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it should therefore be possible to identify CDM haloes in real galaxies by measuring their rotation curves.
it should therefore be possible to identify CDM haloes in real galaxies by measuring their rotation curves.
Over the last 15 years or so. much effort has been put into determining the central mass distribution in. galaxies using their rotation curves. and comparing them with the outcomes of ever more sophisticated numerical simulations.
Over the last 15 years or so, much effort has been put into determining the central mass distribution in galaxies using their rotation curves, and comparing them with the outcomes of ever more sophisticated numerical simulations.
To first order, one can summarize this work as observational determinations yielding slopes a~0. while simulations produce ο~—1 slopes.
To first order, one can summarize this work as observational determinations yielding slopes $\alpha \sim 0$, while simulations produce $\alpha \sim -1$ slopes.
This persistent difference is known as the “core/cusp controversy”. sometimes also described as “the small-scale crisis in cosmology”.
This persistent difference is known as the “core/cusp controversy”, sometimes also described as “the small-scale crisis in cosmology”.
The attempts to reconcile the observations and simulations. either by trying to improve them. or by trying to quantify systematic effects or missing physics. are the subjects of this paper.
The attempts to reconcile the observations and simulations, either by trying to improve them, or by trying to quantify systematic effects or missing physics, are the subjects of this paper.
I give a brief overview of past and present work dealing with the determination of the central dark matter density distribution in galaxies. with an emphasis on the observational efforts.
I give a brief overview of past and present work dealing with the determination of the central dark matter density distribution in galaxies, with an emphasis on the observational efforts.
An overview like this. touching on many different topics in galaxy evolution. cosmology and computational astrophysics. is never complete. and only a small (but hopefully somewhat representative) fraction of the many papers relevant to this topic can be referred to in the limited space available.
An overview like this, touching on many different topics in galaxy evolution, cosmology and computational astrophysics, is never complete, and only a small (but hopefully somewhat representative) fraction of the many papers relevant to this topic can be referred to in the limited space available.
The rest of this paper is organised as follows: Sect.
The rest of this paper is organised as follows: Sect.
2 gives a description of the results that numerical simulations have produced over the years.
2 gives a description of the results that numerical simulations have produced over the years.
Section. 3 deals with the observational determinations of the dark matter. density distribution.
Section 3 deals with the observational determinations of the dark matter density distribution.
Section 4 discusses physical scenarios that have been proposed to reconcile the core and cusp distributions.
Section 4 discusses physical scenarios that have been proposed to reconcile the core and cusp distributions.
Section 5 briefly summarizes the work discussed.
Section 5 briefly summarizes the work discussed.
The presence of a cusp in the centers of CDM_ halos is one of the earliest and strongest results derived from cosmological N-body simulations.
The presence of a cusp in the centers of CDM halos is one of the earliest and strongest results derived from cosmological N-body simulations.
? were among the first to investigate the density profiles of CDM halos and found that the inner parts of these simulated halos could be characterized by a power-law slope a2—1.
\citet{Dubinski:1991p4} were among the first to investigate the density profiles of CDM halos and found that the inner parts of these simulated halos could be characterized by a power-law slope $\alpha = -1$.
They did not rule out the existence of central cores. but noted that these would have to be smaller than the resolution of their simulations (~L4 kpe).
They did not rule out the existence of central cores, but noted that these would have to be smaller than the resolution of their simulations $\sim 1.4$ kpc).
Subsequent simulations. at higher and higher resolutions. made the presence of cores in simulated CDM haloes increasingly unlikely.
Subsequent simulations, at higher and higher resolutions, made the presence of cores in simulated CDM haloes increasingly unlikely.
Asystematic study by 2?. of simulated CDM halos. derived assuming many different sets of cosmological parameters. found that the innermost dark matter mass density distribution could be well described by a characteristic à——1 inner slope for all simulated halos. independent of mass. size or cosmology.
A systematic study by \citet{Navarro:1996p479, Navarro:1997p482} of simulated CDM halos, derived assuming many different sets of cosmological parameters, found that the innermost dark matter mass density distribution could be well described by a characteristic $\alpha = -1$ inner slope for all simulated halos, independent of mass, size or cosmology.
A similar general result was found for the outer mass profile. with a steeper slope of à2—3.
A similar general result was found for the outer mass profile, with a steeper slope of $\alpha=-3$.
