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The oulv one of these they dist as a confined W UAla svstem is ASAS JUSS128]|1953.1. with a period of 0.2178 d. Rucnuski Pribulla (2008) subsequently presented further photoimetrc and spectroscopic observations of this system and argued the case for this svsteni as the shortest period field contact binary.
The only one of these they list as a confirmed W UMa system is ASAS J083128+1953.1 with a period of 0.2178 d. Rucinski Pribulla (2008) subsequently presented further photometric and spectroscopic observations of this system and argued the case for this system as the shortest period field contact binary.
This object is iudeed preseut iu the SuperWASP database (ax 15SWASP JOS3127.87|195303.5). with the same period. but 1e licht curve js poorly samplec (fewer than 1000 data poiuts) iid προς between two bnghtuess levels. presuuably as a result of another star sometimes fallhius within the iotonietrie aperture.
This object is indeed present in the SuperWASP database (as 1SWASP J083127.87+195303.5), with the same period, but the light curve is poorly sampled (fewer than 1000 data points) and jumps between two brightness levels, presumably as a result of another star sometimes falling within the photometric aperture.
It nonetheless displavs a broadly sinusoidal modulation when folded at this period (see Figure Ta).
It nonetheless displays a broadly sinusoidal modulation when folded at this period (see Figure 7a).
Since the modulation profile did uot match hat of a classic contact binary. it was selected as a candidate eclipsing binary by the current exercise.
Since the modulation profile did not match that of a classic contact binary, it was selected as a candidate eclipsing binary by the current exercise.
More recently Pribulla. Vauko ILuubalek (2009) ooked at two of the candidate short period svstenis isted by Ruciuski (2007).
More recently, Pribulla, Vanko Hambalek (2009) looked at two of the candidate short period systems listed by Rucinski (2007).
They found that one of hese stars (ASAS JI130310101.9) in fact has a perioc ouecr than that originally sugeested (0.2710 d instead of 1,2131 d).
They found that one of these stars (ASAS J113031–0101.9) in fact has a period longer than that originally suggested (0.2710 d instead of 0.2131 d).
The other candidate (CASAS JO7TLS290336.7) Was confirmed as a contact binary with a period of 42113 d. As with the object. discussed by. Ruciuksi Pribulla. this oue too is present in the SuperWASP database (as ISWASP JOrIs25.67.033639.5). but the ligh curve is very poorly suupled (fewer than LOO data poiuts).
The other candidate (ASAS J071829–0336.7) was confirmed as a contact binary with a period of 0.2113 d. As with the object discussed by Rucinksi Pribulla, this one too is present in the SuperWASP database (as 1SWASP J071828.67–033639.5), but the light curve is very poorly sampled (fewer than 100 data points).
Nonetheless. when the SuperWASP data are folded at the ASAS period. a profile characteristic of a W UMa star is revealed (sce Figure 7b).
Nonetheless, when the SuperWASP data are folded at the ASAS period, a profile characteristic of a W UMa star is revealed (see Figure 7b).
However. eiven the few data points. this object was not included in the sample that were examined here.
However, given the few data points, this object was not included in the sample that were examined here.
A survey of a 0.25 square deeree region of the Galactic plane using the ESO-2.21 telescope x Miller et al (2010) vielded more than half a million light curves down to YN> 21.5.
A survey of a 0.25 square degree region of the Galactic plane using the ESO-2.2m telescope by Miller et al (2010) yielded more than half a million light curves down to $R \sim 24.5$ .
Amonest this set they found 1318 variable stars. 533 of which were W UM—a stars.
Amongst this set they found 1318 variable stars, 533 of which were W UMa stars.
This included seven candidate contact binaries with periods less than 0.23 ᾱ three of which have periods at or below the period cut-off: V-737 (0.2109 d). V-301 (0.2113 d) and V-1085 (0.2199 d).
This included seven candidate contact binaries with periods less than 0.23 d, three of which have periods at or below the period cut-off: V-737 (0.2109 d), V-301 (0.2143 d) and V-1085 (0.2199 d).
