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In this respect it is not surprising that only the two nearest FR I radio galaxies have been seen in the VHE bband so far. | In this respect it is not surprising that only the two nearest FR I radio galaxies have been seen in the VHE band so far. |
Both Cen A and M87 are too weak to be detected at 100 GeV in the 1.5 yr exposure ofFermi. | Both Cen A and M87 are too weak to be detected at 100 GeV in the 1.5 yr exposure of. |
iis situated in Perseus galaxy cluster at the distance of 80 Mpc. which is a factor of 22 and 5 larger than the distances of Cen A and M 87. respectively. | is situated in Perseus galaxy cluster at the distance of 80 Mpc, which is a factor of 22 and 5 larger than the distances of Cen A and M 87, respectively. |
iis. therefore. by 1-2 orders of magnitude more luminous than that of Cen A and M87. | is, therefore, by 1-2 orders of magnitude more luminous than that of Cen A and M87. |
Besides. lis not classified as a FR [E type radio galaxy. | Besides, is not classified as a FR I type radio galaxy. |
Instead. it 15 à head-tail radio galaxy (Sijbring&deBruyn. 1998)..the type of galaxiesusually foundin galaxy clusters. | Instead, it is a head-tail radio galaxy \citep{sijbring98}, ,the type of galaxiesusually foundin galaxy clusters. |
It possesses | It possesses |
ist of x and ν coordinates for cach object. one set for each stack the object is found. on. | list of x and y coordinates for each object, one set for each stack the object is found on. |
Calculating proper motions is then simply a matter of performing a linear regression it to each object's x and v coordinates as a function of ime. | Calculating proper motions is then simply a matter of performing a linear regression fit to each object's x and y coordinates as a function of time. |
Llowever. erroneous pairing inevitably occurs between stacks. and. we therefore wish to perform some form of bad »oint rejection to reduce contamination by spurious proper motions. | However, erroneous pairing inevitably occurs between stacks, and we therefore wish to perform some form of bad point rejection to reduce contamination by spurious proper motions. |
In order to reject deviant points (and. calculate xwanpeters such as o, and " 47) an estimate of the error associated with cach measure of position is required. | In order to reject deviant points (and calculate parameters such as $\sigma
_{\mu}$ and $\chi^{2}$ ) an estimate of the error associated with each measure of position is required. |
We assume this error is simply a function of magnitude and hat it will vary from stack to stack. but. not across the survey area. | We assume this error is simply a function of magnitude and that it will vary from stack to stack, but not across the survey area. |
This error is calculated. using the deviation of an objects position on a particular stack [rom the mean »osition over the 20 stacks used. and is determined: over 10 magnitudeὃν bins. | This error is calculated using the deviation of an object's position on a particular stack from the mean position over the 20 stacks used, and is determined over 10 magnitude bins. |
A 3e iterative rejection procedure. is implemented. to reject. spurious pairings or high proper motion objects which are not reflecting the true positional errors sought. | A $3\sigma$ iterative rejection procedure is implemented to reject spurious pairings or high proper motion objects which are not reflecting the true positional errors sought. |
The calculated errors are much as one might expect: decreasing for brighter objects until factors such as saturation and blended. images makes positional measures more uncertain. | The calculated errors are much as one might expect: decreasing for brighter objects until factors such as saturation and blended images makes positional measures more uncertain. |
A straight line fit can now be applied to the x and v data for cach object. the gradient. of which is taken to be the measured. proper motion. ji. and fry respectively. | A straight line fit can now be applied to the x and y data for each object, the gradient of which is taken to be the measured proper motion, $\mu_{x}$ and $\mu_{y}$ respectively. |
An example is shown in Figure 2.. the points showing the deviation at each epoch from the average object position with error bars caleulated as above. | An example is shown in Figure \ref{pmplot}, the points showing the deviation at each epoch from the average object position with error bars calculated as above. |
Deviant points arising [rom spurious pairings often lic fu from the other data ancl will give rise to spurious high proper motion detections if not removed. | Deviant points arising from spurious pairings often lie far from the other data and will give rise to spurious high proper motion detections if not removed. |