2. called this the "universal density profile" and it is described by where p; 1s related to the density of the universe at the time of the time of halo collapse and Ας is the characteristic radius of the halo.
\citet{Navarro:1997p482} called this the “universal density profile” and it is described by where $\rho_i$ is related to the density of the universe at the time of the time of halo collapse and $R_s$ is the characteristic radius of the halo.
This kind of profile is also known as the "NEW profile".
This kind of profile is also known as the “NFW profile”.
The corresponding rotation curve is given by with x2r/Rsoo.
The corresponding rotation curve is given by with $x=r/R_{200}$.
This curve curve is parameterized by a radius Aso; and a concentration parameter c=Rooo/R,.
This curve curve is parameterized by a radius $R_{200}$ and a concentration parameter $c=R_{200}/R_s$.
Here Rsoo is the radius at which the density contrast with respect to the critical density of the universe exceeds 200. roughly the virtal radius: Vsoo is the circular velocity at Rago (2)..
Here $R_{200}$ is the radius at which the density contrast with respect to the critical density of the universe exceeds 200, roughly the virial radius; $V_{200}$ is the circular velocity at $R_{200}$ \citep{Navarro:1996p479}.
The parameters c and Vago are tightly related through the assumed cosmology.
The parameters $c$ and $V_{200}$ are tightly related through the assumed cosmology.
Indeed. one can be expressed as a function of the other. with only à small scatter (2)..
Indeed, one can be expressed as a function of the other, with only a small scatter \citep{Bullock:2001p541}.
That 15. the range of (c.Yoo) combinations that describes “real” CDM rotation curves is tightly constrained by the ACDM cosmology.
That is, the range of $(c,V_{200})$ combinations that describes “real” CDM rotation curves is tightly constrained by the $\Lambda$ CDM cosmology.
Simulations by ? indicated an even steeper inner slope.
Simulations by \citet{Moore:1999p517} indicated an even steeper inner slope.
They found that their simulated halos could be best described by a function Le. with an inner slope à2—1.5 and an outer slope à2—3.
They found that their simulated halos could be best described by a function i.e., with an inner slope $\alpha=-1.5$ and an outer slope $\alpha=-3$.
The difference between these two results indicated. that issues such as numerical convergence. initial. conditions. analysis or interpretation could still play a role in defining the inner slope.
The difference between these two results indicated that issues such as numerical convergence, initial conditions, analysis or interpretation could still play a role in defining the inner slope.
As ever more powerful computers and increasingly higher resolution simulations became available. the value and behavior of the inner slope of CDM_ halos has therefore been extensively discussed in the literature.
As ever more powerful computers and increasingly higher resolution simulations became available, the value and behavior of the inner slope of CDM halos has therefore been extensively discussed in the literature.
For example. to give but an incomplete listing of the many papers that have appeared on this topic. ? derived slopes à=—1.5 for their simulated halos.
For example, to give but an incomplete listing of the many papers that have appeared on this topic, \citet{Klypin:2001p46} derived slopes $\alpha = -1.5$ for their simulated halos.
From phase-space density arguments. ? argue that the density profile should resemble an NEW profile. but converging to an inner slope à=—0.75. instead of the a--] value.
From phase-space density arguments, \citet{Taylor:2001p684} argue that the density profile should resemble an NFW profile, but converging to an inner slope $\alpha = -0.75$, instead of the $\alpha = -1$ value.
? investigated low-mass haloes and found that they were best described using NFW profiles (i.e... à=—1).
\citet{Colin:2004p38} investigated low-mass haloes and found that they were best described using NFW profiles (i.e., $\alpha=-1$ ).
? found that CDM halos have cusps with a slope a~—1.2.
\citet{Diemand:2005p13} found that CDM halos have cusps with a slope $\alpha \simeq -1.2$.
Many studies assumed that the central cusp consisted of a region where the mass density behaved as a power-law with a constant slope.
Many studies assumed that the central cusp consisted of a region where the mass density behaved as a power-law with a constant slope.
? and? suggested that this did not have to be the case.
\citet{Navarro:2004p19} and \citet{Hayashi:2004p2} suggested that this did not have to be the case.
They did not find evidence for an asymptotic power-law slope. but instead noted that the slope kept getting shallower towards smaller radii without converging to a single asymptotic value.
