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Following the L99a aud L99b exposition: The scene being inaged has intriusic intensity distribution Or.(LOTS):y). with Fourier trausform OCpy).
Following the L99a and L99b exposition: The scene being imaged has intrinsic intensity distribution $O(x,y)$, with Fourier transform $\tilde O(k_x, k_y)$.
The Fourier transform convention is "System2" of Bracewell The telescope optics convolve theimage with some optical point-spread function (PSF) Pr.9) (which I take to have unit integral).
The Fourier transform convention is “System2” of \citet{Br78}: The telescope optics convolve the image with some optical point-spread function (PSF) $P(x,y)$ (which I take to have unit integral).
With the above couvention for the Fourier trausform. the convolution O«P has transform O-P.
With the above convention for the Fourier transform, the convolution $O\ast P$ has transform $\tilde O \cdot \tilde P$.
The pixelization of theimage by the detector entails two operatious: first. the optical image is convolved with the (PRE) Ar.) (which E normalize to unit integral). aud sampled ou the two-dimensional grid of pixel centers on spacing a.
The pixelization of the image by the detector entails two operations: first, the optical image is convolved with the (PRF) $R(x,y)$ (which I normalize to unit integral), and sampled on the two-dimensional grid of pixel centers on spacing $a$.
The data [rom a sinele array readout are thus the image where HI is the 2d. funetion. Iu the Fourier domain. the pixelated. sampled image is The detected image. therefore. looks like the source image as convolved with anPSF (ePSE) P'=PxFR. and sampled at interval ¢.
The data from a single array readout are thus the image where $\shah$ is the 2d function, In the Fourier domain, the pixelated, sampled image is The detected image, therefore, looks like the source image as convolved with an (ePSF) $P^\prime\equiv P\ast R$, and sampled at interval $a$ .
The sampling mixes power at spatial frequency Aycan down to frequency. Ay. leaving the nature of the original O(k) ambiguous.
The sampling mixes power at spatial frequency $k_x+m\Delta k$ down to frequency $k_x$, leaving the nature of the original $\tilde O({\bf k})$ ambiguous.
Thisaliasing is detrimental to our efforts. as we caunot from a siugle measurement know exactly either the ePSF (from observing poiut-source stars) or the intrinsicscene (0.
This is detrimental to our efforts, as we cannot from a single measurement know exactly either the ePSF (from observing point-source stars) or the intrinsicscene $O$ .
cross sections show noticeable resonant structures down to the lowest energies measured so fav in the laboratory £~2.4 MeV (Ixettner. Lorenz-Wirzba. Rolls 1980).
cross sections show noticeable resonant structures down to the lowest energies measured so far in the laboratory $E \sim 2.4$ MeV (Kettner, Lorenz-Wirzba, Rolfs 1980).
If the resonant structure continues to even lower energies and (he astrophysical reaction rate is due to the contributions of narrow resonances. one then has to consider that the entrance channel width of (hese resonances will be modified in the plasma.
If the resonant structure continues to even lower energies and the astrophysical reaction rate is due to the contributions of narrow resonances, one then has to consider that the entrance channel width of these resonances will be modified in the plasma.
Cussons. Langanke. Liolios (2002) have specifically pointed out the possible importance of the plasma effects on the resonant PC + PC reactions [or a carbon white dwarf environment with T—5x10* IK and p—2x10? gem 7.
Cussons, Langanke, Liolios (2002) have specifically pointed out the possible importance of the plasma effects on the resonant $^{12}$ C + $^{12}$ C reactions for a carbon white dwarf environment with $T = 5 \times 10^{7}$ K and $\rho = 2 \times 10^{9}$ g $^{-3}$.
They have considered a resonance energy interval 0.4—2 MeV. They have specifically discussed. a rather extreme case of the low resonance energv £2, = 400 keV and have estimated the overall enhancement of the resonant AC + PC reaction rates due to the plasma effects for (his case.