Each ofthese stars is relatively faint (2 iuaguitudes in the range ~19 22) and so thev are not amenable to detailed follow-up.
Each of these stars is relatively faint $R$ magnitudes in the range $\sim 19 - 22$ ) and so they are not amenable to detailed follow-up.
Noue of these are present iu the SuperWASP database. owing to their location auc füutness.
None of these are present in the SuperWASP database, owing to their location and faintness.
As noted in the introduction. all of the stars discussed above have recently been superseded as the shortest oeriod eclipsing binary with main sequence coniponeuts x GSC 23110530 (= ISWASP JO022050.85|332017.6) with a period of 0.1926 d (Norton ct al 2007: Dimitrov Igurkchieva 2010). as shown in Figure 3 of the prescut oper.
As noted in the introduction, all of the stars discussed above have recently been superseded as the shortest period eclipsing binary with main sequence components by GSC 2314–0530 (= 1SWASP J022050.85+332047.6) with a period of 0.1926 d (Norton et al 2007; Dimitrov Kjurkchieva 2010), as shown in Figure 3 of the present paper.
As n result of our work. however. Figure 5 aud Table l show that we can now add a further 22 candidates o the set of eclipsing binaries with the shortest periods 200 d 0.225 d).
As a result of our work, however, Figure 5 and Table 1 show that we can now add a further 22 candidates to the set of eclipsing binaries with the shortest periods (0.200 d – 0.225 d).
Four of these are brighter than uaenitude 13. so nüeht have been accessible to ASAS.
Four of these are brighter than magnitude 13, so might have been accessible to ASAS.
Iu addition we fiud a further 30 candidates iu the period range at which the sharp cut-off occurs (0.225 d 1230 d). Ithoueh four of this latter set were already known. two of them had the wroue perio previously recorded.
In addition we find a further 30 candidates in the period range at which the sharp cut-off occurs (0.225 d – 0.230 d), although four of this latter set were already known, two of them had the wrong period previously recorded.
Ruciuski 2007) conunented at the time of ιο ASAS work that the statistics of the sample at such short periods were very limited. despite the fact that ASAS covered around 3/l of the skv anc extended down ο nmaeguitude ~123.
Rucinski (2007) commented at the time of the ASAS work that the statistics of the sample at such short periods were very limited, despite the fact that ASAS covered around 3/4 of the sky and extended down to magnitude $\sim 13$.
SuperWASP covers a similar fraction of the sky (avoiding the Galactic plane) but is scusitive down to magnitude 1S. Le. over sx times faüuter. aud as a result increases the sample by a factor of wore thia six.
SuperWASP covers a similar fraction of the sky (avoiding the Galactic plane) but is sensitive down to magnitude $\sim 15$, i.e. over six times fainter, and as a result increases the sample by a factor of more than six.
Deb Singh (2010) have τοσο presented au analysis of the light curves of 62 binary stars (inostlv contact binaries) from the ASAS-3 survey.
Deb Singh (2010) have recently presented an analysis of the light curves of 62 binary stars (mostly contact binaries) from the ASAS-3 survey.
In particular they show that there is a rather tight correlation between the period and (FIV) colour of the coutact binarics in their sample (sce their Figure 11).
In particular they show that there is a rather tight correlation between the period and $J-K$ ) colour of the contact binaries in their sample (see their Figure 11).
The relationship is parameterized by where P is in days.
The relationship is parameterized by where $P$ is in days.
Although their relationship is well constrained. the majority of their sample have periods around 0.1 d and they iuclude relatively few objects close to the period cut-off.
Although their relationship is well constrained, the majority of their sample have periods around 0.4 d and they include relatively few objects close to the period cut-off.
In Figure 8 we show the 241ÀSS (7 A) colours of our salple. against period. with Equation 1 over-plotted.
In Figure 8 we show the 2MASS $J-K$ ) colours of our sample, against period, with Equation 1 over-plotted.
Although the fit is reasonably eood. there is significant scatter iu the colours of these short period svstenis.
