Fed-Star in Astrophysics & Federated Learning

• Fed-Star is a term applied to diverse, domain-specific feeding mechanisms, covering astrophysical inflows (e.g., stream-fed star formation and wind-fed accretion) and federated learning strategies.
• In astrophysics, Fed-Star describes external feeding processes that regulate star formation and accretion via turbulent energy injection and clump-fed mechanisms, supported by analytic models and simulations.
• In federated learning, FedSTAR encompasses methods like self-training with pseudo-labels and style-aware transformer aggregation that enhance performance under label scarcity and non-IID client data.

“Fed-Star” and “FedSTAR” denote several distinct constructs in astrophysics and federated learning rather than a single unified concept. Across these usages, the recurring motif is external feeding: cold streams feeding galaxies, stellar winds feeding accretion flows, clump-scale inflow feeding massive protostellar fragments, or exchanged representations feeding federated models. In galaxy formation, the term describes delayed star formation in high-redshift stream-fed galaxies (Gabor et al., 2013). In compact-object and stellar contexts, it appears in wind-fed accretion onto the supermassive black hole M31*, clump-fed accretion in high-mass star-forming objects, and wind-fed disks in binaries (Su et al., 5 Jun 2025, Traficante et al., 2023, Kulikova et al., 2019). In machine learning, FedSTAR names both a semi-supervised federated self-training method for audio recognition and a personalized federated-learning framework based on style-aware prototype aggregation (Tsouvalas et al., 2021, Jeon et al., 24 Nov 2025).

1. Nomenclature and scope

Domain	Meaning of “Fed-Star” / “FedSTAR”	Central mechanism
Galaxy formation	Delayed star formation in high-redshift stream-fed galaxies	Inflow-driven turbulence suppresses star formation
SMBH accretion	Stellar-wind feeding of M31*	AGB-star winds build a cool quasi-Keplerian disk
Massive star formation	Clump-fed accretion mechanism	Parsec-scale inflow sustains fragment growth
Binary accretion	Wind-fed accretion disk	Red-giant wind feeds a thin disk around a companion
Federated learning	FEderated Self-TRAining	Pseudo-labeling exploits on-device unlabeled audio
Personalized FL	Federated Style-Aware Transformer Aggregation of Representations	Content–style disentanglement and attention-weighted prototype fusion

The arXiv record therefore uses the same lexical label for unrelated problems. In astrophysics, the term is attached to feeding mechanisms in gaseous systems; in federated learning, it functions as an acronym. This suggests a mnemonic convergence rather than a standardized cross-disciplinary taxonomy.

A common misconception is to treat “Fed-Star” as a singular model family. The cited literature does not support that reading. Instead, each usage is domain-specific, with independent definitions, observables, and mathematical formalisms.

2. Stream-fed suppression of star formation in high-redshift galaxies

In “Delayed star formation in high-redshift stream-fed galaxies,” the Fed-Star mechanism proposes that star formation is delayed relative to the inflow rate in rapidly accreting galaxies at very high redshift because the accreting gas conveys energy into the disk and raises turbulence above the level compatible with gravitational instability (Gabor et al., 2013). The inflowing gas therefore acts simultaneously as fuel and as a stabilizing agent.

The analytic model begins from turbulent energy injection by cold streams. For an inflow rate $\dot M_{\rm inflow}$, infall velocity $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$, and coupling fraction $\epsilon$, the injection rate is

$$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$

which yields the scaling

$$\sigma_{\rm turb}\simeq \bigl[\epsilon\,(\dot M/M_{\rm gas})\bigr]^{1/2}R_{\rm gal}.$$

Internal processes enforce a floor

$$\sigma_{\min}=\frac{Q_{\min}\,\pi G\,\Sigma_{\rm gas}}{\kappa}, \qquad Q_{\min}\approx 0.7,$$

and the actual dispersion is

$\sigma=\max(\sigma_{\min},\sigma_{\rm turb}),$

so that the instantaneous Toomre parameter becomes

$Q=\frac{\kappa\,\sigma}{\pi G\,\Sigma_{\rm gas}}.$

Whenever inflow-driven turbulence dominates, $Q$ rises above unity and the disk is stabilized against fragmentation.