They did not find evidence for an asymptotic power-law slope, but instead noted that the slope kept getting shallower towards smaller radii without converging to a single asymptotic value.
At the smallest resolved radit they derive slopes of ~—1.2 for "galaxy-sized" halos (as measured at ~1.3 kpe). and ~—1.35 for "dwarf galaxy" halos (as measured at ~0.4 kpe).
At the smallest resolved radii they derive slopes of $\sim -1.2$ for “galaxy-sized” halos (as measured at $\sim 1.3$ kpc), and $\sim -1.35$ for “dwarf galaxy” halos (as measured at $\sim 0.4$ kpc).
These values are significantly steeper than the original NFW slope. but not as steep as the ?— value.
These values are significantly steeper than the original NFW slope, but not as steep as the \citet{Moore:1999p517} value.
? introduce a newfitting formula to quantify their results.
\citet{Navarro:2004p19} introduce a newfitting formula to quantify their results.
For reasonable choices of its input parameters. this formula yields an extrapolated slope of à~—0.7 at r~0.01 kpe.
For reasonable choices of its input parameters, this formula yields an extrapolated slope of $\alpha \sim -0.7$ at $r \sim 0.01$ kpc.
? also finds a gradual turn-over in slope towards smaller radii.
\citet{Stoehr:2006p25} also finds a gradual turn-over in slope towards smaller radii.
Though his simulations formally resolve only radit —| kpe (where a slope of à~—1 is measured). an extrapolation of his favoured fitting function towards smaller radit results in à decreasing slope ending up as a flat slope (42 0) around r~0.01 kpe.
Though his simulations formally resolve only radii $\sim 1$ kpc (where a slope of $\alpha \sim -1$ is measured), an extrapolation of his favoured fitting function towards smaller radii results in a decreasing slope ending up as a flat slope $\alpha=0$ ) around $r \sim 0.01$ kpc.
? and? showed that the density distribution presented in could be equally well described by a Sérrsic function.
\citet{Merritt:2005p605} and \citet{Graham:2006p616} showed that the density distribution presented in \citet{Navarro:2004p19} could be equally well described by a Sérrsic function.
In the context of CDM halos they refer to this function as an Einasto model.
In the context of CDM halos they refer to this function as an Einasto model.
For completeness. this profile is given by where 5 determines the shape of the profile. and d, is a function of which enables the use of the density p, measured at the effective2 radius ο.
For completeness, this profile is given by where $n$ determines the shape of the profile, and $d_n$ is a function of $n$ which enables the use of the density $\rho_e$ measured at the effective radius $r_e$ .
The latter is defined as the radius of the volume containing half of the total mass.
The latter is defined as the radius of the volume containing half of the total mass.
In terms of observationally more accessible quantities this can be written as
In terms of observationally more accessible quantities this can be written as
Fornax luminous CSS sub-populations.
Fornax luminous CSS sub-populations.
By eye it appears that there is a concentration of blue (presumably metal-poor) luminous CSSs near the central massive galaxy in each cluster.
By eye it appears that there is a concentration of blue (presumably metal-poor) luminous CSSs near the central massive galaxy in each cluster.
However, while the Virgo red sub-population is too small to give reliable statistics, a 2D K-S test confirms (probability statistic 0.1) that the red and blue luminous CSS sub-populations around NGC 1399 have similar overall spatial distributions.
However, while the Virgo red sub-population is too small to give reliable statistics, a 2D K-S test confirms (probability statistic 0.1) that the red and blue luminous CSS sub-populations around NGC 1399 have similar overall spatial distributions.
Fig.
Fig.
14 compares the SDSS extinction-corrected colour indices of CSSs in the Fornax and Virgo cluster core environments with those of dEs and GCs.
\ref{fig:colmag_6} compares the SDSS extinction-corrected colour indices of CSSs in the Fornax and Virgo cluster core environments with those of dEs and GCs.
To aid our analysis, we separate the brighter and fainter luminous CSSs at an arbitrary deredenned magnitude of M,.o=—11.
To aid our analysis, we separate the brighter and fainter luminous CSSs at an arbitrary deredenned magnitude of $M_{g,0} = -11$.
We also indicate with circles those CSSs outside the magnitude limits (grey box) shown in Fig. 12,,
We also indicate with circles those CSSs outside the magnitude limits (grey box) shown in Fig. \ref{fig:colmag_2},
and exclude them from our analysis.
and exclude them from our analysis.
