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TiC presence of scattered light in tlic NIR nay cause a discrepancy | The presence of scattered light in the NIR may cause a discrepancy |
For fine particles well inside the sub-domain. and for coarse particles well outside the sub-clomain. this makes no difference. but it is important. for particles near the boundary. including coarse particles which are about to be split. | For fine particles well inside the sub-domain, and for coarse particles well outside the sub-domain, this makes no difference, but it is important for particles near the boundary, including coarse particles which are about to be split. |
A second way to implement Particle Splitting is to identify xwticles whose resolution. (amp) is insullicient for he problem in hand. and split them on-the-fly. | A second way to implement Particle Splitting is to identify particles whose resolution, $\sim (m/\rho)^{1/3}$, is insufficient for the problem in hand, and split them on-the-fly. |
To do this. we must recognize what the deterministic physics of the ooblem is. and quantify αν a local function. of state the minimum resolution required to capture this physics. | To do this, we must recognize what the deterministic physics of the problem is, and quantify – as a local function of state -- the minimum resolution required to capture this physics. |
Then. whenever the resolution of a particle approaches the ocal minimunmi resolution. that particle is split. | Then, whenever the resolution of a particle approaches the local minimum resolution, that particle is split. |
As dn Nested Splitting. the velocities. of the new children-particles are evaluated by summing contributions rom the neighbours of the parent particle. | As in Nested Splitting, the velocities of the new children-particles are evaluated by summing contributions from the neighbours of the parent particle. |
Successive generations of splitting iwe possible. vielding particles with masses m/13. 2m/137. m/13"nPy. ote. | Successive generations of splitting are possible, yielding particles with masses $m/13$, $m/13^2$, $m/13^3$, etc. |
and hence linear resolution improved by factors 2.4. 5.5. 13. etc. | and hence linear resolution improved by factors 2.4, 5.5, 13, etc. |
As in Nested Splitting. better smoothing is obtained if the desired. number of neighbours is evaluated according to Eqn. (18)) | As in Nested Splitting, better smoothing is obtained if the desired number of neighbours is evaluated according to Eqn. \ref{NEIB}) ). |
In the context of star formation. gravitational collapse anc fragmentation are crucial physical processes. and herefore the code must always be able to resolve the Jeans mass. | In the context of star formation, gravitational collapse and fragmentation are crucial physical processes, and therefore the code must always be able to resolve the Jeans mass. |
This is called the Jeans Condition. (Lruclove et. al. | This is called the Jeans Condition (Truelove et al. |
1997. Bate Burkert 1977). | 1997, Bate Burkert 1977). |
Since the Jeans mass is a ocal function of state. it is straightforward. to formulate he Jeans Condition. and to split. particles when they are about to violate it. | Since the Jeans mass is a local function of state, it is straightforward to formulate the Jeans Condition, and to split particles when they are about to violate it. |
We explore On-TPhe-Fly Particle Splitting rigecred by the Jeans Condition. in the next two sections. | We explore On-The-Fly Particle Splitting triggered by the Jeans Condition, in the next two sections. |
lTruelove et al. ( | Truelove et al. ( |
1997) have. shown that simulations of protostellar collapse and fragmentation performed. using AMIR only converge when the local Jeans length Ajo,~a(Cp) is resolved. (where a is the local sound. speed) | 1997) have shown that simulations of protostellar collapse and fragmentation performed using AMR only converge when the local Jeans length $\lambda\Jea \sim
a (G\rho)^{-1/2}$ is resolved (where $a$ is the local sound speed). |
Speetlically the linear resolution Ar must everywhere satisfy Otherwise artificial fragmentation can occur and/or real [fragmentation can be suppressed. | Specifically the linear resolution $\Delta r$ must everywhere satisfy Otherwise artificial fragmentation can occur and/or real fragmentation can be suppressed. |