They have considered a resonance energy interval $-$ 2 MeV. They have specifically discussed a rather extreme case of the low resonance energy $E_{r}$ = 400 keV and have estimated the overall enhancement of the resonant $^{12}$ C + $^{12}$ C reaction rates due to the plasma effects for this case.
Cussons. Langanke. Liolios (2002) adopted the method of Salpeter Van llorn (1969) which is based on the lattice model of the dense plasma to calculate (he resonant screening effects.
Cussons, Langanke, Liolios (2002) adopted the method of Salpeter Van Horn (1969) which is based on the lattice model of the dense plasma to calculate the resonant screening effects.
One of the present authors CN. 1.) and his collaborators have caleulated the enhancement of non-resonant thermonuclear reaction rates in extremely dense stellar plasmas (toh. Ixuwashima. Munakata 1990).
One of the present authors (N. I.) and his collaborators have calculated the enhancement of non-resonant thermonuclear reaction rates in extremely dense stellar plasmas (Itoh, Kuwashima, Munakata 1990).
This work is a natural extension of the works of Itoh. Totsuji. Ichimaru (1977) and Hoh et al. (
This work is a natural extension of the works of Itoh, Totsuji, Ichimaru (1977) and Itoh et al. (
1979). and improves upon the accuracy ol (he results of Salpeter Van Horn (1969).
1979), and improves upon the accuracy of the results of Salpeter Van Horn (1969).
Itoh. Ixuwashima. Munakata (1990) have summarized their numerical results by an accurate analvtical fitting Formula which will be readilv implemented in the stellar evolution computations.
Itoh, Kuwashima, Munakata (1990) have summarized their numerical results by an accurate analytical fitting formula which will be readily implemented in the stellar evolution computations.
The aim of the present paper is to extend (he work of Hoh. INwwashima. Munakata (1990) to the case of resonant reactions.
The aim of the present paper is to extend the work of Itoh, Kuwashima, Munakata (1990) to the case of resonant reactions.
The present paper is organized as follows.
The present paper is organized as follows.
Physical conditions relevant to the present calculation are made explicit in 2.
Physical conditions relevant to the present calculation are made explicit in 2.
Calculation of the enhancement factor of the resonant thermonuclear reaction rates is summarized in 3.
Calculation of the enhancement factor of the resonant thermonuclear reaction rates is summarized in 3.
The results are presented in 4+.
The results are presented in 4.
Extension (o the case of ionic mixtures is made in 5.
Extension to the case of ionic mixtures is made in 5.
rg Concluding remarks are given in 6.
Concluding remarks are given in 6.
First we consider thermonuclear reactions which take place in the plasma in the thermodynamic equilibium at temperature 7 composed of one kind of atomic nuclei and electrons. with number densities n; and n», respectively: Ze and AL denote the electric charge and the mass of such an ion.
First we consider thermonuclear reactions which take place in the plasma in the thermodynamic equilibrium at temperature $T$ composed of one kind of atomic nuclei and electrons with number densities $n_{i}$ and $n_{e}$ respectively; $Ze$ and $M$ denote the electric charge and the mass of such an ion.
The conventional parameters which characterize such a plasma are
The conventional parameters which characterize such a plasma are
scenarios.
scenarios.
The so-calledfrostiny models (e... Trager 200) propose sinall amounts of recent star formaion originated these stars;
The so-called models (e.g., Trager 2000) propose small amounts of recent star formation originated these stars.
Ou the other haud.quenching models (e.9.. Dell 200L. Faber 2005) sugecst tliat blue galaxies nügrate to the red sequence after cessaion of star formation. possibly associated with a merecr eveut and/or cuhanced ACN activity.
On the other hand, models (e.g., Bell 2004, Faber 2005) suggest that blue galaxies migrate to the red sequence after cessation of star formation, possibly associated with a merger event and/or enhanced AGN activity.