We therefore iteratively remove points lying 3o from the fitted line. | We therefore iteratively remove points lying $3\sigma$ from the fitted line. |
Εις can occasionally lead to further problems if there are several bad. points associated. with the object. and the result. of several iterations can be a larger spurious motion detection. | This can occasionally lead to further problems if there are several bad points associated with the object, and the result of several iterations can be a larger spurious motion detection. |
This source of contamination is generally eliminated. by insisting sample objects are detected on virtually every stack. | This source of contamination is generally eliminated by insisting sample objects are detected on virtually every stack. |
“Phe validity of the positional error estimation scheme described above has been verified by confirming that scatter plots of log reduced 47 as à function of magnitude cluster around. zero for all magnitudes in all regions of the survey arena. | The validity of the positional error estimation scheme described above has been verified by confirming that scatter plots of log reduced $\chi^{2}$ as a function of magnitude cluster around zero for all magnitudes in all regions of the survey area. |
Instrumental magnitudes are calculated [or every object. detected. on. cach stack in. the. standard COSMOS/SuperCOSMOS. fashion (Beard et al. | Instrumental magnitudes are calculated for every object detected on each stack in the standard COSMOS/SuperCOSMOS fashion (Beard et al. |
1990 ancl references. therein). | 1990 and references therein). |
Brielly. an object. detection is defined by a given number of interconnected: pixels. with intensity above a given threshold (ce. | Briefly, an object detection is defined by a given number of interconnected pixels with intensity above a given threshold (eg. |
S interconnected pixels with intensity above a 2.57 sky noise threshold. for SuperCOSMOS data). | 8 interconnected pixels with intensity above a $2.5\sigma$ sky noise threshold for SuperCOSMOS data). |
An objects instrumental magnitude is then calculated. as the log of the sum of the intensity above background. across the object area. | An object's instrumental magnitude is then calculated as the log of the sum of the intensity above background across the object area. |
This. quantity varies monotonically with true magnitude. and is therefore suitable. for use in constructing calibration curves using a CCD sequence. | This quantity varies monotonically with true magnitude, and is therefore suitable for use in constructing calibration curves using a CCD sequence. |
A sequence of ~200 stars with CCD magnitudes measured. in a variety of passbands exists. in ficld 287 (αννκας et al. | A sequence of $\sim 200$ stars with CCD magnitudes measured in a variety of passbands exists in field 287 (Hawkins et al. |
1998). vielding U. D. V. IG and 1 photometry to a tvpical accuracy of 0.15 magnitudes (see Section 5.2)). | 1998), yielding U, B, V, R and I photometry to a typical accuracy of 0.15 magnitudes (see Section \ref{photom}) ). |
Significantly smaller errors are theoretically obtainable from photographic material. ancl the larger uncertainties we find appear to be caused. by systematic deviations of sequence objects from the calibration curve. | Significantly smaller errors are theoretically obtainable from photographic material, and the larger uncertainties we find appear to be caused by systematic deviations of sequence objects from the calibration curve. |
This is not a colour or field effect. and is probably caused by dillerences in detection media. | This is not a colour or field effect, and is probably caused by differences in detection media. |
The ceatalogue! resulting from the implementation of the procedure deseribed. in the previous section. consists. of astrometric and. photometric measures for. over. 200.000 objects. | The `catalogue' resulting from the implementation of the procedure described in the previous section consists of astrometric and photometric measures for over 200,000 objects. |
Criteria for. inclusion. in this preliminary sample is merely detection in both By ancl 1t passbands. (since these are required. for construction of the reduced. proper motion clagram (RPALD)) and a measure of proper motion in both these passbancs. | Criteria for inclusion in this preliminary sample is merely detection in both $\rm B_{J}$ and R passbands (since these are required for construction of the reduced proper motion diagram (RPMD)) and a measure of proper motion in both these passbands. |
Lt is from these objects that an uncontaminated proper motion sample is to be drawn: and we require well defined universal survey limits so that space densities can be calculated from the final survey. sample. | It is from these objects that an uncontaminated proper motion sample is to be drawn; and we require well defined universal survey limits so that space densities can be calculated from the final survey sample. |