Although the fit is reasonably good, there is significant scatter in the colours of these short period systems.
Gazeas Stepicen (2008) and CGazeas Niarclios (2006) demonstrated that there aro clear correlations between the masses of the components of contact binaries aud their orbital periods. aud also between the radii of the coniponeuts and their orbital periods.
Gazeas Stepien (2008) and Gazeas Niarchos (2006) demonstrated that there are clear correlations between the masses of the components of contact binaries and their orbital periods, and also between the radii of the components and their orbital periods.
Iu particular. the period cut-off of around 0.22 d corresponds to primary aud secondary masses of around 0:55AL. and 0.3A. and radii of around 0.7.AL. and 0.5AZ. respectively.
In particular, the period cut-off of around 0.22 d corresponds to primary and secondary masses of around $0.85~M_\odot$ and $0.3~M_\odot$ and radii of around $0.7~M_\odot$ and $0.5~M_\odot$ respectively.
These masses radii correspond to stars of spectral type I& aud thus match the observed V...AN. colours we see in our sample (Figure 6).
These masses and radii correspond to stars of spectral type K and thus match the observed $V-K$ colours we see in our sample (Figure 6).
The mass aud radius estimates of Gazeas Stepien (2008) are good to around accuracy.
The mass and radius estimates of Gazeas Stepien (2008) are good to around accuracy.
However. the short period cud of these correlations is defined by ouly three stars: CC Com (0.2211 d). V523 Cas (0.2337 d) alu RW Cou (025753 οἱ). and there is considerable scatter between their parameters.
However, the short period end of these correlations is defined by only three stars: CC Com (0.2211 d), V523 Cas (0.2337 d) and RW Com (0.2373 d), and there is considerable scatter between their parameters.
Increasing the number
Increasing the number
Snowdenetal. (1998))). we estimate ils electron density.
\citet{snowden_etal_98}) ), we estimate its electron density.
The temperature is not well known. but most of the lions are likely to be near their ionizational equilibrium temperature (3.2xLO? IK). and the enission equation is relatively insensitive to temperature for 1.x10~T—ο. An VI-rich plasma. with the above column density. a temperature of 6.3xLO? IX. and a thermal pressure of 15.000 IX 7? emits 450 photons 7s J|1 |, which is within the observational 2 sigma upper limit on the intensitv.
The temperature is not well known, but most of the ions are likely to be near their ionizational equilibrium temperature $3.2 \times 10^5$ K), and the emission equation is relatively insensitive to temperature for $1 \times 10^5 \stackrel{<}{\sim} T \stackrel{<}{\sim} 1 \times 10^6$ K. An -rich plasma, with the above column density, a temperature of $6.3 \times 10^5$ K, and a thermal pressure of 15,000 K $^{-3}$ emits 450 photons $^{-2}$ $^{-1}$ $^{-1}$, which is within the observational 2 sigma upper limit on the intensity.
Similar plasmas will temperatures as low as c1x10" Ix emit even less intense rracliiation.
Similar plasmas with temperatures as low as $\sim 1 \times 10^5$ K emit even less intense radiation.
Thus. a (wo phase model could meet (he observational constraints without requiring unobserved astroplivsies or unusual conditions.
Thus, a two phase model could meet the observational constraints without requiring unobserved astrophysics or unusual conditions.
In this section. we examine the multi-phase bubble simulations of and (he evaporating cloud simulations of Slavin(1989)..
In this section, we examine the multi-phase bubble simulations of \citet{smith_cox} and the evaporating cloud simulations of \citet{slavin}.
In the models of (2001).. the Local Bubble resulted rom two or (11ος supernova explosions occurring up lo several million vears ago.
In the models of \citet{smith_cox}, the Local Bubble resulted from two or three supernova explosions occurring up to several million years ago.
The interior of the structure consists of hot (~10" IX). nearly collisional ionizational equilibrium: plasma. while (he periphery of the structure consists of an intermediate temperature transition zone.