In a separate study (IHarker
In a separate study (Harker
The prospects of finding habitable planets orbiting nearby solar-type stars are to a large degree associated with the ultra-precise astrometric instruments under development or construction, such as the SIM Observatory (Shaoetal.2009;Unwin2008;Catan-zariteetal.2006) and Gaia (Casertanoetal.2008).
The prospects of finding habitable planets orbiting nearby solar-type stars are to a large degree associated with the ultra-precise astrometric instruments under development or construction, such as the SIM Observatory \citep{sha,unw,catpa} and Gaia \citep{cas}.
. The Earth orbiting the Sun produces an observable astrometric wobble of 3 wAU (micro-AU) and a radial velocity variation of 0.089s~!,, if seen equator-on (inclination i= 90°).
The Earth orbiting the Sun produces an observable astrometric wobble of 3 $\mu$ AU (micro-AU) and a radial velocity variation of 0.089, if seen equator-on (inclination $i=90\degr$ ).
A pole-on configuration (;=0° or 180°) is optimal for astrometry, because the reflex motion signal is present in both
A pole-on configuration $i=0\degr$ or $180\degr$ ) is optimal for astrometry, because the reflex motion signal is present in both
magnetic buovancy is increased by increasing the control parameter f.
magnetic buoyancy is increased by increasing the control parameter $f$.
In the limit /—0. we get back the model of Chouclhuri. Schüsssler. Dikpati (1995). in which there was no magnetic buovaney and the toroidal field was brought to the surface by the meridional circulation.
In the limit $f=0$, we get back the model of Choudhuri, Schüsssler, Dikpati (1995), in which there was no magnetic buoyancy and the toroidal field was brought to the surface by the meridional circulation.
When f is made sufficiently large (even though it has to remain less than 1). magnetic buovancy is found to dominate and the svstem has a limiting behavior.
When $f$ is made sufficiently large (even though it has to remain less than 1), magnetic buoyancy is found to dominate and the system has a limiting behavior.
oNupared to the double ring method. this method has some attractive features.
Compared to the double ring method, this method has some attractive features.
Firstly. here (he eruption at any instant takes place over a range of latitude rather than at one point as in the double ring method.
Firstly, here the eruption at any instant takes place over a range of latitude rather than at one point as in the double ring method.
This corresponds (o the real Sun more closely.
This corresponds to the real Sun more closely.
It is not easy to extend the double ring method to handle simultaneous [αν eruptions at more than one point.
It is not easy to extend the double ring method to handle simultaneous flux eruptions at more than one point.
If we simultaneously put several double rings in a range of latitudes. then (he positive ring of an intermediate double ring will cancel with tlie negative ring of the next double ring and we shall be left with a positive ring and a negative ring al a wide separation.
If we simultaneously put several double rings in a range of latitudes, then the positive ring of an intermediate double ring will cancel with the negative ring of the next double ring and we shall be left with a positive ring and a negative ring at a wide separation.
Ht follows from (14) that this will mean adding to A over a wide range of latitude.
It follows from (14) that this will mean adding to $A$ over a wide range of latitude.
This would make (he model more similar to the mean field model ancl the special character of the original double ring model would be completely lost.
This would make the model more similar to the mean field model and the special character of the original double ring model would be completely lost.
Also. we now allow for the toroidal {nx to be depleted at the bottom of the convection zone due to magnetic buovancey.
Also, we now allow for the toroidal flux to be depleted at the bottom of the convection zone due to magnetic buoyancy.
As we shall argue later. we believe this to be «uite important.
As we shall argue later, we believe this to be quite important.
In fact. we shall present some results with the double ring method with the toroidal (lux at the bottom depleted parametrically.
In fact, we shall present some results with the double ring method with the toroidal flux at the bottom depleted parametrically.
We now present and compare results obtained by the (wo methods described above.
We now present and compare results obtained by the two methods described above.
As we saw. A" and f happen to be the respective control parameters in these (wo methods.
As we saw, $K'$ and $f$ happen to be the respective control parameters in these two methods.
On selling these control parameters equal to 0. both these methods are reduced to the model ol Choudhliuri. Sehüsssler. Dikpati (1995. hereafter CSD model).