Number count plots from this survey cata increase linearly with increasing magnitude. as shown for the It data in Figure 3.. before dropping precipitously. | Number count plots from this survey data increase linearly with increasing magnitude, as shown for the R data in Figure \ref{Rhist}, before dropping precipitously. |
Εις eut-olf is attributed to the survey detection. limit. and the position of the turnover is used to determine photometric survey limits. | This cut-off is attributed to the survey detection limit, and the position of the turnover is used to determine photometric survey limits. |
The limits used are 21.2 in It and 22.5 in D. The proper motion distribution for all objects in. our survey area detected on at least 15 stacks in both D and 1H is shown in Figure 4.. | The limits used are 21.2 in R and 22.5 in B. The proper motion distribution for all objects in our survey area detected on at least 15 stacks in both B and R is shown in Figure \ref{pmhist}. |
Low proper motions are generally an artifact of measuring machine error. thus the distribution indicates a typical error in measured. proper motions of I0mas/vr. Our criteria for choosing a survey proper motion limit | Low proper motions are generally an artifact of measuring machine error, thus the distribution indicates a typical error in measured proper motions of $ \rm \sim10mas/yr$ Our criteria for choosing a survey proper motion limit |
does not exceed ~30 for the scattering inside the light cylinder. | does not exceed $\sim 30^\circ$ for the scattering inside the light cylinder. |
We have considered the induced Compton scattering by the particles of the ultrarelativistic electron-positron plasma in the presence of a superstrong magnetic field. | We have considered the induced Compton scattering by the particles of the ultrarelativistic electron-positron plasma in the presence of a superstrong magnetic field. |
In. particular. we have examined the scattering. of pulsar. radio beam into background. which takes place in the open field. line tube of a pulsar. | In particular, we have examined the scattering of pulsar radio beam into background, which takes place in the open field line tube of a pulsar. |
Le has been demonstrated that the photons are predominantly scattered. approximately along the ambient magnetic field. | It has been demonstrated that the photons are predominantly scattered approximately along the ambient magnetic field. |
This contrasts with the non-magnetic scattering. in which case the scattered: photons concentrate in the backward. direction. | This contrasts with the non-magnetic scattering, in which case the scattered photons concentrate in the backward direction. |
This dillerence. is solely determined by a specific role of the superstrong magnetic field in the scattering process and does not depen on a detailed form of the particle distribution function. | This difference is solely determined by a specific role of the superstrong magnetic field in the scattering process and does not depend on a detailed form of the particle distribution function. |
InclueecL scattering in a superstrong magnetic Lick transfers the photons from lower to higher. frequencies. py,~d757οon10e,. and if the process is ellicient. the scattered Component may. become as strong as the origina racio. bean. Z,,Gn)~£5,iGh). | Induced scattering in a superstrong magnetic field transfers the photons from lower to higher frequencies, $\nu_b\sim\nu_a\theta^2\gamma^2\sim n\cdot 10\nu_a$, and if the process is efficient, the scattered component may become as strong as the original radio beam, $I_{\nu_b}(\nu_b)\sim
I_{\nu_a}^{(0)}(\nu_a)$. |
As the beam has a decreasing. spectrum. Ρα...τοeoqn(7). the intensity of the scatterec component may dominate the original beam intensity at the same frequency. d£. | As the beam has a decreasing spectrum, $I_{\nu_a}^{(0)}(\nu_a)\gg I_{\nu_a}^{(0)}(\nu_b)$, the intensity of the scattered component may dominate the original beam intensity at the same frequency $\nu_b$. |
For steep enough original spectra. of pulsar radiation. ac2. the induced. scattering in a superstrong magnetic field is most elficient at. distances roughly comparable to the radius of evelotron resonance. | For steep enough original spectra of pulsar radiation, $\alpha>2$, the induced scattering in a superstrong magnetic field is most efficient at distances roughly comparable to the radius of cyclotron resonance. |
Because of rotational aberration. the scattered component appears in the pulse profile as a precursor to the main pulse. | Because of rotational aberration, the scattered component appears in the pulse profile as a precursor to the main pulse. |