The star-formation law is then modified through the density PDF. For a log-normal PDF with width

$$\sigma_{\ln\rho}^2=\ln\bigl[1+b^2(\sigma/c_s)^2\bigr],$$

the efficiency per free-fall time is written

$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$0

This enters a Kennicutt-style law of the form

$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$1

with $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$2–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$3 when turbulence suppresses collapse. The gas fraction is

$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$4

The redshift dependence is central. At $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$5, theoretical accretion rates scale as $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$6, so low-mass galaxies experience very high $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$7. For $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$8 and $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$9–$\epsilon$0, the model gives $\epsilon$1 and hence $\epsilon$2–$\epsilon$3. The star-formation efficiency is reduced by a factor of about three relative to the self-regulated floor, and $\epsilon$4–$\epsilon$5 is maintained down to $\epsilon$6. As $\epsilon$7 drops below $\epsilon$8–$\epsilon$9, the specific inflow rate falls by $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$0 and the geometric coupling factor $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$1 decreases as filaments decouple from the compact disk. Then $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$2, $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$3–$$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$4, $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$5 returns to its canonical $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$6, and $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$7 declines toward $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$8–$$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$9 by $$\sigma_{\rm turb}\simeq \bigl[\epsilon\,(\dot M/M_{\rm gas})\bigr]^{1/2}R_{\rm gal}.$$0.

Idealized hydrodynamic simulations with RAMSES at $$\sigma_{\rm turb}\simeq \bigl[\epsilon\,(\dot M/M_{\rm gas})\bigr]^{1/2}R_{\rm gal}.$$1 resolution down to $$\sigma_{\rm turb}\simeq \bigl[\epsilon\,(\dot M/M_{\rm gas})\bigr]^{1/2}R_{\rm gal}.$$2 support the analytic picture. At $$\sigma_{\rm turb}\simeq \bigl[\epsilon\,(\dot M/M_{\rm gas})\bigr]^{1/2}R_{\rm gal}.$$3, runs with $$\sigma_{\rm turb}\simeq \bigl[\epsilon\,(\dot M/M_{\rm gas})\bigr]^{1/2}R_{\rm gal}.$$4, $$\sigma_{\rm turb}\simeq \bigl[\epsilon\,(\dot M/M_{\rm gas})\bigr]^{1/2}R_{\rm gal}.$$5, and three filamentary streams totaling $$\sigma_{\rm turb}\simeq \bigl[\epsilon\,(\dot M/M_{\rm gas})\bigr]^{1/2}R_{\rm gal}.$$6 yield a coupling efficiency $$\sigma_{\rm turb}\simeq \bigl[\epsilon\,(\dot M/M_{\rm gas})\bigr]^{1/2}R_{\rm gal}.$$7–$$\sigma_{\rm turb}\simeq \bigl[\epsilon\,(\dot M/M_{\rm gas})\bigr]^{1/2}R_{\rm gal}.$$8. At $$\sigma_{\rm turb}\simeq \bigl[\epsilon\,(\dot M/M_{\rm gas})\bigr]^{1/2}R_{\rm gal}.$$9, analogous runs with $$\sigma_{\min}=\frac{Q_{\min}\,\pi G\,\Sigma_{\rm gas}}{\kappa}, \qquad Q_{\min}\approx 0.7,$$0, $$\sigma_{\min}=\frac{Q_{\min}\,\pi G\,\Sigma_{\rm gas}}{\kappa}, \qquad Q_{\min}\approx 0.7,$$1, and $$\sigma_{\min}=\frac{Q_{\min}\,\pi G\,\Sigma_{\rm gas}}{\kappa}, \qquad Q_{\min}\approx 0.7,$$2 give $$\sigma_{\min}=\frac{Q_{\min}\,\pi G\,\Sigma_{\rm gas}}{\kappa}, \qquad Q_{\min}\approx 0.7,$$3. In the $$\sigma_{\min}=\frac{Q_{\min}\,\pi G\,\Sigma_{\rm gas}}{\kappa}, \qquad Q_{\min}\approx 0.7,$$4 simulations, the stream-fed case has $$\sigma_{\min}=\frac{Q_{\min}\,\pi G\,\Sigma_{\rm gas}}{\kappa}, \qquad Q_{\min}\approx 0.7,$$5 and $$\sigma_{\min}=\frac{Q_{\min}\,\pi G\,\Sigma_{\rm gas}}{\kappa}, \qquad Q_{\min}\approx 0.7,$$6, compared with $$\sigma_{\min}=\frac{Q_{\min}\,\pi G\,\Sigma_{\rm gas}}{\kappa}, \qquad Q_{\min}\approx 0.7,$$7 and $$\sigma_{\min}=\frac{Q_{\min}\,\pi G\,\Sigma_{\rm gas}}{\kappa}, \qquad Q_{\min}\approx 0.7,$$8 in the control. The global efficiency $$\sigma_{\min}=\frac{Q_{\min}\,\pi G\,\Sigma_{\rm gas}}{\kappa}, \qquad Q_{\min}\approx 0.7,$$9 drops from $\sigma=\max(\sigma_{\min},\sigma_{\rm turb}),$0 in the control to $\sigma=\max(\sigma_{\min},\sigma_{\rm turb}),$1 in the fed run, while $\sigma=\max(\sigma_{\min},\sigma_{\rm turb}),$2 at fixed $\sigma=\max(\sigma_{\min},\sigma_{\rm turb}),$3.

