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The resu tis R=21.950.01. so that /?—N'=2.663- 0.10. | The result is $R = 21.95 \pm 0.04$, so that $R-K' = 2.66 \pm 0.10$ . |
These values of 2—A" are plotted as a [uuction of time in figure 2.. | These values of $R-K'$ are plotted as a function of time in figure \ref{rkfig}. |
If we assuue a sinele [I—A! color throughout the alterglow. the best fit is 1—A!=2.79. | If we assume a single $R-K'$ color throughout the afterglow, the best fit is $R-K' = 2.79$. |
With 42/d.o.£=2.87 for 5 degrees of [reedoim. this fit is not especially good. | With $\chi^2 / \dof = 2.87$ for $5$ degrees of freedom, this fit is not especially good. |
We cousider three possible explanations for the observed variation of 2—N' with time. | We consider three possible explanations for the observed variation of $R-K'$ with time. |
First. it is possible that the color variations are real. | First, it is possible that the color variations are real. |
Second. as suggestedMD by Masetti et al (2000). it may be that there are no true color variations. but that the afterglow exhibits achromatic f[Iuctuations on time scales short compared to the interval betweenobservatious. | Second, as suggested by Masetti et al (2000), it may be that there are no true color variations, but that the afterglow exhibits achromatic fluctuations on time scales short compared to the interval betweenobservations. |
If so. the fIuctuatious need to | If so, the fluctuations need to |
From the study of X-ray transient svstemis. it ds. possible o determine the masses of stellar-mass black-holes. by observing the cool secondary star during quiescence (see e.g. Charles. 1999). | From the study of X-ray transient systems, it is possible to determine the masses of stellar-mass black-holes, by observing the cool secondary star during quiescence (see e.g. Charles, 1999). |
In this paper we present new observations of he late-type secondary star in the soft X-ray transient GRO J0422|32 (Nova Per 1992/ V5IS Per). | In this paper we present new observations of the late-type secondary star in the soft X-ray transient GRO J0422+32 (Nova Per 1992/ V518 Per). |
Since the discovery of 0422]32 on August 5th. 1992. whilst in outburst. by he "Compton. Gamma Rav Observatory’ (Paciesas ct al. 1992). there have been three subsequent mini! outbursts: December 1992 (Llarmon. Fishman DPaciesas. 1992). August 1993 (Filipenko Matheson. 1993) and. December-January 1993/4 (Zhao et al. 1993). | Since the discovery of J0422+32 on August 5th 1992, whilst in outburst, by the `Compton Gamma Ray Observatory' (Paciesas et al, 1992), there have been three subsequent `mini' outbursts: December 1992 (Harmon, Fishman Paciesas, 1992), August 1993 (Filipenko Matheson, 1993) and December-January 1993/4 (Zhao et al, 1993). |
The svstem has been observed at. L=20.03 (Orosz&Bailvn1995). ancl L=20.22 (Casaresοἱal1995).. but never. as we show in this paper. in absolute quicscence. | The system has been observed at I=20.03 \cite{Oros95} and I=20.22 \cite{Casa95}, but never, as we show in this paper, in absolute quiescence. |
Therefore. observations in the optical have been dominated bv the [lux emitted. from the accretion disc around. the compact object. making observations of the Al-star difficult. | Therefore, observations in the optical have been dominated by the flux emitted from the accretion disc around the compact object, making observations of the M-star difficult. |
Lere we present results showing that JO422|32 was fainter still in December LOOT. thus observations of the secondary were more accessible. due to less contamination from the disc. | Here we present results showing that J0422+32 was fainter still in December 1997, thus observations of the secondary were more accessible, due to less contamination from the disc. |
Previously. Beckman ct al (1997) have determined a minimum mass of the compact object (black hole) of 15M.. from an IL band Dight-curve. | Previously, Beekman et al (1997) have determined a minimum mass of the compact object (black hole) of $_\odot$, from an I band light-curve. |