This elfect provi"sS the main pulse-precursor separationsin longitude rsinc/2ri. which may run up to 30°. | This effect provides the main pulse-precursor separationsin longitude $\Delta\lambda\sim r\sin\zeta/2r_L$ , which may run up to $\sim 30^\circ$. |
Since the length of the scattering region is larger than the height. of the emission region. the intrinsic radiuseto-frequencey. mapping of the radio emission is smeared. | Since the length of the scattering region is larger than the height of the emission region, the intrinsic radius-to-frequency mapping of the radio emission is smeared. |
The cllective height of the scattering region is an extremely weak function of the wave [requeney. so that the main pulse-precursor. separation is practically independent of frequency. just as is observed. | The effective height of the scattering region is an extremely weak function of the wave frequency, so that the main pulse-precursor separation is practically independent of frequency, just as is observed. |
Since the induced scattering in. the superstrong magnetic Ποιά holds only between the ordinary waves. the scattered. component should have complete linear polarization. | Since the induced scattering in the superstrong magnetic field holds only between the ordinary waves, the scattered component should have complete linear polarization. |
This is indeed the main distinctive feature of the observed. precursors. | This is indeed the main distinctive feature of the observed precursors. |
Note that in general &«5][/Ry 6]. ic. in the main pulse and. precursor the position angles of linear. polarization should somewhat cdiller. | Note that in general $[{\bmath k}\times{\bmath b}]\not\parallel[{\bmath
k_1}\times{\bmath b}]$ , i.e. in the main pulse and precursor the position angles of linear polarization should somewhat differ. |
Such a dillerence can be noticed. e... in PSR. DB1822-09 (Fowleretal. 1981). | Such a difference can be noticed, e.g., in PSR B1822-09 \citep{f81}. |
.. Besides that. if the main pulse is dominated by the extraordinary rather than ordinary. polarization. the position angle of the precursor should additionally diller by 90. as is the case in the Vela pulsar. (Ixrishnamohan&Downs 1983). | Besides that, if the main pulse is dominated by the extraordinary rather than ordinary polarization, the position angle of the precursor should additionally differ by $90^\circ$, as is the case in the Vela pulsar \citep{kd83}. |
. As first noted by Fowlerctal.CI981).. the precursor components are met in pulsars with relatively large surface magnetic field. | As first noted by \citet{f81}, the precursor components are met in pulsars with relatively large surface magnetic field. |
Firstly. large D, are necessary for the regime ol superstrong magnetic field to hold well above the emission region. | Firstly, large $B_\star$ are necessary for the regime of superstrong magnetic field to hold well above the emission region. |
Secondly. the scattering ellicieney is proportional to B,. | Secondly, the scattering efficiency is proportional to $B_\star$. |
Short periods ancl large radio luminosities also favour significant scattering. | Short periods and large radio luminosities also favour significant scattering. |
The pulse-to-pulse variations of the incident intensity ancl of the physical parameters in the scattering region may result in strong Uuctuations of the precursor emission. | The pulse-to-pulse variations of the incident intensity and of the physical parameters in the scattering region may result in strong fluctuations of the precursor emission. |
The former variations imply the main pulse-precursor connection. which may have civersiform observational manifestations. | The former variations imply the main pulse-precursor connection, which may have diversiform observational manifestations. |
Lor example. PSR J1826-6700 shows occasional main pulse nullings accompanied by the strong precursor emission. (Wangetal.2007). | For example, PSR J1326-6700 shows occasional main pulse nullings accompanied by the strong precursor emission \citep{w07}. |
This can he interpreted as a consequence of extremely strong scattering. (Lp,iU!ἐν)exp(U)Z91. when the main pulse intensity is almost completely. transferred to the precursor. Z,,—70. laii"Note . | This can be interpreted as a consequence of extremely strong scattering, $(I_{\nu_b}^{(0)}/I_{\nu_a}^{(0)})\exp (\Gamma)\gg 1$, when the main pulse intensity is almost completely transferred to the precursor, $I_{\nu_a}\to 0$, $I_{\nu_b}\to I_{\nu_a}^{(0)}$. |
that this. may happen only ifep the originalM intensity is mainly in the ordinary modo. which is subject to the scattering. | Note that this may happen only if the original intensity is mainly in the ordinary mode, which is subject to the scattering. |