On setting these control parameters equal to 0, both these methods are reduced to the model of Choudhuri, Schüsssler, Dikpati (1995, hereafter CSD model).
All our caleulations are done on a 64x grid.
All our calculations are done on a $64 \times 64$ grid.
We allow the eruptions to take place alter times 7=8.8xLO?
We allow the eruptions to take place after times $\tau = 8.8 \times 10^5$
be older todav: when the Sun coudenused from the nebula in which the planets were also forming. nueht it not have preferred to do so in the eravitatioual poteutial well of a nascent super-Jupiter?
be older today; when the Sun condensed from the nebula in which the planets were also forming, might it not have preferred to do so in the gravitational potential well of a nascent super-Jupiter?
It would have taken only a few Jupiter masses of heavy material to do the trick.
It would have taken only a few Jupiter masses of heavy material to do the trick.
As I iuntimated just now. these suggestions are uulikelv to be correct.
As I intimated just now, these suggestions are unlikely to be correct.
But I mention them to emphasize how important it is to consider possibilities that lie outside the domain sampled by typical AICAIC and genetic algorithms. useful as those procedures are.
But I mention them to emphasize how important it is to consider possibilities that lie outside the domain sampled by typical MCMC and genetic algorithms, useful as those procedures are.
Finally. I ust remind vou that the road ahead is not going to be casy
Finally, I must remind you that the road ahead is not going to be easy.
Jérrouune Ballot. Voroutsov and especially Daniel Reese have given us a glimpse of some of the complexity that we are likely to have to face.
Jérrômme Ballot, Vorontsov and especially Daniel Reese have given us a glimpse of some of the complexity that we are likely to have to face.
We need to have the ορπάσι that we shall be able to extract useful information from the real aud apparent chaos that we shall encounter. a degree of optiuisiu lat was adimurablv demonstrated by Mirkus Both. who colmpited from an artificial solar meridional flow sone weuty-five times faster than he expects to be present in 16 Sun a signal that looks like zero.
We need to have the optimism that we shall be able to extract useful information from the real and apparent chaos that we shall encounter, a degree of optimism that was admirably demonstrated by Markus Roth, who computed from an artificial solar meridional flow some twenty-five times faster than he expects to be present in the Sun a signal that looks like zero.
Aud still he savs wat he hopes to detect the real flow!
And still he says that he hopes to detect the real flow!
That demonstrates 1e kind of optimusi that we need.
That demonstrates the kind of optimism that we need.
Uuless one really tries o achieve the impossible.τσ there can be little chance of actually achieving it.
Unless one really tries to achieve the impossible, there can be little chance of actually achieving it.
the A? variable by some amount, affecting the oscillitory behaviour of the complex components of R(¢).
the $\lambda^2$ variable by some amount, affecting the oscillitory behaviour of the complex components of $R(\phi)$.
In general, the optimal shift is given by the weighted mean value of X?, It is simple to show, as can be seen in the top panel of Figure 3, that this forces OS[R(9)]/Oe|o-o=0, thereby making the evaluation of the polarisation angle using Equation (4)) more accurate.
In general, the optimal shift is given by the weighted mean value of $\lambda^2$, It is simple to show, as can be seen in the top panel of Figure \ref{fig:RMSF}, , that this forces $\partial {\Im[R(\phi)]}/\partial\phi |_{\phi = 0} = 0$, thereby making the evaluation of the polarisation angle using Equation \ref{eq:p and chi}) ) more accurate.
Hence, using Equation (8)), we can now write the derotated polarisation angle as Dropping the ‘obs’ subscript hereafter and making the shift in A, we have where K is defined as before in Equation (15)).
Hence, using Equation \ref{eq:chi}) ), we can now write the derotated polarisation angle as Dropping the `obs' subscript hereafter and making the shift in $\lambda^2$, we have where $K$ is defined as before in Equation \ref{eq:K}) ).
