Burst Fraction in Astrophysics

• Burst Fraction is a quantitative measure that captures the proportion of burst-dominated phenomena in different astrophysical settings, including solar active regions, reionization, and GRB jets.
• Its definition varies by context: as the complement of steady emission in solar heating, a duty cycle in star formation models, a geometric beaming factor in GRBs, and an event-based detection fraction in Fermi observations.
• Analytical methods demonstrate that in solar observations, for example, transient bursts can account for up to 95% of the hot-94 luminosity, influencing models of impulsive plasma heating and star formation histories.

In the cited arXiv literature, burst fraction is not a single invariant quantity but a context-dependent fraction that quantifies burst-dominated behavior in at least three distinct ways. In solar active-region heating it is defined as the complement of a steady-emission fraction, $f_{\mathrm{burst}}(T)=1-f_{\mathrm{steady}}(T)$, with $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$; in semi-analytic reionization models it is the duty cycle of bursty star formation, $$f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}=t_{\mathrm{on}}/T$$; and in gamma-ray-burst jet studies the related symbol $f_b$ denotes the geometric beaming fraction, $1-\cos\theta_j$ (Tiwari et al., 2022, Hartley et al., 2016, Lloyd-Ronning et al., 2020). A distinct but nearby concept in Fermi gamma-ray-burst phenomenology is the LAT detection fraction $$f_{\mathrm{LAT}}\equiv N_{\mathrm{LAT}}/N_{\mathrm{GBM}}\approx0.10\pm0.04$$, which is an event-detection fraction rather than a burst fraction in the temporal or geometric senses (Guetta, 2013).

1. Terminological scope and formal variants

Across these works, the same phrase maps onto different denominators and therefore different physical interpretations. In one case the denominator is total luminosity over a running window; in another it is the cycle period of an intermittent source; in another it is the full sky; and in the Fermi review it is the number of detected bursts. The formal definitions are therefore not interchangeable (Tiwari et al., 2022, Hartley et al., 2016, Lloyd-Ronning et al., 2020, Guetta, 2013).

Context	Symbol	Definition
Solar AR hot-94 heating	$f_{\mathrm{burst}}(T)$	$1-f_{\mathrm{steady}}(T)$, where $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$
Bursty reionization	$f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}$	$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$0
GRB jet geometry	$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$1	$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$2 for $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$3
Fermi GRB detections	$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$4	$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$5

The first two definitions quantify temporal intermittency, but even there they differ structurally. The solar definition is an empirically inferred complementary luminosity fraction and is explicitly a lower bound on the contribution from transient brightenings. The reionization definition is a prescribed duty cycle in a source model. The GRB beaming fraction is instead a geometric selection probability, and the LAT fraction is an instrument-limited detection fraction. A common source of confusion is to read all of them as direct measures of time variability; the GRB beaming case does not support that reading (Lloyd-Ronning et al., 2020).

2. Burst fraction in coronal heating diagnostics

In the analysis of Fe xviii (“hot 94”) emission from a solar active region, Tiwari et al. define a running-window estimator from a sequence of EUV images $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$6 sampled every $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$7 min. For a window of duration $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$8 centered at $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$9, the pixelwise minimum, average, and maximum are

$$f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}=t_{\mathrm{on}}/T$$0

$$f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}=t_{\mathrm{on}}/T$$1

$$f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}=t_{\mathrm{on}}/T$$2

with $$f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}=t_{\mathrm{on}}/T$$3. Summing over the active-region pixel set $$f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}=t_{\mathrm{on}}/T$$4 yields

$$f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}=t_{\mathrm{on}}/T$$5

$$f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}=t_{\mathrm{on}}/T$$6

The steady and burst fractions are then

$$f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}=t_{\mathrm{on}}/T$$7

By construction, $$f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}=t_{\mathrm{on}}/T$$8 is an upper bound on the fraction of total hot-94 luminosity attributable to plasma that remains at least as bright as $$f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}=t_{\mathrm{on}}/T$$9 throughout the window, and $f_b$0 is the complementary lower bound on the fraction arising from transient brightenings (Tiwari et al., 2022).