Relative to traditional bathtub or self-regulated models, this framework predicts a prolonged gas-rich phase, suppressed early stellar-mass build-up, thicker high-$\sigma=\max(\sigma_{\min},\sigma_{\rm turb}),$4 disks with $\sigma=\max(\sigma_{\min},\sigma_{\rm turb}),$5–$\sigma=\max(\sigma_{\min},\sigma_{\rm turb}),$6, and a transition near $\sigma=\max(\sigma_{\min},\sigma_{\rm turb}),$7–$\sigma=\max(\sigma_{\min},\sigma_{\rm turb}),$8 to marginally stable star formation. The paper explicitly frames this as a way to unify high gas fractions, elevated dispersions, delayed star formation, and the later self-regulated regime within one stream-feeding picture.

3. Stellar-wind feeding of M31*

For M31*, Fed-Star denotes a stellar-wind feeding mechanism in which the central supermassive black hole is supplied by collective mass loss from the surrounding nuclear star cluster (Su et al., 5 Jun 2025). The mass-losing population is modeled as $\sigma=\max(\sigma_{\min},\sigma_{\rm turb}),$9 thermally-pulsing AGB stars associated with an $Q=\frac{\kappa\,\sigma}{\pi G\,\Sigma_{\rm gas}}.$0-old, metal-rich population with $Q=\frac{\kappa\,\sigma}{\pi G\,\Sigma_{\rm gas}}.$1 and total mass $Q=\frac{\kappa\,\sigma}{\pi G\,\Sigma_{\rm gas}}.$2.

Each AGB star is assigned a time-averaged mass-loss rate $Q=\frac{\kappa\,\sigma}{\pi G\,\Sigma_{\rm gas}}.$3, wind temperature $Q=\frac{\kappa\,\sigma}{\pi G\,\Sigma_{\rm gas}}.$4, and wind speed $Q=\frac{\kappa\,\sigma}{\pi G\,\Sigma_{\rm gas}}.$5. The ensemble therefore injects $Q=\frac{\kappa\,\sigma}{\pi G\,\Sigma_{\rm gas}}.$6. The stars move on Keplerian orbits around a central SMBH of mass $Q=\frac{\kappa\,\sigma}{\pi G\,\Sigma_{\rm gas}}.$7, sampling orbital elements with $Q=\frac{\kappa\,\sigma}{\pi G\,\Sigma_{\rm gas}}.$8, $Q=\frac{\kappa\,\sigma}{\pi G\,\Sigma_{\rm gas}}.$9, and inclination $Q$0. Winds are injected within a sphere of radius $Q$1 centered on each star and carry both orbital velocity, about $Q$2, and intrinsic wind velocity.