From calculations by line. Ixolb Burderi (1996) based on the ‘cise instability model’. where the mass transfer rate must be below a critical level in order for low mass X-ray binary svstenis to. become transient. à minimum. mass of the compact object can be calculated. | From calculations by King, Kolb Burderi (1996) based on the `disc instability model', where the mass transfer rate must be below a critical level in order for low mass X-ray binary systems to become transient, a minimum mass of the compact object can be calculated. |
H£ their assumptions are correct. Beckman et al (1997) calculated: a minimum. mass of the compact object of 28M... | If their assumptions are correct, Beekman et al (1997) calculated a minimum mass of the compact object of $_\odot$. |
This mass is impossibly large for stellar evolution models. | This mass is impossibly large for stellar evolution models. |
From studying the ellipsoidal modulation alone. we constrain the black-hole to have a much lower minimum mass. which is consistent with the evolution of massive stars that form black holes. | From studying the ellipsoidal modulation alone, we constrain the black-hole to have a much lower minimum mass, which is consistent with the evolution of massive stars that form black holes. |
components in component E was largely an exercise in intuition al the lower frequencies. | components in component E was largely an exercise in intuition at the lower frequencies. |
Therefore. we have fitted (he strongest eniission component in E. calling it E1. but otherwise have not split component E into multiple subcomponents. | Therefore, we have fitted the strongest emission component in E, calling it E1, but otherwise have not split component E into multiple subcomponents. |
El has a steep radio spectrum. with a (wo-point spectral index between 8 and 15 Gllz of ayy,=—1.2Y+0.16 (defining 5S,xv“). | E1 has a steep radio spectrum, with a two-point spectral index between 8 and 15 GHz of $\alpha_{8,15}=-1.27\pm 0.16$ (defining $S_\nu\propto \nu^{+\alpha}$ ). |
In contrast. D3b. the most powerful component of D3. has ayy,=—0.11£0.23. | In contrast, D3b, the most powerful component of D3, has $\alpha_{8,15}=-0.11\pm 0.23$. |
Component D3c has ayy,=—0.48d:0.30. consistent with either a flat or a steep spectrum. while D3a is not clearly separable from D3b at 8 GlIz. and therefore cannot be fitted. | Component D3c has $\alpha_{8,15}=-0.48\pm 0.30$, consistent with either a flat or a steep spectrum, while D3a is not clearly separable from D3b at 8 GHz, and therefore cannot be fitted. |
The second method of analysis was to produce an image of (he spectral index between 5 and 15 GlIz. | The second method of analysis was to produce an image of the spectral index between 8 and 15 GHz. |
This was done by aligning the 8 and 15 GIIz images on the strongest peak of D3. then computing the spectral index at each individual pixel anc generating a new image: (his image was blanked αἱ all points where either of the input images had a [lux clensity less than five (nes (he noise level. | This was done by aligning the 8 and 15 GHz images on the strongest peak of D3, then computing the spectral index at each individual pixel and generating a new spectral-index image; this image was blanked at all points where either of the input images had a flux density less than five times the noise level. |
The spectral indices then. were evaluated along major axis slices through components D3 and E. Figure 4. shows the spectral index along the major axis slice of Component E. There clearly is no flat- or inverted-spectrum component here. so it is unlikely that Component E contains the nucleus of NGC 4151. | The spectral indices then were evaluated along major axis slices through components D3 and E. Figure \ref{fig:sliceE} shows the spectral index along the major axis slice of Component E. There clearly is no flat- or inverted-spectrum component here, so it is unlikely that Component E contains the nucleus of NGC 4151. |
Figure 5 shows the spectral index (top panel) and the 15 GllIz total intensity (bottom panel) along (he major axis (Gin the direction Irom West to East) of Component. D3: the locations of the three sub-components in D3 are indicated in the figure. | Figure \ref{fig:sliceD} shows the spectral index (top panel) and the 15 GHz total intensity (bottom panel) along the major axis (in the direction from West to East) of Component D3; the locations of the three sub-components in D3 are indicated in the figure. |