Fornax luminous CSSs, both brighter and fainter, closely follow the point source (mainly stellar) locus.
Fornax luminous CSSs, both brighter and fainter, closely follow the point source (mainly stellar) locus.
Virgo luminous CSSs (not circled) are distributed on two best fitting lines — the brighter CSS’s (solid line: [g—r]=(0.76+0.33)[rτ](0.40 0.10)) more closely follow the distribution of Virgo dEs, whereas the fainter CSSs (dashed line: [g—r]=(0.232-0.19)[r—i]+ (0.560.06)) appear more widely dispersed like the M87 GCs.
Virgo luminous CSSs (not circled) are distributed on two best fitting lines – the brighter CSS's (solid line: $[g-r] = (0.76\pm0.33)[r-i] + (0.40\pm0.10)$ ) more closely follow the distribution of Virgo dEs, whereas the fainter CSSs (dashed line: $[g-r] = (0.23\pm0.19)[r-i] + (0.56\pm0.06)$ ) appear more widely dispersed like the M87 GCs.
We could interpret the differing distributions of brighter and fainter sub-populations of luminous CSSs in the Virgo Cluster to support the hypothesis that the brightest CSSs come from tidally-stripped dE,N galaxies while the fainter CSSs are the luminous tail of the cluster core GC distribution — however the dashed line for faint CSS and the GC distribution are influenced strongly by the uncertainties in photometry (see typical error bars on the plot) and small number statistics (evidenced by the uncertainties in best fitting line slopes quoted above).
We could interpret the differing distributions of brighter and fainter sub-populations of luminous CSSs in the Virgo Cluster to support the hypothesis that the brightest CSSs come from tidally-stripped dE,N galaxies while the fainter CSSs are the luminous tail of the cluster core GC distribution – however the dashed line for faint CSS and the GC distribution are influenced strongly by the uncertainties in photometry (see typical error bars on the plot) and small number statistics (evidenced by the uncertainties in best fitting line slopes quoted above).
the massive companion.
the massive companion.
The hard X-ray spectrum with photon index Γ«1.5 also points towards dominant adiabatic losses, which imply an injection electron index of o,~2.
The hard X-ray spectrum with photon index $\mathrm{\Gamma}\approx1.5$ also points towards dominant adiabatic losses, which imply an injection electron index of $\mathrm{\alpha_e}\approx2$.
In the regime of dominant adiabatic losses the emitted X-ray flux is proportional to the number of emitting particles, so the orbital dependency of adiabatic losses can be inferred from the X-ray light curve.
In the regime of dominant adiabatic losses the emitted X-ray flux is proportional to the number of emitting particles, so the orbital dependency of adiabatic losses can be inferred from the X-ray light curve.
Dominant adiabatic losses are not uncommon when modelling gamma-ray binaries.
Dominant adiabatic losses are not uncommon when modelling gamma-ray binaries.
They have also been invoked to explain the variations of the X-ray and VHE fluxes in the gamma-ray binaries PSR B1259-63 and LS 5039 by (see also ?)) and?,, respectively.
They have also been invoked to explain the variations of the X-ray and VHE fluxes in the gamma-ray binaries PSR B1259-63 and LS 5039 by (see also ) and, respectively.
We note that, as explained in Sec. ??,,
We note that, as explained in Sec. \ref{sec:icsyn},
the IC emission will be additionally modulated by the changes in seed photon density along the orbit as well as the geometrical effects of the location of the emitter along the orbit.
the IC emission will be additionally modulated by the changes in seed photon density along the orbit as well as the geometrical effects of the location of the emitter along the orbit.
For the sake of simplicity, and given how little we know about the exact properties of the particle accelerator and the non-thermal emitter inLS1+61303,, we will here assume a constant particle injection spectrum and emitter magnetic field along the orbit.
For the sake of simplicity, and given how little we know about the exact properties of the particle accelerator and the non-thermal emitter in, we will here assume a constant particle injection spectrum and emitter magnetic field along the orbit.
We will also adopt one leptonic population cooling down under variable adiabatic losses at the location of the emitter, which will be approximated as homogeneous and point-like.
We will also adopt one leptonic population cooling down under variable adiabatic losses at the location of the emitter, which will be approximated as homogeneous and point-like.
The orbital parameters were adopted from with an inclination angle of i=45°.