The methodology used 24 h of SDO/AIA 94 Å images at 3 min cadence. The Warren et al. (2012) algorithm was applied to isolate the Fe xviii component by subtracting scaled contributions of the cooler 171 Å and 193 Å channels. Standard AIA calibration and pointing corrections (aia_prep) were applied. To avoid spurious fluctuations, the analysis estimated the $f_b$1 noise in quiet corner regions of each hot-94 image and discarded all pixels whose brightness ever fell below that threshold. Running windows of $f_b$2 h, $f_b$3 h, $f_b$4 h, $f_b$5 h, $f_b$6 h, $f_b$7 h, $f_b$8 h, $f_b$9 h, and $1-\cos\theta_j$0 h were stepped through the full sequence in 3 min increments, with no additional spatial smoothing beyond the hot-94 isolation (Tiwari et al., 2022).

The resulting fractions are:

Window $1-\cos\theta_j$1	$1-\cos\theta_j$2	$1-\cos\theta_j$3
24 h	$1-\cos\theta_j$4	$1-\cos\theta_j$5
20 h	$1-\cos\theta_j$6	$1-\cos\theta_j$7
16 h	$1-\cos\theta_j$8	$1-\cos\theta_j$9
12 h	$$f_{\mathrm{LAT}}\equiv N_{\mathrm{LAT}}/N_{\mathrm{GBM}}\approx0.10\pm0.04$$0	$$f_{\mathrm{LAT}}\equiv N_{\mathrm{LAT}}/N_{\mathrm{GBM}}\approx0.10\pm0.04$$1
8 h	$$f_{\mathrm{LAT}}\equiv N_{\mathrm{LAT}}/N_{\mathrm{GBM}}\approx0.10\pm0.04$$2	$$f_{\mathrm{LAT}}\equiv N_{\mathrm{LAT}}/N_{\mathrm{GBM}}\approx0.10\pm0.04$$3
5 h	$$f_{\mathrm{LAT}}\equiv N_{\mathrm{LAT}}/N_{\mathrm{GBM}}\approx0.10\pm0.04$$4	$$f_{\mathrm{LAT}}\equiv N_{\mathrm{LAT}}/N_{\mathrm{GBM}}\approx0.10\pm0.04$$5
3 h	$$f_{\mathrm{LAT}}\equiv N_{\mathrm{LAT}}/N_{\mathrm{GBM}}\approx0.10\pm0.04$$6	$$f_{\mathrm{LAT}}\equiv N_{\mathrm{LAT}}/N_{\mathrm{GBM}}\approx0.10\pm0.04$$7
1 h	$$f_{\mathrm{LAT}}\equiv N_{\mathrm{LAT}}/N_{\mathrm{GBM}}\approx0.10\pm0.04$$8	$$f_{\mathrm{LAT}}\equiv N_{\mathrm{LAT}}/N_{\mathrm{GBM}}\approx0.10\pm0.04$$9
0.5 h	$f_{\mathrm{burst}}(T)$0	$f_{\mathrm{burst}}(T)$1

These values imply that, over a full 24 h, no more than $f_{\mathrm{burst}}(T)$2 of the hot-94 luminosity comes from plasma that never dims below its 24 h minimum, so at least $f_{\mathrm{burst}}(T)$3 of the heating must occur in bursts shorter than one day. Even on 30 min timescales the steady fraction is only $f_{\mathrm{burst}}(T)$4, so at least $f_{\mathrm{burst}}(T)$5 of the heating is delivered in bursts briefer than half an hour. The monotonic rise of $f_{\mathrm{burst}}(T)$6 as $f_{\mathrm{burst}}(T)$7 decreases suggests that typical hot-loop lifetimes lie in the tens of minutes to hour range. The paper therefore concludes that the bulk of the $f_{\mathrm{burst}}(T)$8–$f_{\mathrm{burst}}(T)$9 MK plasma heating in the studied active region is impulsive, or burst-dominated, on timescales $1-f_{\mathrm{steady}}(T)$0 min (Tiwari et al., 2022).

3. Burst fraction as duty cycle in reionization models

In semi-analytic models of the epoch of reionization, the burst fraction is introduced as the duty cycle of star formation. The notation is

• $1-f_{\mathrm{steady}}(T)$1 = period between the start of successive bursts,
• $1-f_{\mathrm{steady}}(T)$2 = duration of each burst,
• $1-f_{\mathrm{steady}}(T)$3,
• $1-f_{\mathrm{steady}}(T)$4.