It appears that (here is [Iat-spectirum emission near all three radio components. | It appears that there is flat-spectrum emission near all three radio components. |
We note. however (hat the major axis slice does not intersect the peak of D3c. which is considerably north of the major axis defined bx the (wo stronger components. | We note, however that the major axis slice does not intersect the peak of D3c, which is considerably north of the major axis defined by the two stronger components. |
The apparent [at spectrum near D3c appears to be an artifact of the fact that the slice actually intersects (he edge of the radio component. where (he SNI is quite low and the spectrum probably is affected by component registration errors that are a small fraction of a beam in size. | The apparent flat spectrum near D3c appears to be an artifact of the fact that the slice actually intersects the edge of the radio component, where the SNR is quite low and the spectrum probably is affected by component registration errors that are a small fraction of a beam in size. |
We (hen nist assess the spectra of D3a and D3b Irom the spectral slices. | We then must assess the spectra of D3a and D3b from the spectral slices. |
The image was produced by aligning the 8 GlIz and 15 Gllz peaks. in D3b. to an accuracy ol about 1 microaresecond. | The spectral-index image was produced by aligning the 8 GHz and 15 GHz peaks, in D3b, to an accuracy of about 1 microarcsecond. |
Assuming that this alignment is correct. the spectral index at the exact location of D3b can be taken [rom the spectral-index slice. and is found to be Axis=£0.16+0.23. consistent with the value of —0.11£0.23 that was found [rom the eaussian fitting, | Assuming that this alignment is correct, the spectral index at the exact location of D3b can be taken from the spectral-index slice, and is found to be $\alpha_{8,15}=+0.16\pm 0.23$, consistent with the value of $-0.11\pm 0.23$ that was found from the gaussian fitting. |
At D3a. we [ind a similar spectral index of n&445=+0.20+0.24. | At D3a, we find a similar spectral index of $\alpha_{8,15}=+0.20\pm 0.24$. |
This cannot be compared wilh the value [rom a gaussian [it because (he 8-Gllz image has no distinct peak at the location of D3a. | This cannot be compared with the value from a gaussian fit because the 8-GHz image has no distinct peak at the location of D3a. |
We note that taking single values [rom a spectral index slice is a perilous endeavor. because this process amounts to assuming both infinite resolution and perfect alignment of the two input images. | We note that taking single values from a spectral index slice is a perilous endeavor, because this process amounts to assuming both infinite resolution and perfect alignment of the two input images. |
The beam size along (he major axis of D3 is 0.5 mas. as plotted in Figure 5.. and the absolute alienment of (he 8 GlIIz aud 15 Gllz images probably is known to no better (han 0.10.2 mas. | The beam size along the major axis of D3 is 0.5 mas, as plotted in Figure \ref{fig:sliceD}, and the absolute alignment of the 8 GHz and 15 GHz images probably is known to no better than 0.1–0.2 mas. |
Therefore. the best conclusions that can | Therefore, the best conclusions that can |
while stars (hat clissipate their disks late. end up as slow rotators Herbst&Mundt| 2005). | while stars that dissipate their disks late, end up as slow rotators \citep{bouvier94,rebull04,HM05}. |
. Most rotational evolution models explore only (he case of a single strs interaction with its disk. | Most rotational evolution models explore only the case of a single star's interaction with its disk. |
Comparing the (v;—IN) index with the rotation period of single stars enables us lo compare our results to the theoretical frameworks which use single star + disk scenarios. | Comparing the $(K_{s}-N)$ index with the rotation period of single stars enables us to compare our results to the theoretical frameworks which use single star + disk scenarios. |
Furthermore. there is some evidence that the presence of a companion affects the evolution of a disk around a PAIS object (Meyeretal.1997b:Ghez1994). | Furthermore, there is some evidence that the presence of a companion affects the evolution of a disk around a PMS object \citep{meyer97b,ghez94}. |