Since a correlator samples signals discretely, we must use discrete rather than continuous Fourier transformations.
Since a correlator samples signals discretely, we must use discrete rather than continuous Fourier transformations.
To do this, consider a correlator which provides No, frequency channels over a bandpass of width B centred on ro, such that each channel has a width Av=Β/Να.
To do this, consider a correlator which provides $N_{\rm ch}$ frequency channels over a bandpass of width $B$ centred on $\nu_0$, such that each channel has a width $\Delta\nu = B/N_{\rm ch}$.
The correlator gives signal measurements for equally spaced channels in frequency, however, the parameter we wish to work with is A?.We make the following approximations for the channel centres and widths.
The correlator gives signal measurements for equally spaced channels in frequency, however, the parameter we wish to work with is $\lambda^2$.We make the following approximations for the channel centres and widths.
The central A? of channel j is given by where v; is the central frequency of the channel.
The central $\lambda^2$ of channel $j$ is given by where $\nu_j$ is the central frequency of the channel.
The width of channel j is given by If dA<1 for all channels, Equations (15)), (19)) and (20)) can be approximated by the following Fourier sums: where Ai is the central A? of channel j as approximated by Equation (21)).
The width of channel $j$ is given by If $\phi\Delta\lambda^2 \ll 1$ for all channels, Equations \ref{eq:K}) ), \ref{eq:F}) ) and \ref{eq:R}) ) can be approximated by the following Fourier sums: where $\lambda_j^2$ is the central $\lambda^2$ of channel $j$ as approximated by Equation \ref{eq:lambda2c}) ).
In the lag-domain, the correlator we consider corresponds to Ναι lag channels covering a total time At= 1/Av, with lag spacing δέ=1/B.
In the lag-domain, the correlator we consider corresponds to $N_{\rm ch}$ lag channels covering a total time $\Delta t = 1/\Delta\nu$ , with lag spacing $\delta t = 1/B$.
By choosing to perform the frequency-lag conversion with respect to the central frequency, the recorded lag spectrum will be centred on t=0.
By choosing to perform the frequency-lag conversion with respect to the central frequency, the recorded lag spectrum will be centred on $t = 0$.
This enables the recovery of Faraday depths of either sign in the Faradaydispersion function.
This enables the recovery of Faraday depths of either sign in the Faradaydispersion function.
Unlike the complex linear polarisation, P, the non-polarised Stokes I signal is a real-valued function.
Unlike the complex linear polarisation, $P$ , the non-polarised Stokes $I$ signal is a real-valued function.
Despite this, it is possible to calculate an effective Stokes J Faraday dispersion
Despite this, it is possible to calculate an effective Stokes $I$ Faraday dispersion
Almost all the analysis of the decaying light-curves of Gamma-Ray Burst (GRB) afterglows is done within the framework of external shocks driven into the circumburst medium by ultrarelativistic ejecta (Mésszárros Rees 1997) whose kinetic energy is the same in all directions'.
Almost all the analysis of the decaying light-curves of Gamma-Ray Burst (GRB) afterglows is done within the framework of external shocks driven into the circumburst medium by ultrarelativistic ejecta (Mésszárros Rees 1997) whose kinetic energy is the same in all directions.
. A non-isotropic distribution of the energy per solid angle within the outflow is a natural extension of the afterglow model.
A non-isotropic distribution of the energy per solid angle within the outflow is a natural extension of the afterglow model.
Fireballs whose kinetic energy and initial Lorentz factor fall-off with angle are power-laws have been considered for the first time by Mésszárros. Rees Wijers (1998). who studied the effect of such distributions on the afterglow light-curve decay.
Fireballs whose kinetic energy and initial Lorentz factor fall-off with angle are power-laws have been considered for the first time by Mésszárros, Rees Wijers (1998), who studied the effect of such distributions on the afterglow light-curve decay.
The faster dimming of the afterglow emission that a structured outflow can produce has been used by Dar Gou (2001) to explain the steep fall-off of the optical light-curve of the afterglow 991208.