Eqn. (19)) | Eqn. \ref{JCAMR}) ) |
is the formulation of the Jeans Condition appropriate for. ED Codes. | is the formulation of the Jeans Condition appropriate for FD codes. |
Bate Burkert (1997) have shown that there is a similar Jeans Condition for SPL simulations: the minimum resolvable mass 2,m must evervwhere be less than the local Jeans mass ~(6777pa. or Otherwise artificial [fragmentation may occur. and real fragmentation will definitely be suppressed. | Bate Burkert (1997) have shown that there is a similar Jeans Condition for SPH simulations: the minimum resolvable mass $\sim 2 {\cal N}\nei m$ must everywhere be less than the local Jeans mass $\sim G^{-3/2} \rho^{-1/2} a^3$, or Otherwise artificial fragmentation may occur, and real fragmentation will definitely be suppressed. |
Whitworth (1998). has shown that provided the smoothing kernel is sullicienth centrally peaked. SPILL simulations of collapse are unlikely to sulfer from artificial fragmentation. | Whitworth (1998) has shown that provided the smoothing kernel is sufficiently centrally peaked, SPH simulations of collapse are unlikely to suffer from artificial fragmentation. |
However. it remains the case that Eqn. (20)) | However, it remains the case that Eqn. \ref{JCSPH}) ) |
must be satisfied if real fragmentation is to be modelled. | must be satisfied if real fragmentation is to be modelled. |
AX standard test of star formation codes is the one propose by Boss Bodenheimer (1979: hereafter. το). | A standard test of star formation codes is the one proposed by Boss Bodenheimer (1979; hereafter BB79). |
In this test the initial conditions are a uniforme-density. isotherma loud having mass AJ.. radius 0.02 pe. density 2.10ix e " and isothermal sound speed. 0.17 km (so the ratio of thermal to gravitational energy is 0.26). | In this test the initial conditions are a uniform-density isothermal cloud having mass $M_\odot$, radius $0.02\,$ pc, density $2 \times
10^{-18}$ g $^{-3}$ and isothermal sound speed $0.17 $ km $^{-1}$ (so the ratio of thermal to gravitational energy is $\sim\,$ 0.26). |
Ehe clou then has an m=2 azimuthal density perturbation with amplitude imposed. on it. ancl it is set to rotate a uniformip angular speed. -.7.210915 rad 1 (so the ratio. of rotational to gravitational energy is 0.16). | The cloud then has an $m = 2$ azimuthal density perturbation with amplitude imposed on it, and it is set to rotate at uniform angular speed $7.2 \times 10^{-13}$ rad $^{-1}$ (so the ratio of rotational to gravitational energy is $\sim\,$ 0.16). |
There is no external pressure. and the gas remains isothermal during the subsequent evolution. until the density rises above a critical value Pept. When it switches to being adiabatic and therefore reals up. | There is no external pressure, and the gas remains isothermal during the subsequent evolution, until the density rises above a critical value $\rho\cri$, when it switches to being adiabatic and therefore heats up. |
ιο et. al. ( | Klein et al. ( |
1999). have performed this test. with Pepi=50P g using AMIR. and shown that he cloud should collapse to produce an elongated structure. | 1999) have performed this test, with $\rho\cri
= 5 \times 10^{-14}$ g $^{-3}$, using AMR, and shown that the cloud should collapse to produce an elongated structure. |
This structure then evolves into a binary svstem with a xw between the components. | This structure then evolves into a binary system with a bar between the components. |
As long as the gas remains isothermal. the binary components and the inter-connecting xw condense into filamentary singularities. and the bar does not fragment Clruelove et al. | As long as the gas remains isothermal, the binary components and the inter-connecting bar condense into filamentary singularities, and the bar does not fragment (Truelove et al. |
1998). | 1998). |
Phese results agree with he theoretical analysis of Inutsuka Alivama (1992). but contrast with earlier. simulations of the BB79 test. using yoth FD and SPIEL codes. | These results agree with the theoretical analysis of Inutsuka Miyama (1992), but contrast with earlier simulations of the BB79 test using both FD and SPH codes. |