. Jensenοἱal.(1996) and Jensen&Mathieu(1997) present convincing evidence that disks [rom 50-100AU. are allected by the presence of binary companions over this range of separationis. | \citet{jensen96} and \citet{JM97} present convincing evidence that disks from 50-100AU are affected by the presence of binary companions over this range of separations. |
In Figure 4 we combine data points from Figure 1 with (A,—V) and period data available for unresolved. binaries (listed in Table 4). | In Figure 4 we combine data points from Figure 1 with $(K_{s}-N)$ and period data available for unresolved binaries (listed in Table 4). |
Performing a linear correlation test similar to the one described in section 3. for (A,—.N) vs. rotation period. resulted in a correlation coefficient of 0.35. for 38 data points (not including upper-limits). | Performing a linear correlation test similar to the one described in section 3, for $(K_{s}-N)$ vs. rotation period, resulted in a correlation coefficient of 0.35, for 38 data points (not including upper-limits). |
This means the probability Chat linear correlation is random is <0.05. suggesting that the presence of unresolved. binaries in a sample of voung stars could anv expected correlation. | This means the probability that linear correlation is random is $< 0.05$, suggesting that the presence of unresolved binaries in a sample of young stars could any expected correlation. |
We also conducted a two sided IxX-5 test on the objects in Figure d. separating them into disked and clisk-less stars. as described in section 3. | We also conducted a two sided K-S test on the objects in Figure 4, separating them into disked and disk-less stars, as described in section 3. |
We derived a D-statistic of 0.45. which indicates that the probability the distributions are similar is 7.0%. | We derived a D-statistic of 0.45, which indicates that the probability the distributions are similar is $7.0\%$. |
This is a 1.06 result. | This is a $1.0\sigma$ result. |
So we cannot confidently assert that the period distributions were drawn [rom different parent populations. | So we cannot confidently assert that the period distributions were drawn from different parent populations. |
Ghezetal.(1993) estimate that 2/3 of the stars in the Tan-Aur region are multiple svstenms. | \citet{ghez93} estimate that 2/3 of the stars in the Tau-Aur region are multiple systems. |
Yet. the linear correlation test only includes 13 binary svstems compared to the 25 sinele svstenms (excluding upperlimits). | Yet, the linear correlation test only includes 13 binary systems compared to the 25 single systems (excluding upperlimits). |
This is because (here is a lack of photometric period data for approx 85'& of the stars surveved to be multiple svstems. compared to 50% for those survved to be single svstems (Ghezetal.1993:LeinertSimon1995:Bouvieral.1993.1995:Osterlohet 1996). | This is because there is a lack of photometric period data for approx $\%$ of the stars surveyed to be multiple systems, compared to $\%$ for those survyed to be single systems \citep{ghez93,leinert93,simon95,bouvier93,bouvier95,osterloh96}. |
. This is probably due to the difficulty in interpreting data from binary svstems. | This is probably due to the difficulty in interpreting data from binary systems. |
Most. photometric period survevs that include binaries do nol resolve companions and hence it is hard (ο know for certain a) which component's period was determined (if the stars are of comparable brightness) or b) i£ the light. fluctuations were due to orbital motions (eclipsing) or actual rotational motion (spots). | Most photometric period surveys that include binaries do not resolve companions and hence it is hard to know for certain a) which component's period was determined (if the stars are of comparable brightness) or b) if the light fluctuations were due to oribital motions (eclipsing) or actual rotational motion (spots). |
In addition. components in a close binary system are hard to resolve in the infrared. | In addition, components in a close binary system are hard to resolve in the infrared. |
The source of flux rom an unresolved. multiple svstem cannot be easily determined. | The source of N-band flux from an unresolved multiple system cannot be easily determined. |