The interior of the structure consists of hot $\sim 10^6$ K), nearly collisional ionizational equilibrium plasma, while the periphery of the structure consists of an intermediate temperature transition zone.
Although the simulations do not explicitly: include the transition zones surrounding the embedded cool elouds. such zones are thought to harbor lions (Slavin1989:Oegerleetal.2002) and so should contribute to the Πακ of rresonance line photons.
Although the simulations do not explicitly include the transition zones surrounding the embedded cool clouds, such zones are thought to harbor ions \citep{slavin,oegerle_etal} and so should contribute to the flux of resonance line photons.
Here. we will use the simulations of Slavin(1989).. which predict their intensity.
Here, we will use the simulations of \citet{slavin}, which predict their intensity.
After performing several hvdrocwnamical simulations. Smith&Cox(2001) decided that the best model lies intermediate between their various simulations.
After performing several hydrodynamical simulations, \citet{smith_cox} decided that the best model lies intermediate between their various simulations.
Their simulations vield predicted intensities ranging from 190 to more than 9.500 photons 7s | LH (alter the extra [actor ol 4x im their Figure 19 (Smith. private communication) is divided out).
Their simulations yield predicted intensities ranging from 190 to more than 9,500 photons $^{-2}$ $^{-1}$ $^{-1}$, (after the extra factor of $4\pi$ in their Figure 19 (Smith, private communication) is divided out).
To this intensity must be added the intensity emitted by the transition zone on the local cloud aud. possibly. transition zones on other cool clouds embedded in the Local Bubble.
To this intensity must be added the intensity emitted by the transition zone on the local cloud and, possibly, transition zones on other cool clouds embedded in the Local Bubble.
For the transition zone on the cool cloud surrounding (he Sun. Slavin(1989). predicted an ddoublet intensity of 250 photons 7s | |.
For the transition zone on the cool cloud surrounding the Sun, \citet{slavin} predicted an doublet intensity of 250 photons $^{-2}$ $^{-1}$ $^{-1}$.
The sium of the Smith&Cox(2001). and Slavin(1989). predictions (440 to >9800 photons 7s ! 1) marginally overlaps the 2 signia upper limit reported in this paper.
The sum of the \citet{smith_cox} and \citet{slavin} predictions (440 to $\geq9800$ photons $^{-2}$ $^{-1}$ $^{-1}$ ) marginally overlaps the 2 sigma upper limit reported in this paper.
If other cool clouds reside along the line of sight. then the predicted intensity would be even higher.
If other cool clouds reside along the line of sight, then the predicted intensity would be even higher.
Thus. this particular model is heavily constrained by the null results.
Thus, this particular model is heavily constrained by the null results.
We would have lost confidence in this (vpe of model had it not been possible to
We would have lost confidence in this type of model had it not been possible to
We thank Lars Hernquist. Chris McKee. Phil Chang. Pedro Aarronetti. Leo Blitz. Jay Gallagher. Risa. Wechsler. and Daniel Holz for helpful discussions.
We thank Lars Hernquist, Chris McKee, Phil Chang, Pedro Marronetti, Leo Blitz, Jay Gallagher, Risa Wechsler, and Daniel Holz for helpful discussions.
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LLL respectively). and the closely-related76-11 sample.
III respectively), and the closely-related sample.
“Phis lattor sample was selected at a frequency of 151 MlZ using the 7€ survey data of Lacy et ((1995) with a flux limi chosen to maximise overlap with the SC sample.
This latter sample was selected at a frequency of 151 MHz using the 7C survey data of Lacy et (1995) with a flux limit chosen to maximise overlap with the 8C sample.
Phis sample is defined in Section 2. and will be described in more detai in a future paper (Lacy et 11999h).
This sample is defined in Section 2, and will be described in more detail in a future paper (Lacy et 1999b).
A further paper wil discuss spectroscopy of the GC sample of Eales (1985). ane details of the and samples are presented in Willot (1998. ancl papers in preparation).