The orbital parameters were adopted from with an inclination angle of $i=45^\circ$.
We will also discuss other possibilities regarding the inclination angle, in particular the extremes of the allowed range 15?Si60? considered by?.
We will also discuss other possibilities regarding the inclination angle, in particular the extremes of the allowed range $15^\circ \la i \la 60^\circ$ considered by.
. One of the remarkable features of the VHE light curve of lis the lack of detectable emission during periastron(?).
One of the remarkable features of the VHE light curve of is the lack of detectable emission during periastron.
. IC emission is very effective at this phase owing to the high stellar photon density, so the lack of emission means that either it is strongly absorbed or that the number of accelerated electrons falls drastically.
IC emission is very effective at this phase owing to the high stellar photon density, so the lack of emission means that either it is strongly absorbed or that the number of accelerated electrons falls drastically.
However, the angular dependency of the pair production process, ος(1—cosy) for the interaction rate and «(1—cos)! for the gamma-ray energy threshold (where y is the interaction angle of the incoming photons), implies that absorption is very low for phases immediately after periastron even under the dense seed photon field of the star (see Sec.
However, the angular dependency of the pair production process, $\propto (1-\cos\psi)$ for the interaction rate and $\propto (1-\cos\psi)^{-1}$ for the gamma-ray energy threshold (where $\psi$ is the interaction angle of the incoming photons), implies that absorption is very low for phases immediately after periastron even under the dense seed photon field of the star (see Sec.
3.5 for details on the calculation of pair production absorption).
\ref{sec:gg} for details on the calculation of pair production absorption).
Inverse Compton emission also has a dependency on the angle between the seed photon and the accelerated electron and would be lower during these phases, but this effect is diminished at higher energies when scattering takes place in the deep Klein-Nishina regime?).
Inverse Compton emission also has a dependency on the angle between the seed photon and the accelerated electron and would be lower during these phases, but this effect is diminished at higher energies when scattering takes place in the deep Klein-Nishina regime.
. A reduction in the number of accelerated particles is required to explain the lack of VHE emission around periastron.
A reduction in the number of accelerated particles is required to explain the lack of VHE emission around periastron.
Although the X-ray flux during periastron is around half of the peak flux during the periodic outbursts, the non-detection of VHE emission during periastron places an upper limit at only a of the outburst peak VHE flux.
Although the X-ray flux during periastron is around half of the peak flux during the periodic outbursts, the non-detection of VHE emission during periastron places an upper limit at only a of the outburst peak VHE flux.
Taking all this into account, we will assume that in the context of one electron population the X-ray/VHE correlation is only valid for the X-ray excess flux above a certain quiescent or pedestal emission.
Taking all this into account, we will assume that in the context of one electron population the X-ray/VHE correlation is only valid for the X-ray excess flux above a certain quiescent or pedestal emission.
Therefore, the number of emitting particles responsible for this excess X-ray flux is lower and their VHE emission remains at non-detectable levels around periastron.
Therefore, the number of emitting particles responsible for this excess X-ray flux is lower and their VHE emission remains at non-detectable levels around periastron.
This reasoning is supported by the X-ray/VHE correlation found by?,, in which the constant term of the linear correlationis (12.2+°observations00)x107?ergcm"?s-!.
This reasoning is supported by the X-ray/VHE correlation found by, in which the constant term of the linear correlation is $(12.2^{+0.9}_{-1.0})\times10^{-12}\,\ergcms$.
Considering that some of the X-ray are below this constant term, we took the pedestal flux value to be Fyeq=11.5x107?ergcm? sl.
Considering that some of the X-ray observations are below this constant term, we took the pedestal flux value to be $F_\mathrm{ped}=11.5\times10^{-12}\,\ergcms$ .
As mentioned above, a scenario with dominant adiabatic losses is a very good candidate mechanism to explain the correlation characteristics of the source.
As mentioned above, a scenario with dominant adiabatic losses is a very good candidate mechanism to explain the correlation characteristics of the source.
When considering a system with constant magnetic field and constant injected electron spectrum, the only modulation in synchrotron radiation is due to the modulation of the dominant cooling process that results in a modulation of the (evolved) stationary electron distribution.
When considering a system with constant magnetic field and constant injected electron spectrum, the only modulation in synchrotron radiation is due to the modulation of the dominant cooling process that results in a modulation of the (evolved) stationary electron distribution.