Contributions from each stellar component. the dust disk around each object. or a circumbinary disk could add to the N-band fIux. | Contributions from each stellar component, the dust disk around each object, or a circumbinary disk \citep{ghez94} could add to the N-band flux. |
In investigating star-disk interactions in the PAIS. | In investigating star-disk interactions in the PMS, |
the same as the correction in the A-band )). | the same as the correction in the $K$ -band ). |
In summary. the svstematic errors in our I-band red clump distance to the LAIC are probably of order (photometric calibration) and (population correction). and thus comparable to the statistical error obtained. | In summary, the systematic errors in our $K$ -band red clump distance to the LMC are probably of order (photometric calibration) and (population correction), and thus comparable to the statistical error obtained. |
The A-band red clump distance to (he LAIC is in agreement will previously reported {- results (811). | The $K$ -band red clump distance to the LMC is in agreement with previously reported $I$ -band results 1). |
The only serious cliscrepancy is (ae short cdistauce result of Udalski (2000) based on OGLE II data, | The only serious discrepancy is the short distance result of Udalski (2000) based on OGLE II data. |
Udalski (2000) finds a mean dereddened red. clamp brightness of fy=17.94440.014, for 9 fields in the LMC halo. | Udalski (2000) finds a mean dereddened red clump brightness of $I_0 = 17.944 \pm 0.014_r$ for 9 fields in the LMC halo. |
Those fields are on average 27.12 from the center of the LMC. nearly perpendicular to the line-of-nocles. and on (henear sile of the inclined clisk (wan der Marel et al. | Those fields are on average $2^{\circ}.12$ from the center of the LMC, nearly perpendicular to the line-of-nodes, and on the side of the inclined disk (van der Marel et al. |
2002). | 2002). |
Correcting lor the 0.02 mag zero-point offset between OGLE II and our /WE data. and for geometric projection. the Udalski (2000) red clump brightness becomes fy=18.024 mag (at LAIC center). | Correcting for the 0.02 mag zero-point offset between OGLE II and our /WF data, and for geometric projection, the Udalski (2000) red clump brightness becomes $I_0 = 18.024$ mag (at LMC center). |
For comparison. our dereddened red clump brightness (Table 1) corrected to the LAIC center is J)=18.019 mag. | For comparison, our dereddened red clump brightness (Table 1) corrected to the LMC center is $I_0 = 18.019$ mag. |
The results ol this work. Udalski (2000). ancl Romaniello et al. ( | The results of this work, Udalski (2000), and Romaniello et al. ( |
2000) therefore all agree. which lends strong support to the accuracy of the A-band red clamp distance to the LMC. | 2000) therefore all agree, which lends strong support to the accuracy of the $K$ -band red clump distance to the LMC. |
Bretthorst(1998) describes the Bayesian theory of modelling data using a sinusoid plus white noise. | \cite{bre98} describes the Bayesian theory of modelling data using a sinusoid plus white noise. |
In this formalism, our metric can be shown to be is the squared magnitude of the discrete Fourier transform of the data. | In this formalism, our metric can be shown to be is the squared magnitude of the discrete Fourier transform of the data. |
N is the number of data points in the light curve, R(w)=Σιcos(wt;), Ι(ω)=Σιsin(wt;) and στπις is the rms scatter of the data. | $N$ is the number of data points in the light curve, $R(\omega) = \sum_{i=1}^{N}\cos(\omega t_i)$, $I(\omega) = \sum_{i=1}^{N}\sin(\omega t_i)$ and $\sigma_{\mbox{rms}}$ is the rms scatter of the data. |
We have assumed the data is mean-subtracted. | We have assumed the data is mean-subtracted. |
As discussed in B11, the choice of a threshold for periodicity detection is best made empirically. | As discussed in B11, the choice of a threshold for periodicity detection is best made empirically. |
We selected S=0.3 as an appropriate threshold between clear-cut periodic variability and non-periodic or ambiguous cases. | We selected $S = 0.3$ as an appropriate threshold between clear-cut periodic variability and non-periodic or ambiguous cases. |
This threshold is slightly lower than the value of S=0.4 used in the Monitor project (seee.g.Irwinetal.2006),, which is reasonable given the vastly superior time-sampling of the data. | This threshold is slightly lower than the value of $S=0.4$ used in the Monitor project \citep[see e.g.][]{irw06}, which is reasonable given the vastly superior time-sampling of the data. |