The simulations solve the Euler equations with source terms for wind mass, momentum, and energy injection, together with external gravity and radiative heating/cooling:

$Q$3

$Q$4

$Q$5

Here $Q$6 with $Q$7, and $Q$8 is derived from CLOUDY-based lookup tables.

The numerical setup uses PLUTO 4.4 on a Cartesian grid of $Q$9 with $$\sigma_{\ln\rho}^2=\ln\bigl[1+b^2(\sigma/c_s)^2\bigr],$$0 zones, corresponding to $$\sigma_{\ln\rho}^2=\ln\bigl[1+b^2(\sigma/c_s)^2\bigr],$$1. The innermost $$\sigma_{\ln\rho}^2=\ln\bigl[1+b^2(\sigma/c_s)^2\bigr],$$2 cells define an effective accretion radius $$\sigma_{\ln\rho}^2=\ln\bigl[1+b^2(\sigma/c_s)^2\bigr],$$3. The fiducial and point-mass runs use outflow boundaries, whereas the inflow run adds an isotropic inflow of $$\sigma_{\ln\rho}^2=\ln\bigl[1+b^2(\sigma/c_s)^2\bigr],$$4, $$\sigma_{\ln\rho}^2=\ln\bigl[1+b^2(\sigma/c_s)^2\bigr],$$5, and $$\sigma_{\ln\rho}^2=\ln\bigl[1+b^2(\sigma/c_s)^2\bigr],$$6. The evolution is followed for $$\sigma_{\ln\rho}^2=\ln\bigl[1+b^2(\sigma/c_s)^2\bigr],$$7 with a time step of about $$\sigma_{\ln\rho}^2=\ln\bigl[1+b^2(\sigma/c_s)^2\bigr],$$8.

By $$\sigma_{\ln\rho}^2=\ln\bigl[1+b^2(\sigma/c_s)^2\bigr],$$9, the slow and cold AGB winds have collided, shock-heated, radiatively cooled, and settled into a flattened eccentric disk in the mean orbital plane. The disk extends to $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$00–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$01, with $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$02–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$03 and $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$04–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$05. It is embedded in a hot halo with $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$06–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$07 and $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$08–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$09. The surface density declines roughly as $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$10 and peaks near $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$11 at small radii. The scale height obeys

$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$12

giving $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$13–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$14 for $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$15 across $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$16–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$17.

The accretion rate through $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$18 approaches a quasi-steady value. The point-mass run yields $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$19, or $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$20 of $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$21; the fiducial run gives $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$22, or $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$23 of $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$24; and the inflow run reaches $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$25, or $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$26 of $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$27. The non-axisymmetric NSC potential increases $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$28 by about $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$29, and short-term fluctuations of order $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$30 track stars passing pericenter on $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$31 timescales.

The predicted observables include an X-ray luminosity $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$32–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$33 from hot plasma within $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$34, consistent with the Chandra range $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$35–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$36. The synthetic spectrum would appear very soft if fitted by a power law, with photon index $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$37. Photoionization of the cool disk gives $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$38, comparable to the observed $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$39, and predicts optical forbidden lines and IR lines potentially accessible to JWST. The paper concludes that old-star winds can dominate SMBH fueling in quiescent nuclei and argues that cosmological and galaxy-evolution simulations should include NSC wind feeding as a sub-grid source term.

4. Clump-fed accretion in high-mass star-forming objects

Within the SQUALO project, Fed-Star refers to a clump-fed mechanism for the formation of massive stars (Traficante et al., 2023). The observational basis is an ALMA Band 6 and Band 3 continuum survey of 13 massive clumps selected from the Hi-GAL and MALT90 catalogues for having blue-asymmetric HCO$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$40(1–0) or HNC(1–0) profiles indicating infall. The selection requires $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$41, $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$42, $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$43, and relative isolation. Three additional $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$44-quiet clumps with $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$45 and infall signatures extend the sample over $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$46–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$47.