A further paper will discuss spectroscopy of the 6C sample of Eales (1985), and details of the and samples are presented in Willott (1998, and papers in preparation).
The properties of these samples are sunimarised in Table 1. and the coverage of the racio-Iuminositv — redshift plane is illustrated in. of Blundell. Rawlines Willott (1999).
The properties of these samples are summarised in Table 1, and the coverage of the radio-luminosity – redshift plane is illustrated in, 1 of Blundell, Rawlings Willott (1999).
Samples such as these are very valuable for studying the cosmic evolution of the radio source population. in particular the question of the evolution of source size. D. with recdshilt.
Samples such as these are very valuable for studying the cosmic evolution of the radio source population, in particular the question of the evolution of source size, $D$, with redshift.
As redshift ancl radio luminosity are always strongly correlated within any flux-limited. sample. partial correlation cocllicients need to be calculated to investigate whether the true correlation is with redshift or luminosity.
As redshift and radio luminosity are always strongly correlated within any flux-limited sample, partial correlation coefficients need to be calculated to investigate whether the true correlation is with redshift or luminosity.
Studies based. on low-frequeney selected: samples. typically show a weak correlation with redshift only. parameterised as Dx(112)" where nzLT1.0 tthe ος sample. Eales 1985: Neeser ct 11995. and the and samples: Blundell. Rawlings Willott 1999). although samples selected at higher frequencies seem to have stronger dependences. with ;g3 (Oort. Ixatgert Windhorst LOST: Ixapahi 1989) combined with a radio luminosity dependence DxL' with e20.3.
Studies based on low-frequency selected samples typically show a weak correlation with redshift only, parameterised as $D\propto (1+z)^{-\eta}$ where $\eta \approx 1.7-1.9$ the 6C sample, Eales 1985; Neeser et 1995, and the and samples; Blundell, Rawlings Willott 1999), although samples selected at higher frequencies seem to have stronger dependences, with $\eta \approx 3$ (Oort, Katgert Windhorst 1987; Kapahi 1989) combined with a radio luminosity dependence $D\propto L^{\epsilon}$ with $\epsilon \approx 0.3$.
In Paper Lb we performed. a partial rank correlation on the SC-NEC sample using photometric redshifts.
In Paper II we performed a partial rank correlation on the 8C-NEC sample using photometric redshifts.
This showed that the funcamental correlation was between size and redshift. but having only recshilts based. on /-bancl magnitudes we were forced to only consider ο<1.1 objects and to accept the uncertainties associated with photomoetric redshifts.
This showed that the fundamental correlation was between size and redshift, but having only redshifts based on $R$ -band magnitudes we were forced to only consider $z<1.1$ objects and to accept the uncertainties associated with photometric redshifts.
We also investigated. the four-way correlation coefficients. of redshift. luminosity. size and spectral index. finding that spectral index at 2GIz in the rest frame was most stronely correlated with redshift. ancl any correlation with luminosity was weak.
We also investigated the four-way correlation coefficients of redshift, luminosity, size and spectral index, finding that spectral index at 2GHz in the rest frame was most strongly correlated with redshift, and any correlation with luminosity was weak.
Since then. independent: work on the GC. and surveys has been carried. out by Blunclell et (1999) with results that are broadly consistent with he Neeser et rresult and. the results of Paper LL.
Since then, independent work on the 6C, and surveys has been carried out by Blundell et (1999) with results that are broadly consistent with the Neeser et result and the results of Paper II.
They ind. however. a significant correlation of spectral index with uminosity at a rest-frame frequenev of 1 GllIz. although a correlation with redshift is seen at higher frequencies.
They find, however, a significant correlation of spectral index with luminosity at a rest-frame frequency of 1 GHz, although a correlation with redshift is seen at higher frequencies.
In his paper we use the redshifts for the and SC-NEC samples to study the size-redshift relation further.
In this paper we use the redshifts for the and 8C-NEC samples to study the size-redshift relation further.
We assume Jo=50kms!Mpe. and an Einstein. — de Sitter (O3;=1. O4— 0) cosmology unless otherwise stated.