In case of a moderately strong scattering. VIPiii1)exp(E)H~I. the main. pulse intensity. Z5,iU. is almost unchanged. whereas the precursor grows exponentially with I5 | In case of a moderately strong scattering, $(I_{\nu_b}^{(0)}/I_{\nu_a}^{(0)})\exp (\Gamma)\sim 1$, the main pulse intensity $I_{\nu_a}^{(0)}$ is almost unchanged, whereas the precursor grows exponentially with $I_{\nu_a}^{(0)}$. |
Phovefore even weak lucetuations of the latter quantity mav alleet the scattered. component. dramatically. | Therefore even weak fluctuations of the latter quantity may affect the scattered component dramatically. |
La some pulsars the precursors are indeed met only in strong pulses (Llankins&Cordes1981:Ciletal.1994:Weltevredect2006).. and one can expect that the transient. precursors are much more abundant in the pulsar population and are vet to be studied observationally. | In some pulsars the precursors are indeed met only in strong pulses \citep{hc81,g94,welt06}, and one can expect that the transient precursors are much more abundant in the pulsar population and are yet to be studied observationally. |
The precursor component can Uuetuate not only in intensity. but. also in pulse longitude. | The precursor component can fluctuate not only in intensity but also in pulse longitude. |
In the Vela pulsar. stronger precursors exhibit larger separations [rom the main pulse. which is thought to result. from the fluctuations of the physical. parameters in the scattering region. | In the Vela pulsar, stronger precursors exhibit larger separations from the main pulse, which is thought to result from the fluctuations of the physical parameters in the scattering region. |
Larger separations imply Larger scattering heights. AAxor. in which case the angle of incidence of the photons is also larger. @x r. and at a fixed. [requeney. v, the precursor is formed by the photonscoming from. lower frequencies We=anf0ye? which are more numerous ancl stimulate stronger scattering. | Larger separations imply larger scattering heights, $\Delta\lambda\propto r$ , in which case the angle of incidence of the photons is also larger, $\theta\propto r$ , and at a fixed frequency $\nu_b$ the precursor is formed by the photonscoming from lower frequencies $\nu_a=\nu_b/\theta^2(r)\gamma^2$ , which are more numerous and stimulate stronger scattering. |
continu exposures were used to subtract the coufinmmun contribution to the oenüsson line nuaees. | continuum exposures were used to subtract the continuum contribution to the emission line images. |
Iu addition. we have long-slit. high-cispersion echelle spectra of the SNR caudidates from the CTIO bu telescope. | In addition, we have long-slit, high-dispersion echelle spectra of the SNR candidates from the CTIO 4m telescope. |
The data have a tan pixel size that corresponds to 0.082 ((3.65 1)) along the dispersion axis aud ~0726 along the skv. | The data have a $\mu$ m pixel size that corresponds to 0.082 (3.65 ) along the dispersion axis and $\sim 0\farcs 26$ along the sky. |
The spatial coverage alone the slit is roughly 3". limited by the optics of the calcera. | The spatial coverage along the slit is roughly $3'$, limited by the optics of the camera. |
Details of the reduction of these data can be found in Sinithetal.(2011.iupreparation). | Details of the reduction of these data can be found in \citet{inprep}. |
. The echelle profile of the liue is broadened along the dispersion axis both by the inherent iustrunental profile (—Ll 13) aud by thermal Doppler broadening (~IS at N-rav teniperatures). | The echelle profile of the line is broadened along the dispersion axis both by the inherent instrumental profile $\sim14$ ) and by thermal Doppler broadening $\sim18$ at X-ray temperatures). |
To assign au expansion velocity to cach remmaut. we extract profiles aloug the dispersion axis where he enuüsson shows the greatest dispersion aud neasure the peaks of the profile. | To assign an expansion velocity to each remnant, we extract profiles along the dispersion axis where the emission shows the greatest dispersion and measure the peaks of the profile. |
Using the fux-calibrated AICELS nuages. we ueasured an ssurtace brightness aud the average radius AP aud hickuess AR of the shell. | Using the flux-calibrated MCELS images, we measured an surface brightness and the average radius $R$ and thickness $\Delta R$ of the shell. |
For a uniform spherical shell. the ereatest line of sight through the shell is L=2\/R?(R AR). | For a uniform spherical shell, the greatest line of sight through the shell is $\mathcal{L}=2\sqrt{R^{2}-(R-\Delta R)^{2}}$ . |
The measured surface xiehtuess then duplics au enuüssion nmieasure EM=u2£5«1075OSB where SD is the surface briehtuess i ces units aud arcsecouds. | The measured surface brightness then implies an emission measure $\mathrm{EM}\equiv n^{2}_{e}\mathcal{L}=5\times10^{17}\times \mathrm{SB}$ where SB is the surface brightness in cgs units and arcseconds. |