Equivalently,

$1-f_{\mathrm{steady}}(T)$5

The instantaneous ionizing-photon rate is modeled as a periodic top-hat,

$1-f_{\mathrm{steady}}(T)$6

where $1-f_{\mathrm{steady}}(T)$7 is the time-averaged ionizing-photon rate over one cycle, and

$1-f_{\mathrm{steady}}(T)$8

with $1-f_{\mathrm{steady}}(T)$9–$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$0 parameterizing spectral hardness (Hartley et al., 2016).

The physical rationale is that concentration of luminosity into short bursts produces large H II regions, while recombination in relic H II regions proceeds slowly because recombination rates scale as $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$1. During the on-phase, the radial ionization profile around an isolated source is fitted by

$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$2

and after star formation ceases,

$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$3

with $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$4 and $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$5 (Hartley et al., 2016).

The explored variability scales were $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$6 Myr and $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$7 Myr, implying $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$8, together with a continuous case $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$9. For runs calibrated to complete reionization at $f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}$0 and reproduce the Planck optical depth $f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}$1, the reported escape-fraction requirements were:

• continuous star formation: $f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}$2–$f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}$3,
• modest bursting ($f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}$4, $f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}$5 Myr, $f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}$6 Myr): $f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}$7–$f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}$8,
• extreme bursting ($f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}$9, $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$00 Myr, $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$01 Myr): $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$02–$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$03.

For the fiducial $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$04 Myr suite with no ISM pre-absorption, the tabulated values are $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$05, $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$06, $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$07, and $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$08 for $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$09, $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$10, $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$11, and $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$12, respectively; the corresponding $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$13 values are $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$14, $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$15, $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$16, and $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$17 (Hartley et al., 2016).

Within this framework, lowering the burst fraction in the duty-cycle sense increases peak luminosity at fixed time-averaged production, leaves behind long-lived relic partial ionization, increases $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$18, and reduces the escape-fraction budget needed to match $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$19 and $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$20. The paper therefore cautions against using the simple photon-budget equation

$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$21

without accounting for broadened ionization fronts and relic partial ionization in a bursty mode of star formation (Hartley et al., 2016).

4. Burst fraction as GRB beaming fraction

In long-GRB population studies, the burst fraction is a geometric beaming correction rather than a temporal intermittency measure. For a conical jet with half-opening angle $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$22, the solid angle of one jet is

$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$23

Assuming two jets, the total beamed solid angle is $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$24. The burst fraction is then defined as the fraction of the sky covered by one jet divided by $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$25,

$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$26

and for $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$27 rad,

$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$28

These relations assume that progenitor spins and magnetic axes are randomly oriented, so jets point in all directions uniformly (Lloyd-Ronning et al., 2020).

Lloyd-Ronning et al. (2019, 2020), as summarized in the cited paper, compile a sample of $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$29 long GRBs with measured jet breaks and redshifts and, after correcting for Malmquist bias and selection effects, find a statistically significant anti-correlation between jet opening angle and $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$30. They parameterize it as

$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$31

with $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$32 and $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$33 rad at $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$34. Substituting this into the beaming fraction yields

$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$35

and for $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$36 rad,

$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$37

Defining the local burst fraction $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$38, one obtains

$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$39

Numerically, taking $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$40 rad and $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$41 gives $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$42 at $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$43, $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$44 at $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$45, and $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$46 at $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$47 (Lloyd-Ronning et al., 2020).

This redshift dependence enters directly into rate corrections. If $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$48 is the observed differential GRB rate, then the true progenitor rate is

$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$49

or, in comoving form,

$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$50

If a fraction $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$51 of massive stars produce long GRBs, then

$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$52

The paper reports that including evolving $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$53 can raise the inferred star-formation-rate density at $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$54 by up to an order of magnitude above canonical estimates, while also diminishing the apparent excess at low redshift. It also emphasizes degeneracies involving the normalization of $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$55, the $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$56 uncertainty on $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$57, selection biases in $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$58 measurements, and possible redshift dependence of $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$59 due to metallicity or binarity (Lloyd-Ronning et al., 2020).

5. Related observational fractions in Fermi gamma-ray-burst theory

The Fermi review introduces a quantitatively different fraction, $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$60, that is closely related to burst detectability but is not itself a burst fraction in the duty-cycle or beaming sense. For long GRBs with GBM $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$61 s,

$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$62

Here “35” is the number of long GRBs with high-energy photons reported by the LAT up to September 2012, and “350” is an estimate of the GBM long-burst sample in the same period. The paper states that only a small minority of long GRBs seen by the GBM at $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$63 keV–$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$64 MeV produce detectable emission above $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$65 MeV in the LAT, and that LAT-detected bursts were among the brightest GBM bursts (Guetta, 2013).