We assume $H_0=50 \, {\rm kms^{-1}Mpc^{-1}}$ and an Einstein – de Sitter $\Omega_{\rm M} = 1$, $\Omega_{\Lambda}=0$ ) cosmology unless otherwise stated.
Positions are all epoch D1950.0.
Positions are all epoch B1950.0.
Spectral indices. both radio and optical are defined in the sense that S,x&mm
Spectral indices, both radio and optical are defined in the sense that $S_{\nu} \propto \nu^{-\alpha}$.
The SC-NEC sample has been formally defined in Paper I. ancl some refinements described in Paper HI.
The 8C-NEC sample has been formally defined in Paper I, and some refinements described in Paper III.
It consists of all raclio sources with 38-Mllz flux densities Saszc1.3 Jv within 3° o£ 18 OO. 66° in the survey of Rees (1990).
It consists of all radio sources with 38-MHz flux densities $S_{38} \geq 1.3$ Jy within $3^{\circ}$ of $^{\rm h}$ $^{\rm m}$ $^{\circ}$ in the survey of Rees (1990).
Phe sample consists of all TC objects from the mini-survey of Lacy et ((1995) with 151-MllIz ux densities Sy.)20.5 Jy. again within 3° of 18"00". |667.
The sample consists of all 7C objects from the mini-survey of Lacy et (1995) with 151-MHz flux densities $S_{151}\geq 0.5$ Jy, again within $3^{\circ}$ of $18^{\rm h} 00^{\rm m}$ $+66^{\circ}$.
Lhe Dux density limits are deliberately chosen to maximise the overlap of the two samples. thus most sources are common to both samples.
The flux density limits are deliberately chosen to maximise the overlap of the two samples, thus most sources are common to both samples.
The sample will be more fully described in Lacey et 11999b. but the positions and radio properties of the objects which are present only in the sample (and which therefore have not appeared in Papers LLL) are presented in Table 2. for completeness.
The sample will be more fully described in Lacy et 1999b, but the positions and radio properties of the objects which are present only in the sample (and which therefore have not appeared in Papers I-III) are presented in Table 2, for completeness.
number of objects have been excluded: from the samples on the basis of confusion by bright nearby sources.
A number of objects have been excluded from the samples on the basis of confusion by bright nearby sources.
7€ 1732|6715 (SC. 1732|672) has avery bright radio source nearby and has been temporarily removed from the sample pending an improved radio image.
7C 1732+6715 (8C 1732+672) has a very bright radio source nearby and has been temporarily removed from the sample pending an improved radio image.
το 182116442 (SC 1821646) has a star on top of the ID position MMeMahon.. personal. communication).
7C 1821+6442 (8C 1821+646) has a star on top of the ID position McMahon, personal communication).
TC 182716517 (SC 18271652) has been excluded as there is à very bright. star nearby.
7C 1827+6517 (8C 1827+652) has been excluded as there is a very bright star nearby.
TFhis has reduced. the total number of objects in the sample from 57 to 54 and for the SC-NEC sample from 61 to 58.
This has reduced the total number of objects in the sample from 57 to 54 and for the 8C-NEC sample from 61 to 58.
As these objects were removed for reasons unconnected with their intrinsic radio or optical properties their omission should not alfect the sample statistics.
As these objects were removed for reasons unconnected with their intrinsic radio or optical properties their omission should not affect the sample statistics.
Apart [rom these. all objects in the sample have spectroscopic data.
Apart from these, all objects in the sample have spectroscopic data.
In the SC-NISC sample. there are three objects for which spectra are vet to be obtained.
In the 8C-NEC sample, there are three objects for which spectra are yet to be obtained.
Alost of the objects in both the SC-NEC ancl North Ecliptic Cap samples were observed. with the ISLS spectrograph on the Willian Llerschel Telescope. (IUE).
Most of the objects in both the 8C-NEC and North Ecliptic Cap samples were observed with the ISIS spectrograph on the William Herschel Telescope (WHT).
