Constraints on the Synchrotron Emission Mechanism in GRBs
We reexamine the general synchrotron model for GRBs’ prompt emission and determine the regime in the parameter phase space in which it is viable. We characterize a typical GRB pulse in terms of its peak energy, peak flux and duration and use the latest Fermi observations to constrain the high energy part of the spectrum. We solve for the intrinsic parameters at the emission region and find the possible parameter phase space for synchrotron emission. Our approach is general and it does not depend on a specific energy dissipation mechanism. Reasonable synchrotron solutions are found with energy ratios of , bulk Lorentz factor values of , typical electrons’ Lorentz factor values of and emission radii of the order cm. Most remarkable among those are the rather large values of the emission radius and the electron’s Lorentz factor. We find that soft (with peak energy less than 100KeV) but luminous (isotropic luminosity of ) pulses are inefficient. This may explain the lack of strong soft bursts. In cases when most of the energy is carried out by the kinetic energy of the flow, such as in the internal shocks, the synchrotron solution requires that only a small fraction of the electrons are accelerated to relativistic velocities by the shocks. We show that future observations of very high energy photons from GRBs by CTA, could possibly determine all parameters of the synchrotron model or rule it out altogether.
GRBs’ non thermal spectrum has lead to the suggestion that both the prompt and the afterglow are produced by synchrotron emission of relativistic electrons (Katz, 1994; Rees and Mészáros, 1994; Sari et al., 1996, 1998). There is reasonable agreement between the predictions (Sari et al., 1998) of the synchrotron model and afterglow observations (Wijers and Galama, 1999; Granot et al., 1999, 2002; Panaitescu and Kumar, 2001; Nousek et al., 2006; Zhang et al., 2006). However, the situation concerning the prompt emission is more complicated.
The observations of many bursts whose lower spectral slope is steeper than the “synchrotron line of death” (Crider et al., 1997; Preece et al., 1998, 2002) pose a serious problem to synchrotron emission. This has motivated considerations of photospheric thermal emission (Eichler Levinson, 2000; Mészáros Rees, 2000; Peer et al., 2006; Rees Mészáros, 2005; Giannios, 2006; Thompson et al., 2007; Ryde and Peer, 2009), in which thermal energy stored in the bulk is radiated in the prompt phase at the Thomson photosphere and the high energy tail is produced by inverse Compton. While this model yields a consistent low energy slope it has its own share of problems. Most notably Vurm et al. (2012) have recently shown that unless the outflow is moving very slowly no known mechanism can produce the needed soft photons. Furthermore, it is not clear how the high energy GeV emission can be produced within an optically thick regime (Vurm et al., 2012a). On the other hand, observations of some bursts, e.g. GRB 080916C in which there is a strong upper limit on the thermal component (Zhang et al., 2009), or GRBs: 100724B, 110721A, 120323A, where a thermal component was possibly detected but with only a small fraction () of the total energy in the thermal component (Guiriec et al., 2011, 2012; Axelsson et al., 2012) and other bursts whose spectral slopes is consistent with synchrotron suggest that for some bursts synchrotron might be a viable model. In fact, in all cases that involve Poynting flux outflows that are Poynting flux dominated at the emitting region synchrotron is inevitable and possibly dominant (Beniamini Piran, 2013). As such it is worthwhile to explore the conditions needed to generate the observed prompt emission via synchrotron111Note that the synchrotron-self Compton (Waxman, 1997; Ghisellini and Celotti, 1999) have been ruled out by the high energy LAT observations (Piran et al., 2009; Zou et al., 2009), while external inverse Compton (Shemi, 1994; Brainerd, 1994; Shaviv and Dar, 1995; Lazzati et al., 2003) require a strong enough external source of soft photons, which is not available in GRBs. .
To examine the conditions for synchrotron to operate, we consider a general model that follows the spirit of Kumar McMahon (2008). We don’t make any specific assumptions concerning the details of the energy dissipation process or the particle acceleration mechanism. Instead we assume that relativistic electrons and magnetic fields are at place within the emitting region, that is moving relativistically towards the observer. We examine the parameter space in which these electrons emit the observed radiation via synchrotron while satisfying all known limits on the observed prompt emission. We use a simple “single-zone” calculations that do not take into account the blending of radiation from different angles or from different emitting regions (reflecting matter moving at different Lorentz factors), but for the purpose of obtaining the rough range of relevant parameters this is sufficient.