The ALMA data combine 12 m and 7 m arrays in single-pointing mosaics, with typical synthesized beam $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$48–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$49, corresponding to $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$50–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$51 or $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$52–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$53, and rms noise $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$54–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$55. All clumps have single-dish infall rates $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$56–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$57.

The fragment mass is derived from the $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$58 continuum via

$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$59

with $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$60. Surface density is

$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$61

Thermal Jeans scales are written as

$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$62

and

$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$63

The clump-formation efficiency is

$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$64

and the virial parameter is

$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$65

The survey identifies 55 fragments in 13 clumps, with $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$66. All three $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$67-quiet clumps already contain $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$68–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$69 fragments, which the authors interpret as evidence that massive “starless” cores are rare. One source, HIGALBM343.7560–0.1629, with $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$70, hosts a single $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$71 object. The fragment and clump properties are correlated: $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$72 with $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$73, $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$74 with $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$75, total fragment mass correlates weakly with $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$76 with $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$77, and $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$78 with $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$79.

Fragment spacing evolves systematically. The minimum projected separation $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$80 decreases as $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$81 increases: in early clumps $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$82, whereas in evolved systems fragments reach separations of order $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$83. Jeans analysis shows that the thermal Jeans ratio $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$84 in young clumps and approaches unity in evolved clumps, while the non-thermal Jeans ratio $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$85 in almost all clumps. The observational interpretation is therefore staged. Early fragmentation is “gravo-turbulent,” with large-scale turbulence and gravity producing a small number of massive fragments at scales larger than the thermal Jeans length. As collapse proceeds, turbulence dissipates or infall accelerates, separations shrink, and fragmentation approaches the thermal Jeans scale. Magnetic support is invoked for the non-fragmenting source as a special case.

The proposed clump-fed scenario has five steps: parsec-scale gas inflow with $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$86 drives global collapse; turbulence seeds a handful of massive fragments at $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$87–$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$88; continuous accretion from the clump raises fragment mass and surface density; over $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$89 turbulence is damped and fragments contract to separations of $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$90; embedded protostars then continue to accrete from the common clump reservoir along filaments. The paper explicitly contrasts this hierarchical, multi-scale accretion picture with a pure core-fed model.

5. Wind-fed accretion disks and planet migration in binaries

In binary-star accretion, Fed-Star denotes a wind-fed disk formed when a secondary captures part of the slow dense wind of a red-giant companion through Bondi–Hoyle accretion (Kulikova et al., 2019). The analysis assumes that the disk viscous time is shorter than the wind-variation time, allowing a quasi-steady $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$91-disk treatment.

The disk is geometrically thin and Keplerian, with

$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$92

scale height

$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$93

and viscosity

$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$94

Two feeding geometries are considered. In the standard disk, matter is supplied at the outer edge and the accretion rate is radially constant:

$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$95

Angular-momentum conservation gives

$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$96

and radiative balance yields

$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$97

Far from $v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$98, the standard scalings are

$v_{\rm infall}\simeq \sqrt{2}\,v_{\rm halo}$99

$\epsilon$00

In the distributed wind-fed case, material settles over all radii $\epsilon$01 at a rate

$\epsilon$02

so that the same $\epsilon$03 relation holds but with modified $\epsilon$04. In the regime $\epsilon$05,

$\epsilon$06

The only formal difference from the standard solution is therefore an extra factor $\epsilon$07 in $\epsilon$08 and $\epsilon$09 in $\epsilon$10.