The faster dimming of the afterglow emission that a structured outflow can produce has been used by Dai Gou (2001) to explain the steep fall-off of the optical light-curve of the afterglow 991208.
Postnov. Prokhorov and Lipunov (2001) have suggested that GRB outflows may have a universal angular structure. the observed distribution of isotropic 7-ray outputs (which has a width of 3 dex) being due to the observer location.
Postnov, Prokhorov and Lipunov (2001) have suggested that GRB outflows may have a universal angular structure, the observed distribution of isotropic $\gamma$ -ray outputs (which has a width of 3 dex) being due to the observer location.
Rossi. Lazzati Rees (2002) and Zhang Mésszárros (2002) have proposed that the light-curve breaks seen in several GRB afterglows and the narrow distributions of the GRB energy release (Frail 2001) and jet kinetic energy (Panaitescu Kumar 2002) may be due to the angular structure of fireballs.
Rossi, Lazzati Rees (2002) and Zhang Mésszárros (2002) have proposed that the light-curve breaks seen in several GRB afterglows and the narrow distributions of the GRB energy release (Frail 2001) and jet kinetic energy (Panaitescu Kumar 2002) may be due to the angular structure of fireballs.
In this work. we present an analytical treatment. of the afterglow light-curves from structured fireballs refanalytical)). focusing on. axially symmetric fireballs endowed with a power-law distribution of the energy refplaw)).
In this work, we present an analytical treatment of the afterglow light-curves from structured fireballs \\ref{analytical}) ), focusing on axially symmetric fireballs endowed with a power-law distribution of the energy \\ref{plaw}) ).
In refcriteria we give criteria which can be used to assess from the afterglow properties when structure and collimation of GRB outflows is required by observations. and apply these criteria to the afterglows whose optical light-curves exhibited a break.
In \\ref{criteria} we give criteria which can be used to assess from the afterglow properties when structure and collimation of GRB outflows is required by observations, and apply these criteria to the afterglows whose optical light-curves exhibited a break.
Section presents the numerical modeling of two GRB afterglows. 990510 and 000301c. in the framework of structured jets. leading to a few important conclusions about the role of structure in these two cases.
Section presents the numerical modeling of two GRB afterglows, 990510 and 000301c, in the framework of structured jets, leading to a few important conclusions about the role of structure in these two cases.
In this section. we calculate the evolution of the afterglow light-curve index ο(1). defined as the logarithmic derivativewith respect to the observer time £ of the received flux Ε, Fxf ")fora fireball endowed with structure.
In this section we calculate the evolution of the afterglow light-curve index $\alpha (t)$, defined as the logarithmic derivativewith respect to the observer time $t$ of the received flux $F_\nu$ $F_\nu \propto t^{-\alpha}$ ) for a fireball endowed with structure.
Our aim is to obtain the dependence of of α(1) and its asymptotic values at early and late times on the fireball's angular structure. the sharpness of the afterglow light-curve break that the structure can produce.
Our aim is to obtain the dependence of of $\alpha (t)$ and its asymptotic values at early and late times on the fireball's angular structure, the sharpness of the afterglow light-curve break that the structure can produce.
The calculation of the afterglow light-curve requires the following ingredients: dynamics of the fireball. spectrum of its emission. and integration over the equal photon-arrival-time surface.
The calculation of the afterglow light-curve requires the following ingredients: dynamics of the fireball, spectrum of its emission, and integration over the equal photon-arrival-time surface.
For analytical calculations. we shall ignore the tangential motions and mixing in the fireball and consider a simplified scenario where a fluid patch travels as if it were part of a uniform fireball.
For analytical calculations, we shall ignore the tangential motions and mixing in the fireball and consider a simplified scenario where a fluid patch travels as if it were part of a uniform fireball.
For simplicity. in our analytical caleulations of the light-curve index we consider only adiabatic GRB remnants. a case encountered when electrons acquire a negligible fraction of the post-shock energy or if they cool on a timescale longer than the dynamical time.