We characterize the conditions within the emitting region by six parameters: the co-moving magnetic field strength, , the number of relativistic emitters, , the ratio between the magnetic energy and the internal energy of the electrons, , the bulk Lorentz factor of the emitting region, , the minimal electrons’ Lorentz factor in the source frame, and the ratio between the shell crossing time and the angular timescale, . We then characterize a single, “typical”, GRB pulse, which is the building block of the GRB lightcurve, by three basic quantities: the peak (sub-MeV) frequency, the peak flux and the duration. We compare the first two, the observed peak (sub-MeV) frequency and the peak flux with the predictions of the synchrotron model and we use the observed duration of the pulse to limit the angular time scale (Sari and Piran, 1997a). The synchrotron solution is incomplete without a determination of the accompanying synchrotron self-Compton (SSC) emission, which influences the efficiency of the synchrotron emission. Furthermore, the observations of the high energy SSC emission poses additional constraints on the emitting region. We describe, therefore, a self-consistent synchrotron self-Compton solution.
Three additional constraints should be taken into account (see also Daigne et al., 2011). First, energy considerations pose a strong lower limit on the efficiency. The energy of the observed sub-MeV flux, is huge and already highly constraining astrophysical models. Thus, to avoid an “energy crisis”, the emission process must be efficient and emit a significant fraction of the available energy in the sub-MeV band. Second, the emission region cannot be optically thick. Additionally, the LAT band (100 MeV- 300 GeV) GeV component is significantly weaker in most GRBs than the sub-MeV peak (Granot et al., 2009, 2010; Beniamini et al., 2011; Guetta et al., 2011; Ackermann et al., 2012). We combine these three conditions and constrain the possible phase space in which synchrotron can produce the prompt emission.
The paper is organized as follows. We describe in §2 the basic concepts of the model, discussing the observations in §2.1 and the structure of the emitting region and the parameter phase space in §2.2. In §3 we describe the synchrotron equations (§3.1), their solutions (§3.2), the constraints from the accompanying SSC (§3.3), the implications of the results (§3.4) and energy and efficiency constraints (§3.5). We continue with a discussion of specific issues concerning the (i) expected spectral slopes and possibilities to alleviate the issue of the “synchrotron line of death” (§3.6) and (ii) the narrow distribution (§3.7). In §4, we deviate from the general philosophy of the paper and we discuss synchrotron emission within the context of the popular internal shocks framework. We turn, in §5, to the implications of the pair creation limits from observed GeV emission for the synchrotron model. In §6 we briefly discuss the possibilities of detecting GRBs with CTA, and their implications on the model at hand.
2 The Global Picture
2.1 The observations
Limits on the optical (Roming et al., 2006; Yost et al., 2007; Klotz, 2009), x-ray (O’Brien et al., 2006), GeV (Ando et al., 2008; Guetta et al., 2011; Beniamini et al., 2011; Ackermann et al., 2012) and TeV (Atkins et al., 2005; Albert et al., 2006; Aharonian et al., 2009) emission during the prompt phase of GRBs, demonstrate that the observed sub-MeV peak carries most of the GRB’s energy. The only (unlikely) possibilities that have not been ruled out are either an extremely strong and sharp peak between 10-100eV or a peak at extremely high energies above the TeV range. Therefore, we associated the sub-MeV peak with the peak of the synchrotron emission.
We consider the typical observations after redshift corrections, i.e. in the host galaxy frame which we denote as “the source frame” (as opposed to “the observer frame”). The basic observables are: the peak frequency: (300KeV), the duration of a typical pulse, (0.5 sec) and the peak flux, (). With these definitions we remove the dependence from the equations, and the only remaining dependence on redshift is through the luminosity distance, relating the flux and luminosity. All three quantities vary from pulse to pulse and the values in brackets denote the canonical values that we use here. These three observations together with a typical luminosity distance, cm (or ) correspond to an isotropic equivalent luminosity of erg/sec. It has been suggested that and satisfy: erg/sec (Yonetoku et al., 2010), we therefore choose and such that they are compatible with this relation. In §3.7 we explore the dependence of the synchrotron model on this choice and look at 4 representative “GRB types”. An additional observation that we use is the limit on the total flux observed in the LAT band (30 MeV-300 GeV), which is at most 0.13 of the GBM (8KeV-40MeV) flux (Beniamini et al., 2011; Ando et al., 2008; Guetta et al., 2011; Ackermann et al., 2012).