Planet migration is treated in the classical Type I/II framework. For Type I migration in a three-dimensional isothermal disk with $\epsilon$11, the torque is

$\epsilon$12

which implies

$\epsilon$13

and $\epsilon$14. Gap opening and Type II migration are described by the criterion

$\epsilon$15

with $\epsilon$16 expressed in terms of the Reynolds number $\epsilon$17, after which the drift rate is

$\epsilon$18

and

$\epsilon$19

For red-giant mass-loss rates $\epsilon$20–$\epsilon$21 and binary separations $\epsilon$22–$\epsilon$23, the capture rate is $\epsilon$24–$\epsilon$25. With $\epsilon$26–$\epsilon$27, the disk lifetime is set by the red-giant phase, $\epsilon$28. In standard edge-fed disks, Type I migration at $\epsilon$29 is $\epsilon$30–$\epsilon$31 for $\epsilon$32–$\epsilon$33 if $\epsilon$34; for lower accretion rates the disk is too tenuous for migration within the disk lifetime. The Type I–Type II transition occurs at $\epsilon$35–$\epsilon$36, and Type II migration is generally faster in these low-mass disks. Jupiter-mass planets can merge within $\epsilon$37–$\epsilon$38 for $\epsilon$39 and $\epsilon$40, whereas lower-mass planets may survive if $\epsilon$41 or $\epsilon$42.

The disk surface densities are much lower than in protoplanetary disks, about $\epsilon$43–$\epsilon$44 at $\epsilon$45 versus $\epsilon$46–$\epsilon$47, which slows Type I migration and raises the critical mass for gap opening. Yet the longer wind-fed disk lifetime means that substantial migration remains possible. The merger energy $\epsilon$48–$\epsilon$49 motivates the transient interpretation discussed in the paper.

6. FEderated Self-TRAining for semi-supervised audio recognition

In machine learning, FedSTAR was introduced as “FEderated Self-TRAining” for semi-supervised audio recognition (Tsouvalas et al., 2021). The method addresses federated learning with scarce labeled audio and abundant unlabeled audio distributed across devices. The goal is to train a single global model while keeping raw audio local and exploiting pseudo-labeling on each client.

The per-round workflow is straightforward. The server maintains global parameters $\epsilon$50, samples a fraction $\epsilon$51 of clients, and sends $\epsilon$52 to each selected client. Client $\epsilon$53 performs $\epsilon$54 local epochs using labeled minibatches from $\epsilon$55 and unlabeled minibatches from $\epsilon$56. The local objective combines supervised cross-entropy with pseudo-label-based unsupervised cross-entropy, where low-confidence pseudo-labels are discarded through a dynamic threshold $\epsilon$57. Updated local models are then aggregated by weighted FedAvg:

$\epsilon$58

The global optimization problem is

$\epsilon$59

with local loss

$\epsilon$60

The supervised term is categorical cross-entropy,

$\epsilon$61

while the pseudo-label is obtained from temperature-scaled logits

$\epsilon$62

and the unsupervised term is

$\epsilon$63

The framework optionally initializes the model with a self-supervised encoder trained on a large unlabeled corpus such as FSD-50K using an InfoNCE-style objective on paired segments from the same clip:

$\epsilon$64

This pretrained encoder becomes $\epsilon$65 and is reported to reduce the number of required federated rounds by $\epsilon$66–$\epsilon$67 for the same accuracy.

Experiments use Ambient Acoustic Context, Speech Commands v2, and VoxForge, with audio resampled to $\epsilon$68 and represented as $\epsilon$69 log-Mel spectrograms with $\epsilon$70 Mel bins. The model has four convolutional blocks, each comprising a timewise $\epsilon$71D convolution, a frequencywise $\epsilon$72D convolution, concatenation, a $\epsilon$73 convolution, GroupNorm, ReLU, $\epsilon$74 weight decay $\epsilon$75, spatial dropout $\epsilon$76, and max-pooling $\epsilon$77 between blocks, followed by global average pooling and a dense softmax head. Training uses Adam with $\epsilon$78 and client batch size about $\epsilon$79. The federation parameters span $\epsilon$80 clients, $\epsilon$81–$\epsilon$82, $\epsilon$83–$\epsilon$84, labeled fraction $\epsilon$85, unlabeled fraction $\epsilon$86, $\epsilon$87, $\epsilon$88, and a cosine-rising threshold $\epsilon$89 from $\epsilon$90 to $\epsilon$91.