The total mass of the warm. ionized shell is AL=L2vYnangVaar where Vac is the volhune of the shell aud assuuiug singly ionized elim. | The total mass of the warm ionized shell is $M=1.27n_{{e}}m_{{p}}\Vshell$, where $\Vshell$ is the volume of the shell and assuming singly ionized helium. |
Expausonu velocities (a4, have been determined from echelle spectra of the ecnissiou-Iue. aud thus we may determine kinetic euergies for the warin shells by A=2. | Expansion velocities $v_\mathrm{exp}$ have been determined from echelle spectra of the emission-line, and thus we may determine kinetic energies for the warm shells by $K=Mv^{2}_{\mathrm{exp}}/2$. |
It we assuune that T=101 K. we can also Mey,calculate the pressure in the shell as Pop=2».kT2.16«10 στης. | If we assume that $T=10^{4}$ K, we can also calculate the pressure in the shell as $P_{\mathrm{shell}}=2n_{{e}}kT=2.76\times 10^{-12}n_{{e}}$ . |
Tn calculating the volume of the shell. we assuned an cllipsoidal ecometry. mneasudues two axes from the projected face of the shell aud takine the third line-ofsielit axis to be the average of these two. | In calculating the volume of the shell, we assumed an ellipsoidal geometry, measuring two axes from the projected face of the shell and taking the third line-of-sight axis to be the average of these two. |
We then took the mucertaiutv in the radius along this third axis to be the deviation of the first two axes from their iiem. | We then took the uncertainty in the radius along this third axis to be the deviation of the first two axes from their mean. |
We also considered the iustruuieutal aud Doppler broadening of our echelle spectra as an additional source of error. | We also considered the instrumental and Doppler broadening of our echelle spectra as an additional source of error. |
Couvolving the two profiles. we found an uncertainty of 11L. | Convolving the two profiles, we found an uncertainty of 11. |
Calculated quantities are listed in Table 2.. | Calculated quantities are listed in Table \ref{snrtable}. |
The age of the SNR may be estimated using the analytic expressions of the Sedov-Tavlor solution for blast wave expansion. iu which the shell radius is given as a fiction of time by rt)=LATERps where E ds the explosion euergv and p is the ambicut density. | The age of the SNR may be estimated using the analytic expressions of the Sedov-Taylor solution for blast wave expansion, in which the shell radius is given as a function of time by $ r(t)=1.17(Et^2/\rho)^{1/5}$, where $E$ is the explosion energy and $\rho$ is the ambient density. |
TaXiug 1ο time derivative aud applying the BRaukiuc-IIugoniot conditions for a strong shock to relate ie blast wave velocity to the post-shock eas velocity 4. we find a(f)=0.351(E/pt?H. | Taking the time derivative and applying the Rankine-Hugoniot conditions for a strong shock to relate the blast wave velocity to the post-shock gas velocity $u$, we find $u(t)=0.351(E/\rho t^3)^{1/5}$. |
Solving js system vields f=θα). | Solving this system yields $t=0.3(r/u)$. |
We then obtain ie ages bv plugeiugOO in the radius observed iu ie optical mages and the velocities from the echelle spectra. | We then obtain the ages by plugging in the radius observed in the optical images and the velocities from the echelle spectra. |
These procedures apply best to remnants that are still in the Sedov-Tavlor phase of evolution. since post-Sedov-phase roemuauts continue to expand at a slower rate. | These procedures apply best to remnants that are still in the Sedov-Taylor phase of evolution, since post-Sedov-phase remnants continue to expand at a slower rate. |
The Sedov-Tavlor phase is expected to cud when the post-shock gas velocity drops to ~190 kin |. | The Sedov-Taylor phase is expected to end when the post-shock gas velocity drops to $\sim190$ km $^{-1}$. |
Tnhomogencitics i the ambicut medium also affect the expansion rate. | Inhomogeneities in the ambient medium also affect the expansion rate. |
Iun addition. the observed velocities aud radii are found for individual parts of the SNRs aud. due to asviuuetries in the SNRs may uot adequately represent the whole objects. | In addition, the observed velocities and radii are found for individual parts of the SNRs and, due to asymmetries in the SNRs may not adequately represent the whole objects. |
All of these effects combine to make the ages estimated from t=0:3(c0/4) approximate. | All of these effects combine to make the ages estimated from $t = 0.3(r/u)$ approximate. |
The suaulatious cescribed previously also provide an independent numerical estimate of the age. | The simulations described previously also provide an independent numerical estimate of the age. |
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