2.2 The emitting region
The general model requires a relativistic jet, traveling with a Lorentz factor with respect to the host galaxy with a half opening angle of , at a distance from the origin of the explosion and with a kinetic energy in the source frame. The source can be treated as spherically symmetric as long as which is expected to hold during the prompt phase, in which (Fenimore et al., 1993; Woods and Loeb, 1995; Piran, 1995; Sari and Piran, 1999). We therefore do not solve for and use isotropic equivalent quantities throughout the paper. An upper limit on the emitting radius is given by the variability time scale, . We define a dimensionless parameter such that is the shell’s width:
With this definition, the pulse duration is:
Clearly, for the pulse width is determined by angular spreading.
We consider a single-zone model in which the different properties of the emitting region are constant throughout the zone. Notice that the single-zone approach cannot test spectral evolution, and becomes less accurate as increases. Initially, the emitting region consists of either an electron-proton or an electron-positron plasma. The numbers of protons and positrons are denoted by and respectively. Due to electrical neutrality, the number of electrons is . After the gamma-rays are produced, annihilating high energy photons may result in extra electrons and positrons. We denote the number of pairs created in this way by (typically these particles are less energetic than the original population of emitting particles). Altogether, there are electrons and positrons in the flow. the number of electrons is always larger than the number of positrons except for the case of an initial pair dominated outflow, where they are equal. Therefore, to simplify the presentation, we use hereafter ”electrons” instead of the more accurate but cumbersome ”electrons and positrons”. In the fast cooling regime, the typical synchrotron cooling time of a relativistic electron is shorter than the pulse duration and thus the instantaneously emitting particles are much less than the total number of emitting particles during one pulse (see Fig.1). The emitting electrons are characterized by which is defined to be the ratio between the number of instantaneously emitting electrons and the overall number of relativistic electrons emitting during the entire pulse (A concise definition of all these particle numbers is given in table 1).
|The total number of electrons (and positrons)|
|The number of pairs created by annihilating high energy photons 11footnotemark: 1|
|The number of protons in the initial flow|
|The number of positrons in the initial flow|
|The number of relativistic electrons (and positrons) radiating during one pulse|
|Ratio between instantaneously emitting electrons (and positrons) and|
Excluding the original pairs that may reside in the initial flow and are accounted for by
The relativistic electrons have a power-law distribution of energies:
(where C is a normalization constant) which holds for . Based on theory, we expect to be of the order (Achterberg et al., 2001; Bednarz et al., 1998; Gallant Achterberg, 1999). Indeed this result has been confirmed observationally both from the GRB prompt and afterglow phases (Sari and Piran, 1997b; Panaitescu and Kumar, 2001). arises, most likely, due to Synchrotron losses at the energy where the acceleration time equals to the energy loss time (de Jager, 1996):
where is a numerical constant of order unity which encompasses the details of the acceleration process and is measured in Gauss. In a shock-acceleration scenario this depends on the amount of time the particle spends in the downstream and upstream regions (Piran Nakar, 2010; Barniol Duran Kumar, 2011). The total internal energy of the electrons is then (primes denote the co-moving frame):
where is the energy of the flow dissipated into internal energy and the ratio of the total energy in these relativistic electrons to the total internal energy is denoted by . Notice that does not necessarily reflect the ratio of instantaneous energy of the relativistic electrons to the total energy, which could be much smaller.
The magnetic field, , is assumed to be constant over the entire emitting region. Its energy is:
where is the thickness of the shell in the co-moving frame and is the fraction of magnetic to internal energy. In this case, for a Baryoninc dominated flow, . Whereas, in a Poyinting flux dominated jet, one can imagine a situation in which at first the energy is stored within the magnetic fields, and only later it is dissipated and transferred to the electrons. Within the definitions we use here, this means that .