For simplicity, in our analytical calculations of the light-curve index we consider only adiabatic GRB remnants, a case encountered when electrons acquire a negligible fraction of the post-shock energy or if they cool on a timescale longer than the dynamical time.
Radiative losses could be important during the early afterglow. in which case the results presented in this section should be re-derived taking into account their effect on the evolution of the fireball Lorentz factor ~ (1))].
Radiative losses could be important during the early afterglow, in which case the results presented in this section should be re-derived taking into account their effect on the evolution of the fireball Lorentz factor $\gamma$ \ref{gm}) )].
This effect is included in the numerical calculations presented in and ref030]..
This effect is included in the numerical calculations presented in \\ref{0510} and \\ref{0301}. .
For an adiabatic fireball. energy conservation gives that the Lorentz factor of a fluid patch moving in the direction (à. c) (6
For an adiabatic fireball, energy conservation gives that the Lorentz factor of a fluid patch moving in the direction $\delta, \psi$ ) $\delta$
The direction for b proposed by Opheretal(2009).. (A2157.37337) did not emerge from our unbiasedanalvsis of the £ data as one of the directions for closer examination.
The direction for $\hat{b}$ proposed by \cite{Opher09}, $(\lambda \simeq 15^{\circ},\beta \simeq 33^{\circ})$ did not emerge from our unbiasedanalysis of the $\xi$ data as one of the directions for closer examination.
The values of R(a=0.3) and ία=0.5) For a direction in the range of possible directions chosen bx Opheretal(2009).. (A=11.0.53341) are given in Table 2.
The values of $R(a=0.3)$ and $R(a=0.5)$ for a direction in the range of possible directions chosen by \cite{Opher09}, $(\lambda=17.7^{\circ},\beta=34.1^{\circ})$ are given in Table 2.
The value ol Rea=0.3)1.462:0.33 is not statistically significant. but is not inconsistent. with an anisotropy of the velocity fIuctuations <0.4—0.5.
The value of $R(a=0.3) = 1.46 \pm 0.33$ is not statistically significant, but is not inconsistent with an anisotropy of the velocity fluctuations $\leq 0.4 - 0.5$.
Interestingly. (he anisotropy for (e=0.5) is larger. but not in agreement with the results for the smaller binning interval.
Interestingly, the anisotropy for $R(a=0.5)$ is larger, but not in agreement with the results for the smaller binning interval.
As noted in Section 2. in the solar corona and solar wind there is not a single temperature which is valid for all ions. as written in Equation (1).
As noted in Section 2, in the solar corona and solar wind there is not a single temperature which is valid for all ions, as written in Equation (1).
Indeed. the temperature increases Lor ions with larger Larmor radii.
Indeed, the temperature increases for ions with larger Larmor radii.
The reason for adopting Equation (1) in application to the Local Clouds is the simple fact that it vields entirely satisfactory. [its to the spectral line width data for lines from as many as 5 different atoms and ions (see Figure 1 of (2004))).
The reason for adopting Equation (1) in application to the Local Clouds is the simple fact that it yields entirely satisfactory fits to the spectral line width data for lines from as many as 8 different atoms and ions (see Figure 1 of \cite{Redfield04}) ).
The analysis of this section will be in the nature of establishing an upper limit to the ion mass dependence of the ion temperature in the Local Clouds.
The analysis of this section will be in the nature of establishing an upper limit to the ion mass dependence of the ion temperature in the Local Clouds.
If evelotvon resonant. heating is occurring. one would expect a modification of Equation (1).
If cyclotron resonant heating is occurring, one would expect a modification of Equation (1).
A plausible candidate form is where Zi and mi are (he temperature and mass of the lightest atom or ion analysed. and im is the mass of (he more massive atom or ion.
A plausible candidate form is where $T_0$ and $m_0$ are the temperature and mass of the lightest atom or ion analysed, and $m$ is the mass of the more massive atom or ion.