This threshold can be compared to a power spectrum cut, since the power spectrum is given by PS(w)=4C(w)/N where, following Kjeldsen&Bedding(1995) we have normalised the power spectra such that a sinusoidal oscillation of amplitude A gives rise to a peak of height A? in the power spectrum. | This threshold can be compared to a power spectrum cut, since the power spectrum is given by $\mbox{PS}(\omega) = 4C(\omega) / N$ where, following \cite{kje95} we have normalised the power spectra such that a sinusoidal oscillation of amplitude $A$ gives rise to a peak of height $A^2$ in the power spectrum. |
Our threshold is then equivalent to a power spectrum threshold of Sps= 0.6775. | Our threshold is then equivalent to a power spectrum threshold of $S_{\mbox{PS}} = 0.6\sigma^2_{\mbox{rms}}$ . |
By comparison Kjeldsen&Bedding(1995) give an expression for the noise level in the power spectrum of OCPS=4orms/N. | By comparison \cite{kje95} give an expression for the noise level in the power spectrum of $\sigma_{\mbox{PS}} = 4 \sigma_{\mbox{rms}} / N$. |
In our data, N>1, so our threshold is conservative when comparing to white noise. | In our data, $N \gg 1$, so our threshold is conservative when comparing to white noise. |
However, white noise is not the dominant factor defining our ability to detect periodicities. | However, white noise is not the dominant factor defining our ability to detect periodicities. |
To test the appropriateness of our periodicity threshold on data with realistic noise properties, we ran a set of simulations where we injected periodic signals into actual light curves. | To test the appropriateness of our periodicity threshold on data with realistic noise properties, we ran a set of simulations where we injected periodic signals into actual light curves. |
We randomly selected 1000 Q1 light curves with low variability (Ryar< Rvarsun) and no significant period (S« 0.25), and injected sinusoidal signals into them, with random periods uniformly distributed between 2 and 16 days,and random amplitudes ranging from 0.1 to 10 times the | We randomly selected 1000 Q1 light curves with low variability $R_{\rm var} < R_{\rm var,Sun}$ ) and no significant period $S < 0.25$ ), and injected sinusoidal signals into them, with random periods uniformly distributed between 2 and 16 days,and random amplitudes ranging from 0.1 to 10 times the |
where W(E) is the relative velocity distribution and O,; accounts for reactions of like particles. | where $\psi(E)$ is the relative velocity distribution and $\delta_{\alpha\beta}$ accounts for reactions of like particles. |
Because we will be dealing with the correction due to dynamic sereening (a ratio between the unscreened and screened reaction rates), we can ignore the density factor and focus on the reaction rate per pair ofparticles. where jis the reduced mass of the pair. and the Maxwell- distribution is used for (E). | Because we will be dealing with the correction due to dynamic screening (a ratio between the unscreened and screened reaction rates), we can ignore the density factor and focus on the reaction rate per pair ofparticles, where $\mu$ is the reduced mass of the pair, and the Maxwell--Boltzmann distribution is used for $\psi(E)$. |
The cross section o(E) can be defined as a product of three separate energy-dependent factors where b=31282,Z;A!?keV!7, with Z, and Z,; being the charges of the interacting ions and A is the reduced atomic weight. | The cross section $\sigma(E)$ can be defined as a product of three separate energy-dependent factors where $b= 31.28 Z_{\alpha}Z_{\beta}A^{1/2} \;\rm{keV}^{1/2}$, with $Z_{\alpha}$ and $Z_{\beta}$ being the charges of the interacting ions and $A$ is the reduced atomic weight. |
The exponential factor in this expression comes from the barrier penetratior probability. the inverse energy dependence comes from the quantum-mechanical interaction betwee the two particles. and SCE) contains the intrinsically nuclear parts of the probability for a nuclear reactior to occur. | The exponential factor in this expression comes from the barrier penetration probability, the inverse energy dependence comes from the quantum–mechanical interaction between the two particles, and $S(E)$ contains the intrinsically nuclear parts of the probability for a nuclear reaction to occur. |