Overall, we find that six independent parameters define the synchrotron solution 222Kumar McMahon (2008) set the pulse duration to be equal to the angular time scale. We consider here a more general approach in which this is only a lower limit. These are: the co-moving magnetic field strength, , the number of relativistic electrons, , the ratio between the magnetic energy and the kinetic energy of the electrons, , the bulk Lorentz factor of the jet with respect to the GRB host galaxy, , the minimum electrons’ Lorentz factor in the source frame, , and the ratio between the shell crossing time and the angular timescale, . The first three observables: the spectral peak, the peak luminosity and the pulse duration, described in §2.1, reduce the number of free parameters from six to three, which we choose as: (). The other constraints limit the allowed regions within this parameter space.
3 Detailed method and results
3.1 The synchrotron equations
where is the Lorentz factor of the bulk and is the Thomson cross-section. When the electrons have a power-law distribution of energies above some there are two typical breaks in the spectrum. The first is the synchrotron frequency of a typical electron, . The second is the cooling frequency, the frequency at which electrons cooling on the pulse time-scale, radiate:
where is the relative energy loss by IC as compared with synchrotron for .
The peak of the synchrotron is at (Sari et al., 1998). We identify this peak with the observed sub-MeV peak. For , the ”typical” electrons are fast cooling and rapidly dispose of all their kinetic energy, while for the electrons are slow cooling and the typical electrons do not emit all their energy which is eventually lost by adiabatic cooling. We consider fast cooling solutions (or marginally fast) for the prompt emission.
For , this is the flux at and the peak of . However, in the fast cooling regime the flux between and varies as and the peak of is at :
where the factor accounts for the fact that only of the relativistic electrons are instantaneously emitting and only of their energy loss is by synchrotron (the rest is by the accompanying SSC). In the fast cooling regime, the electrons cool on a shorter timescale than the pulse duration and several consecutive cooling processes may be needed to account for the comparatively long pulse duration. Eq. 11 can be understood in terms of the cooling time of the electrons, , the time it takes the electrons to radiate (by synchrotron and IC) their kinetic energy:
is declining as a function of . Therefore it is maximal for , and after a time all of the electrons have cooled significantly. Combining Eqs. 7, 9 and 12 gives: . It follows that: , which tells us that indeed only the instantaneously synchrotron emitting electrons contribute to the peak of .
Given the spectrum, the source must be optically thin for scatterings:
where is the total number of electrons and positrons in the flow. is composed of two components. The first, , is calculated from the peak flux. Notice that in principle there could be additional non-relativistic electrons in the initial flow that although they don’t contribute to the radiation, they can be important due to their effect on the optical depth. We return to this point in §3.2. The second, , is the electron-positron pairs created by annihilation of the high energy photons 333The pairs are created with energies of order half that of the high energy photon. This causes the pairs to have a steep power law of random Lorentz factors extending up to very high energies. These pairs radiate by synchrotron and contribute mainly at the optical band. Their radiation in the sub-MeV band is sub-dominant.. is derived by considering the optical depth for pair creation (Lithwick Sari (2001)):
where is the number of photons in the source with enough energy to annihilate the photon with frequency, . The frequency , is defined as the (source frame) photon frequency at which . Therefore, equals the number of photons above .
A more detailed calculation of the pair creation opacity has to take care of the details of the radiation field in the source (Granot et al., 2008; Hascoët et al., 2012). There could in principle be a numerical factor in Eq. 14 of order (Granot et al., 2008; Hascoët et al., 2012; Vurm et al., 2012) This in turn, would lower the minimum requirement on . However: , which means that the change in the lower limit of in the exact calculation, will only be of order 0.5-0.7.
3.2 A synchrotron solution for the MeV peak
We solve the synchrotron Eqns. described in §3.1 and reduce the 6D parameter space described in §2.2 to a 3D parameter space defined by (). We then consider three constraints that limit the allowed regions within this parameter space. In §5 we discus further constraints on this parameter space that arise from pair creation opacity limits.
We proceed to consider the limits on the parameter space () that arise from (i) efficiency (fast cooling), (ii) the size of the emitting radius and (iii) the opacity.
Efficiency and fast cooling require (§3.1), which yields:
The emitting radius must be smaller than , the deceleration radius (Panaitescu and Kumar, 2004; Kumar McMahon, 2008; Zou et al., 2009). The latter depends on the circum-burst density profile: a wind with or a constant ISM with . The limit on the emitting region can be written, in these two cases, as:
where erg is the total (isotropic) energy of the burst (and not just the energy emitted in one pulse), sec is the total duration of the burst, is the particle density in and .