Quantitatively, with only $\epsilon$92 labels and $\epsilon$93, performance improves from $\epsilon$94 to $\epsilon$95 on Ambient Context, from $\epsilon$96 to $\epsilon$97 on Speech Commands, and from $\epsilon$98 to $\epsilon$99 on VoxForge. Averaged across tasks and client counts, the method improves recognition by up to $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$00 over fully supervised federated learning at $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$01. Under extreme non-IIDness, where each client sees only $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$02 classes on $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$03, supervised federated learning remains below $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$04, whereas FedSTAR still reaches about $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$05–$$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$06 for $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$07–$$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$08. After $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$09 federated rounds with $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$10 and $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$11, SSL initialization improves Speech Commands from about $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$12 to about $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$13.

The paper characterizes the method as a lightweight extension of FedAvg because clients need only add a pseudo-label cross-entropy term with tunable $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$14, $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$15, and $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$16. The principal claim is not personalization but better use of on-device unlabeled data under label scarcity.

7. Style-aware transformer aggregation in personalized federated learning

A distinct 2025 usage, “Federated Style-Aware Transformer Aggregation of Representations,” also abbreviated FedSTAR, targets personalized federated learning under domain heterogeneity, data imbalance, and communication constraints (Jeon et al., 24 Nov 2025). The central claim is that client embeddings entangle task-relevant content with client-specific style and that uniform averaging of class-wise prototypes suppresses minority-client signals.

Each client extracts features $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$17 through a shared encoder and maintains, for every class $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$18, a mean feature prototype

$$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$19

together with a personal residual parameter $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$20. Relative to the current global prototype $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$21, the residual is decomposed into content and style. The content projection is

$$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$22

while the orthogonal style residual is

$$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$23

The full local prototype is

$$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$24

For communication, clients send only the content portion $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$25, or equivalently just $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$26 when $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$27 is shared. The style vectors remain local and are used for FiLM-based personalization during inference.

On the server, class-wise content prototypes from $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$28 participating clients are stacked into a tensor $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$29. Tokens are formed as

$$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$30

with learned client and class embeddings. A standard Transformer encoder is then applied:

$$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$31

$$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$32

A second class-driven attention computes

$$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$33

and the updated global prototype is

$$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$34

Clients then fuse the global prototypes with local residual parameters through a learned gating network.

Communication efficiency is a primary design goal. Rather than exchanging full model weights of size $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$35, each client sends $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$36 content prototypes of dimension $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$37, and optionally $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$38 style vectors, for total communication $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$39. The ratio

$$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$40

is reported as typically $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$41, amounting to one to two orders of magnitude less communication than full-model exchange in typical settings.

The evaluation uses Fashion-MNIST, CIFAR-100, DomainNet, and Office-31 under severe non-IID Dirichlet splits with $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$42 plus Gaussian noise. The reported results are: on Fashion-MNIST, FedProto achieves $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$43 accuracy, the attention-only ablation reaches $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$44, and FedSTAR reaches $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$45 with $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$46 and convergence in $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$47 rounds; on CIFAR-100, performance increases from $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$48 for FedProto to $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$49 for the ablation and $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$50 for FedSTAR; on DomainNet, from $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$51 to $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$52 to $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$53; and on Office-31, from $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$54 to $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$55 to $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$56. Ablations attribute a $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$57–$$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$58 percentage-point gain to replacing uniform averaging with Transformer attention alone and a further $$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$59–$$\dot E_{\rm in}=\tfrac12\,\epsilon\,\dot M_{\rm inflow}\,v_{\rm infall}^2,$$60 percentage-point gain to adding style-aware FiLM personalization.

This framework differs sharply from the audio self-training FedSTAR despite the identical acronym. One addresses semi-supervised learning with pseudo-labels and a single global model; the other addresses personalized federated learning through explicit content–style disentanglement and attention-weighted prototype aggregation. The shared acronym does not indicate methodological continuity.