With this substitution for c&(E). Equation 2 can be re-written as In the non-resonant reaction case. S(E) is slowly varying with £. so we can treat if as a constant So evaluated at the energy where exp(-E/kpT—b/E) is maximum. | With this substitution for $\sigma(E)$, Equation \ref{eq:lambda1} can be re-written as In the non-resonant reaction case, $S(E)$ is slowly varying with $E$, so we can treat it as a constant $S_0$ evaluated at the energy where ${\rm{exp}}(-E/k_BT -b/E^{1/2})$ is maximum. |
Then the reaction rate per pair of particles (without screening) can be computed as Salpeter(1954) developed a treatment to include the effect of static. electron. screening. on. nuclear reaction rates. | Then the reaction rate per pair of particles (without screening) can be computed as \citet{Salpeter_1954} developed a treatment to include the effect of static electron screening on nuclear reaction rates. |
Here we summarize his method which we will use in Section 2 as the inspiration for our calculation of the dynamic sereening correction. | Here we summarize his method which we will use in Section \ref{sect:method} as the inspiration for our calculation of the dynamic screening correction. |
We begin by writing the total interaction energy as a combination of the bare Coulomb potential and a contribution from the plasma: consider a case in which the classical impact parameter r. 1s very small compared with the charge cloud radius Ry and the nuclear radius 75, 1s much smaller than r,. | We begin by writing the total interaction energy as a combination of the bare Coulomb potential and a contribution from the plasma: Then consider a case in which the classical impact parameter $r_{\rm{c}}$ is very small compared with the charge cloud radius $R_D$ and the nuclear radius $r_{\rm{n}}$ is much smaller than $r_{\rm{c}}$. |
Then the barrier penetration factor for ry <¢<r.depends only on the expression For distances larger than r,. the barrier penetration factor hardly depends on the potential. | Then the barrier penetration factor for $r_{\rm{n}}<r<r_{\rm{c}}$ depends only on the expression For distances larger than $r_{\rm{c}}$, the barrier penetration factor hardly depends on the potential. |
OG") must be small for distances greater than Αρ and approach a constant value C, of the order of magnitude of ZiZose[/Rp for small r. | $U(r)$ must be small for distances greater than $R_D$ and approach a constant value $U_0$ of the order of magnitude of $Z_1Z_2e^2/R_D$ for small $r$. |
Then. where £i, i5 the relative kinetic energy for which the integrand in Equation 2. reaches à sharp maximum. | Then, where $E_{\rm{max}}$ is the relative kinetic energy for which the integrand in Equation \ref{eq:lambda1} reaches a sharp maximum. |
If this inequality ts satisfied. U(r) can be replaced by the potential at the origin Uy. | If this inequality is satisfied, $U(r)$ can be replaced by the potential at the origin $U_0$. |
By examining expression 8.. we can see that the screening potential has effectively increased. the kinetic energy by a magnitude of Up. so the cross section factors for Uia=Uca+Vo are equivalent to the unscreened factors with energy E—Uo. | By examining expression \ref{eq:Edep}, we can see that the screening potential has effectively increased the kinetic energy by a magnitude of $U_0$, so the cross section factors for $U_{\rm{total}}=U_{\rm{Coulomb}}+U_0$ are equivalent to the unscreened factors with energy $E-U_0$. |
Equation 2. can then be replaced by With the change of variables E’=E—Uy and the approximation (E+Ug)= E'. the reaction rate per pair of particles becomes the penetration factor for E! =—Ugis so small. the lower limit of the integral can be set to zero without significantly changing the value of the integral. | Equation \ref{eq:lambda1} can then be replaced by With the change of variables $E'=E-U_0$ and the approximation $(E'+U_0) \approx E'$ , the reaction rate per pair of particles becomes Because the penetration factor for $E'=-U_0$ is so small, the lower limit of the integral can be set to zero without significantly changing the value of the integral. |
We then see that | We then see that |
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