The flow is optically thin for scattering. Typically, we find that is dominated by the pairs created within the flow by the annihilation of the high energy part of the spectrum. The main contribution to the number of photons available for pair creation is from IC scatterings of synchrotron photons to energies above the pair creation threshold. Solving Eqns. 13, 14 and 19 we obtain:
The criterion leads to a lower limit on . It should be noted that in many models one may expect a large amount of passive (i.e. non relativistic) electrons in the flow (Bošnjak et al., 2009). In this case the optical depth may become dominated by these electrons. We discuss this option in §4.
So far we discussed only the synchrotron signal itself, and we haven’t yet addressed the SSC contribution. We now turn to describe the SSC emission, and explore its effect on the solutions.
3.3 The SSC component
Synchrotron is essentially accompanied by SSC. We describe here a single-zone analytic solution of this component which is expected to reproduce the general characteristics of the true solution. However, we note that a full description of the up-scattered flux can only be done with more detailed calculations (Nakar et al., 2009; Bošnjak et al., 2009) and those might differ somewhat from the simplified analysis. If a SSC signal is detected, we can measure both the peak frequency and the flux of the SSC component. In this case, we obtain two additional equations that allow us to determine the parameters up to a single free parameter which we choose as . This may be the situation in the future, when CTA begins observing in the very high energy range. We return to this intriguing possibility in §6. But even if the SSC signal is not detected (as is the current observational situation) we can use the non-detection as a limit on the parameter space. Note that the signal could be undetected either because it is very weak or because it peaks at sufficiently high frequencies and cannot be observed by current detectors.
The typical SSC photon frequency is given by:
The IC process is suppressed by the Klein-Nishina effect and the photons are up-scattered only to , if the up-scattered photon’s energy in the electron’s rest frame exceeds the electron rest mass energy:
This condition is satisfied if:
Thus, for typical parameters the up-scattered photons are most likely in the KN regime. We introduce the parameter (Ando et al., 2008) to take the KN effect into account:
The total SSC flux (integrated over frequencies), , is related to the total flux ascosiated with the synchrotron peak, , by:
is the Compton parameter (Sari et al., 1996) .
As mentioned above, a possible future detection of an SSC component above the LAT range can, in principle, measure: and . This provides two extra equations, allowing us to solve for and . Using Eqns. 28 29 we can write:
In addition, detection of a SSC signal above tens of GeV, implies that the spectral breaks observed by the LAT are not due to an optical depth. If the particles are shock accelerated then these breaks would be associated with the maximal synchrotron frequency: Hz(de Jager, 1996) where depends on the acceleration mechanism, and is expected to be of order unity (Piran Nakar, 2010; Barniol Duran Kumar, 2011). If we can also observe we obtain another equation that would allow us to solve for and either obtain the full solution or rule it out if no solution is found:
As opposed to similar observations in Blazars, current observations in the GeV range do not detect a significant high energy component in typical GRBs. Current detections of a GeV component are at the level of 0.03 of the MeV emission and upper limits for most other bursts are at the level of 0.1. In a few cases (e.g. GRB 090926A (Ackermann et al., 2011)) an additional high energy component has been detected, and may be associated with SSC, however, even in this case one does not see a clear GeV peak, like the one observed in Blazars. In this case, the upper limits on the SSC signal further constrain the possible parameter space. First, the photon with largest observed frequency, must satisfy: , where the latter is the frequency at which the flow becomes optically thick to pair creation. Above , the photons could (but do not have to) be optically thick for pair creation. The condition leads to lower limits on . However, this condition is somewhat model dependent (Granot et al., 2008; Zou et al., 2009, 2011; Hascoët et al., 2012), and the limits on from this consideration may be alleviated in case of a two zone model. We therefore separate the discussion of this limit from the main analysis, and return to it later on in §5. Second, Beniamini et al. (2011) have shown that for a typical GBM burst the total flux which is observed in the LAT band (30 MeV-300 GeV) is at most 0.13 of the GBM (8KeV-40MeV) flux of the same burst (see also Ando et al. 2008, Guetta et al. 2011). More recent studies, using extra noise cuts applied by the LAT team (Ackermann et al., 2012) give more constraining upper limits which are lower by a factor . We define as the upper limit on the fractional LAT flux: (where ). Below , the photons are necessarily optically thin for pair creation and therefore in order for the LAT signal to be sufficiently low, either the total up-scattered flux is low (less than of the GBM flux), or most of the up-scattered flux is at and only a small fraction can be observed (for an illustration of these possibilities see Fig. 2). This leads to lower limits on and on .