For dominant adiabatic losses and a constant injection rate, the X-ray flux will be proportional to the adiabatic cooling time fg.
For dominant adiabatic losses and a constant injection rate, the X-ray flux will be proportional to the adiabatic cooling time $t_\mathrm{ad}$.
A consistent calculation of tag would require knowledge of the exact nature of the source and the (magneto)hydrodynamical processes that take place there.
A consistent calculation of $t_\mathrm{ad}$ would require knowledge of the exact nature of the source and the (magneto)hydrodynamical processes that take place there.
One can however infer f,q along the orbit by taking it proportional to the excess flux above the pedestal Fx—Fea, which is independent of the electron energy.
One can however infer $t_\mathrm{ad}$ along the orbit by taking it proportional to the excess flux above the pedestal $F_\mathrm{X}-F_\mathrm{ped}$, which is independent of the electron energy.
For the phases covered by the X-ray observations we derived a smooth curve
For the phases covered by the X-ray observations we derived a smooth curve
wherea is the inaxiumu allowed probability of a false S S*:S*S*(0) a 2)). à Terror: L..
where$\alpha$ is the maximum allowed probability of a false $\thrtest$ $\thresh$$\thresh$$\thresh(\alpha)$ $\alpha$ \ref{fig:thresh}) $\alpha$ \ref{eq:alpha}.
à. 3)). 5. Ag>0.
$\alpha$ \ref{fig:alfabetillus}) \ref{eq:beta} $\lamS>0$
and ο, but with somewhat different amplification factors.
and 6, but with somewhat different amplification factors.
Thus (he presence of resonance of some significant aniplitude is not strongly dependent on the precise value of the meridional flow.
Thus the presence of resonance of some significant amplitude is not strongly dependent on the precise value of the meridional flow.
This means in effect (hat equation (44) more closely determines the resonance than does equation (43).
This means in effect that equation (44) more closely determines the resonance than does equation (43).
la keeping with this inference. if we choose a value of a, onlv a few percent away from that predicted to eive resonance. (he resonance practically disappears no malter what speed of meridional How is taken.
In keeping with this inference, if we choose a value of $\alpha_L$ only a few percent away from that predicted to give resonance, the resonance practically disappears no matter what speed of meridional flow is taken.
Figure 7 displavs the amplitude of the induced toroidal field in the lower laver as a [function of meridional [low speed ο in the upper laver. lor the a, values predicted. [or resonance lor the same parameter choices as in Figures 5 and 6.
Figure 7 displays the amplitude of the induced toroidal field in the lower layer as a function of meridional flow speed $v_U$ in the upper layer, for the $\alpha_L$ values predicted for resonance for the same parameter choices as in Figures 5 and 6.
We see that [or 2),=.03. the peak field does not occur al ep=—12.56. al which equation (43) is satisfied. but rather al values above and below (hat (blue and green. curves. respectivelv). depending on the a, chosen.
We see that for $P_{\eta L}=.03$, the peak field does not occur at $v_U=-12.86$, at which equation (43) is satisfied, but rather at values above and below that (blue and green curves, respectively), depending on the $\alpha_L$ chosen.
While resonance still occurs for the predicted a, values. it is a [actor of five to ten smaller (han at (he peaks shown.
While resonance still occurs for the predicted $\alpha_L$ values, it is a factor of five to ten smaller than at the peaks shown.
This is again evidence (hat the closeness of equation (44) to being satisfied is the determining factor in closeness to resonance.
This is again evidence that the closeness of equation (44) to being satisfied is the determining factor in closeness to resonance.
This is what makes the denominator of the algebraic expressions for ABL and AAL smallest.
This is what makes the denominator of the algebraic expressions for $ABL$ and $AAL$ smallest.
This departure from the predicted points of resonance occurs because of the amplitude ol the cliffusivity assumed for the lower laver.
This departure from the predicted points of resonance occurs because of the amplitude of the diffusivity assumed for the lower layer.
This is evident [rom Figure 7 because the peak toroidal fields produced for Py,=0.003 (red and vellow curves) occur very. close to the à, values predicted [or resonance.
This is evident from Figure 7 because the peak toroidal fields produced for $P_{\eta L}=0.003$ (red and yellow curves) occur very close to the $\alpha_L$ values predicted for resonance.