - (1 - w_0/U_0) \exp [ -(x-1)^m / \varepsilon ] }\right\}$$

with $w_0 \ll 1$ the initial velocity, $\varepsilon \ll 1$ the acceleration phase width, and $U_0$ set to match maximum velocity to $U^\infty$. Both velocity laws are specified in units of $v_\mathrm{th}$.

These velocity profiles interact with gravitational redshift and Doppler effects. The observed line frequency shift for a photon escaping to infinity is given by:

$$y^\infty = \mu u(x) + \Phi(x) - \frac{u(x)^2}{2\zeta}$$

where $\mu$ is the angle cosine between photon direction and local velocity, $\Phi(x)$ is the dimensionless gravitational potential, and $\zeta = c/v_\mathrm{th}$. This equation combines longitudinal Doppler, gravitational, and transverse Doppler shifts.

When the wind forms within $\lesssim 20–30\, r_g$ (with $r_g = 2GM/c^2$), gravitational redshift can dominate over Doppler blueshift, fundamentally altering the shape and shift of observed spectral lines.

2. Impact on Spectral Line Morphologies

Radial velocity profiles, in combination with gravitational effects, produce a spectrum of emergent line shapes (Dorodnitsyn, 2011):

• Distorted P Cygni Profiles: Classical P Cygni (blueshifted absorption, red emission) become distorted by gravity, leading to reduced separation and shifted features.
• W-Shaped Profiles: When resonance conditions (equal frequency surfaces, EFS) are split across the flow, triple-peaked profiles can arise, typically when the opacity is centrally concentrated.
• Inverted P Cygni Profiles: In strongly decelerating or "failed" outflows under deep gravitational potential, blue-to-red absorption inversion occurs even for an outflow, mimicking infall (normally associated with accretion), provided line formation occurs at small radii.

These morphological distinctions directly encode the velocity law, gravitational field, and opacity distribution. The particular form of opacity (e.g., whether $\tau_r \propto (1-w^k)$ or $\tau_r \propto w^k$) can emphasize line features tied to specific flow zones.

3. Decelerated vs. Failed Outflows: Dynamics and Observational Consequences

• Decelerated winds (pure $1/r^m$ decline) have their maximum velocity close to the launch radius, still deep in the potential well, yielding classic or W-shaped line profiles.
• Failed winds (rapid acceleration, then steep deceleration) can, under conditions of strong gravity and large $m$, present unusual inverted P Cygni profiles. The existence of such profiles in an outflowing medium—rather than an accreting one—is a distinct consequence of the velocity structure combined with gravitational redshift.

These behaviors make it clear that, especially for compact object winds, the precise form of the velocity profile and location of peak velocity relative to the potential are as important as the total flow speed.

4. Gravitational Redshift and the Role of Launch Radius

The launch radius (parameterized by $g_0 = R_c / r_g$) plays a deterministic role in shaping velocity profiles (Dorodnitsyn, 2011):

• For winds launched at large radii ($g_0 \gg 1$), gravitational effects are subdominant, and Doppler shifts retain classical dominance.
• For winds originating at $R_c \lesssim 20–30 \, r_g$, gravitational redshift can exceed Doppler shifts, leading to the observed signature reversals (e.g., "inverted" line profiles).

The equal frequency surfaces (EFS)—locations where the rest-frame line is resonant with a given frequency observed at infinity—are strongly warped when the gravitational potential is deep, which in turn dictates which parts of the flow contribute to either emission or absorption at particular observed frequencies.

5. Diagnostic Implications for Observed Systems

The interplay of velocity law, gravitational field, and opacity structure allows for direct diagnosis of physical properties in observed compact object outflows:

Profile Type	Wind Model/Condition	Gravitational Impact	Observational Signature
Distorted P Cygni	Decelerating ($m\sim1–2$)	Modest (larger launch radius)	Emission/absorption shifted or blended
W-shaped	Opacity peaks centrally, either wind model	Moderate to strong	Triple-peaked (abs–em–abs) profile
Inverted P Cygni	Rapid deceleration, strong gravity	Launch radius $\lesssim 20 r_g$	Redshifted absorption with blue emission

Application to real data enables inferences on wind launching mechanism (steady wind, failed wind), mass loss rates, and proximity of the wind acceleration zone to the central object, thereby constraining accretion and feedback models.

6. Theoretical and Observational Extensions

The velocity models organized by (Dorodnitsyn, 2011) lay a framework extensible beyond neutron stars and black holes:

• In AGN and X-ray binaries, the same interplay of velocity law and gravitational field determines absorber kinematics and line shapes, as long as line formation occurs close to the event horizon.
• These results also establish the caution that apparent “inverted” absorption features can arise from an outflow rather than uniquely signaling inflow, provided the gravitational and velocity conditions are met.

Future extensions involve incorporating full general relativistic transfer and more complex velocity/opacity laws tailored to multidimensional flows or MHD winds, facilitating direct comparison with high-resolution, velocity-resolved X-ray and UV spectra from current and next-generation observatories.

7. Summary

Radial velocity profiles of outflows near compact objects are determined by the interplay of acceleration/deceleration laws and gravitational redshift. Both analytical and numerical modeling demonstrate that the velocity field dictates not only the location and magnitude of Doppler shifts but, in strong gravitational potentials, fundamentally alters the mapping between spatial and spectral observables. Correct interpretation of observed line profiles—especially for systems close to a black hole or neutron star—must account for these effects to avoid misattributing inflow vs. outflow and to accurately diagnose the physical state and dynamics of the source. The framework established supports robust inverse modeling of emission and absorption profiles to infer wind parameters and launching conditions in a broad range of compact astrophysical systems (Dorodnitsyn, 2011).