Let denote the photon spectral index below , i.e. . Most GRBs are phenomenologically well fit by a Band function, and for these . Defining as the fraction of up-scattered flux observed in the LAT window, relative to the total up-scattered flux, the ratio between the expected flux up to and the sub-MeV flux, is:
If and if is rising below (as happens for ) then the total flux observed in the LAT window is dominated by . can be written as (Ando et al., 2008):
Plugging in (i.e. assuming ) yields:
We require that the SSC contribution will be sufficiently low to agree with the LAT observations. We separate the possible solutions to two cases.
If , then, and the total up-scattered flux (and not just the fraction within the considered band) is below the observed limit. Otherwise, the total ratio of up-scattered to synchrotron flux is larger than , but it peaks at and thus may be partially absorbed. As increases, the up-scattered flux peaks at higher frequencies. This means, that for large enough values of , the up-scattered tail below is small enough that it is compatible with the observations. Therefore, in this regime of , one can obtain a lower limit on in terms of arising from the requirement that the total up-scattered flux up to be less than the flux observed by LAT:
Notice that these are conservative limits as we only use the spectral range below where we know the conditions are optically thin. One must bear in mind, that efficiency considerations limit the amount of energy that may be carried by the up-scattered flux by virtue of Eq. 41. This limits but not .
Figs. 3, 4 depict the different GRB parameters superimposed on the allowed space in the plane arising from the above conditions. We plot here results for and . The condition that the SSC flux resides below the observational limits within the frequency range up to the maximal observed photon energy is plotted for GeV (source frame) corresponding to the highest (prompt) energy photon to date, from 080916C. They are drawn for two cases of the lower spectral slope: and . We choose to take the most extreme values for this limit, in order to show that even in this case, there is still reasonable parmeter space in the plane. We return to this extreme case in greater detail in §5.
Several characteristic features can be seen in these figures. First, is directly related to the pulse duration and bulk Lorentz factor via Eq. 2, and it is independent of . For the canonical observed parameters we use here, spans two orders of magnitude, , and it is relatively large. is the most fluctuating parameter spanning almost 4 orders of magnitude: . The lines of constant are almost parallel to the line, which means that the value of is almost a direct representative of , the fraction of instantaneously emitting electrons relative to the total number of electrons. The allowed range for is: . These high values of (compared with ) are extremely significant for GRB models in which the electrons are heated by shocks and the initial energy resides in protons. In these cases, it is necessary that only a small fraction of electrons will be heated to relativistic velocities in order to allow them to reach such high energies (Daigne Mochkovitch, 1998; Bošnjak et al., 2009). We show this explicitly for the internal shocks model in §4. In addition, we observe that is expected to lie in the range: Therefore, even with no knowledge of the LAT observations, we could have expected that the SSC does not peak in the GeV but typically at least two orders of magnitude above. This is a direct consequence of the large values of required for this solution. , is expected to lie between: .
Even for the highest observed photon energy of 71 GeV, we see that the lower limits on that arise from the SSC flux in the GeV are less constraining than the lower limits that arise from the optical depth. However, the SSC limits become very strong for small values of the lower spectral slope, as expected in the fast cooling synchrotron regime. For the expected (which is found for of the GRBs in the GBM and BATSE samples) for ( for ). In addition, very negative values of , push solutions with relatively low towards the marginally fast solution () as discussed in §3.6. By Eq. 23, the lower limit on varies as and therefore increases for . The upper limit that arises from scales as , and also increases for . similarly the allowed parameter space increases with 444The less rigid upper limit on the parameter space arising from also increases with , either as for an ISM or as for a wind environment. .
The main effect of increasing is to increase the allowed values for . The allowed range for is roughly for and for . This range is narrowed down by considering the pair creation opacity limits and it will be addressed again in §5. Interestingly, increasing does not change significantly other parameters of the model, . implies that the lower limit on the radius scales as while the upper limit scales as . Thus, even for relatively large one expects a comparable range of emission radii to the one obtained for . For the magnetic field, the lower limit scales as while the upper limit depends on