Fluence-ratio studies support this low detection fraction. For bright GBM bursts not seen by LAT, the $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$66-confidence upper limit on the ratio of LAT-band ($f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$67 MeV) to GBM-band ($f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$68 keV–$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$69 MeV) fluence is

$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$70

whereas for LAT-detected bursts the mean ratio is $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$71. The LAT effective area and background rate imply a minimum high-energy fluence threshold $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$72–$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$73, or equivalently a photon-flux threshold $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$74 above $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$75 MeV. Bursts with predicted Band-function extrapolations below this threshold remain LAT-dark (Guetta, 2013).

Several model classes are identified to explain why most long GRBs do not shine in the GeV. In internal-shock synchrotron plus SSC pictures, the non-detection of a second SSC peak above $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$76 MeV in $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$77 of bursts requires either a low Compton-$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$78 parameter or Klein–Nishina suppression, with $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$79 in most bursts. In $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$80–$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$81 opacity models, requiring internal optical depth $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$82 at $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$83 MeV implies

$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$84

so bursts with $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$85 below a few hundred will be self-opaque above $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$86 MeV and thus LAT-dark. Photospheric or Comptonization models are also discussed as ways to produce the prompt $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$87MeV Band spectrum without a strong GeV tail (Guetta, 2013).

Almost every LAT-detected GRB also shows a systematic delay $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$88 between the onset of GBM emission and the first LAT photons, with typical values $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$89–$f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$90 s and a tail extending to $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$91 s. In the review this delay is used both in source-model discussions and in Lorentz-invariance-violation searches. The key point for the present topic is that $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$92 is governed by instrumental thresholding and source physics, not by the same construction as $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$93 or $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$94 (Guetta, 2013).

6. Comparative interpretation and methodological cautions

The cited papers show that “burst fraction” can denote four different mathematical objects: a complementary luminosity fraction, a temporal duty cycle, an angular covering fraction, and an observational detection fraction. Their denominators are, respectively, total luminosity in a time window, cycle duration, full sky, and event counts. As a result, transferring intuition from one use to another without restating the definition is not justified (Tiwari et al., 2022, Hartley et al., 2016, Lloyd-Ronning et al., 2020, Guetta, 2013).

A second distinction concerns whether the quantity is inferred or prescribed. In the solar active-region application, $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$95 is inferred from image sequences and is explicitly a lower bound on transient heating because $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$96 is only an upper bound on truly steady emission. In the reionization application, $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$97 is a source-model input that controls the periodic top-hat luminosity and thereby the relic-ionization history. In the GRB beaming application, $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$98 is a geometric probability entering the inversion from observed to true event rates. In the Fermi case, $f_{\mathrm{steady}}(T)=L_{\min}(T)/L_{\avg}(T)$99 is the output of a thresholded detection process (Tiwari et al., 2022, Hartley et al., 2016, Lloyd-Ronning et al., 2020, Guetta, 2013).

A common misconception is to treat all burst fractions as statements about intrinsic intermittency. Only the solar and reionization definitions directly encode intermittency in time. The GRB beaming fraction can decrease with redshift even if the central engine were not more intermittent, because it depends on $$f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}=t_{\mathrm{on}}/T$$00. Likewise, the LAT fraction can remain low because of fluence thresholds, low SSC yield, Klein–Nishina suppression, $$f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}=t_{\mathrm{on}}/T$$01–$$f_{\mathrm{burst}}\equiv f_{\mathrm{duty}}=t_{\mathrm{on}}/T$$02 opacity, or photospheric emission, without implying any single duty-cycle interpretation (Lloyd-Ronning et al., 2020, Guetta, 2013).

Taken together, these usages suggest that the term is best understood as a compact label for the fraction of a relevant measure occupied by burst-dominated states, where the relevant measure must be specified explicitly: luminosity, time, solid angle, or detected events. That specification determines what the fraction constrains physically—impulsive coronal heating, episodic ionizing emissivity, GRB beaming corrections, or high-energy detectability—and therefore determines how it should be compared across models and datasets.