Most observations were mace on the nights of 1995 July 28 to 31. though observations of a few objects were made by on the nights of 1993 August 20-21 and 19983 June 15-19.
Most observations were made on the nights of 1995 July 28 to 31, though observations of a few objects were made by on the nights of 1993 August 20-21 and 1993 June 18-19.
Observations of brighter objects in the sample were mace with the IGI on the MeDonalel Observatory LOT telescope on the nights of 1993 July 27-28.
Observations of brighter objects in the sample were made with the IGI on the McDonald Observatory $^{''}$ telescope on the nights of 1993 July 27-28.
Pwo objects were observed using the Ixast spectrograph on the Shane 32m telescope at
Two objects were observed using the Kast spectrograph on the Shane 3-m telescope at
According to the theory of cliffusive shock acceleration or SNRs. the maximum attainable energy for cosmic ravs is determined by the size of accelerator. the magnetic field of the CSAL and the energy. losses resulting from acdiabatic oocesses and svnchrotron. processes.
According to the theory of diffusive shock acceleration for SNRs, the maximum attainable energy for cosmic rays is determined by the size of accelerator, the magnetic field of the CSM and the energy losses resulting from adiabatic processes and synchrotron processes.
The size depends on he explosion evolution of SNR.
The size depends on the explosion evolution of SNR.
According to Ixirk(1904).. he explosion includes three phases: frec-expansion. phase. Sedov-Tavlor. phase. ancl snow-plough phase.
According to \cite{Kirk1994}, the explosion includes three phases: free-expansion phase, Sedov-Taylor phase and snow-plough phase.
During the rec-expansion phase. the kinetic energy of ejecta remains untapped. and the particle acceleration is not significant.
During the free-expansion phase, the kinetic energy of ejecta remains untapped, and the particle acceleration is not significant.
Once the mass swept-up by the shock becomes comparable o the mass of the ejecta Adje. the explosion enters he Secov-Tavlor phase.
Once the mass swept-up by the shock becomes comparable to the mass of the ejecta $M_{\rm ejc}$, the explosion enters the Sedov-Taylor phase.
The acceleration ellicieney is the lighest in this phase.
The acceleration efficiency is the highest in this phase.
The maximum energy for protons is approximately eiven by (Schurectal.2010) where Zis the charge number. e is the elementary electric charge. e is the speed of light and £5 is à relation between the compression ratio of the density and magnetic field.
The maximum energy for protons is approximately given by \citep{Schure2010} where $Z$is the charge number, $e$ is the elementary electric charge, $c$ is the speed of light and $\xi_{\sigma}$ is a relation between the compression ratio of the density and magnetic field.
Here. we assume that the magnetic field is parallel to the shock normal. which means £4=20.
Here, we assume that the magnetic field is parallel to the shock normal, which means $\xi_{\sigma}=20$.
D is the magnetic lield strength. and fp~Repτμ is the duration for which the particles stay in the Sedov-TFavlor phase. where Vy is the shock speed and Rep is the radius where the shock sweeps up the matter whose mass is equal to that of the ejecta.
$B$ is the magnetic field strength, and $t_{\rm ST}\sim R_{\rm ST}/V_{\rm sh}$ is the duration for which the particles stay in the Sedov-Taylor phase, where $V_{\rm sh}$ is the shock speed and $R_{\rm ST}$ is the radius where the shock sweeps up the matter whose mass is equal to that of the ejecta.
1n general. for SNRs. the magnetic Ποιά D of the CSAIL is pCGauss. ancl fer is ~ pe due to the high mass of the ejecta (~ AL.) and the low particle density of CSM (Melee&"Truelove. 1995).
In general, for SNRs, the magnetic field $B$ of the CSM is $\sim \mu$ Gauss, and $R_{\rm ST}$ is $\sim$ pc due to the high mass of the ejecta $\sim M_\odot$ ) and the low particle density of CSM \citep{McKee1995}.
.. Typical fp is several hundred: vears.
Typical $t_{\rm ST}$ is several hundred years.