In. particular. the simulations reported by Burkert Bodenheimer (1993) resulted in the xw fragmenting. | In particular, the simulations reported by Burkert Bodenheimer (1993) resulted in the bar fragmenting. |
The reason for this appears to have been hat their simulations did not satisfy the Jeans Condition. | The reason for this appears to have been that their simulations did not satisfy the Jeans Condition. |
Bate Burkert (1997) have subsequently repeated the DDT9 test with pcri=10HM gem . using SPL and 80.000 xwiicles. sullicient to satisfy. the Jeans Condition up to a density ~pes. | Bate Burkert (1997) have subsequently repeated the BB79 test with $\rho\cri = 10^{-14}$ g $^{-3}$, using SPH and 80,000 particles, sufficient to satisfy the Jeans Condition up to a density $\sim \rho\cri$. |
ον now find essentially the sane result as Truelove et al. ( | They now find essentially the same result as Truelove et al. ( |
1998) and Wlein ct al. ( | 1998) and Klein et al. ( |
1999). confirming hat the earlier results of Burkert Bocenheimer (1993) were compromised. by violating the Jeans condition. | 1999), confirming that the earlier results of Burkert Bodenheimer (1993) were compromised by violating the Jeans condition. |
Thus he BB79 test appears to be à stringent test of whether a code is satisfving the Jeans Condition. | Thus the BB79 test appears to be a stringent test of whether a code is satisfying the Jeans Condition. |
We have performed three SPLIT simulations of the BB79 est. | We have performed three SPH simulations of the BB79 test. |
We have required the gas to remain isothermal up to a density poy,=5-102 & "7. | We have required the gas to remain isothermal up to a density $\rho\cri =
5 \times 10^{-12}$ g $^{-3}$ . |
This is two orders of magnitude higher than the value used. by Klein et al. ( | This is two orders of magnitude higher than the value used by Klein et al. ( |
1999). ancl so in our simulations the Jeans mass and length all to values 10 times smaller. Le. Mjeans~101M. and Hyeans2~ AU. | 1999), and so in our simulations the Jeans mass and length fall to values 10 times smaller, i.e. $M\Jea \sim 10^{-4} M_\odot$ and $R\Jea \sim 2$ AU. |
Thuswe are subjecting our code to an even | Thuswe are subjecting our code to an even |
We have examined the power of a reioulzation model. given its many Cosmological and astrophysical Parameters. to coustrain these input quautities when combined with parameter extraction from the CMB. | We have examined the power of a reionization model, given its many cosmological and astrophysical parameters, to constrain these input quantities when combined with parameter extraction from the CMB. |
In the case of the well-known clegeueracy between 7 aud A in their effects on the CMB. we have found that this can be alleviated by the complementary information [rom a reiouizatiou moclel. aud that this remains a useful cross-coustraint even when allowing for the astropliysical uucertainty in 7. | In the case of the well-known degeneracy between $\tau$ and $A$ in their effects on the CMB, we have found that this can be alleviated by the complementary information from a reionization model, and that this remains a useful cross-constraint even when allowing for the astrophysical uncertainty in $\tau$. |
When we eliminate 7 aud perform a more general Fisher matrix analysis. we fiud that the astroplivsical details of reiouization be useful in further coustraiuLE.ig the CMB's limits ou cosinological parameters. eveu in the case of the expected temperature data from PLANCAN. | When we eliminate $\tau$ and perform a more general Fisher matrix analysis, we find that the astrophysical details of reionization be useful in further constraining the CMB's limits on cosmological parameters, even in the case of the expected temperature data from $PLANCK$. |
We lave shown that independent limits ou the astrophysical inputs to reionizatiou. despite the current uncertainty iu their values. reduce the errors [or cosmological parameters by a factor of at least ~ 2. | We have shown that independent limits on the astrophysical inputs to reionization, despite the current uncertainty in their values, reduce the errors for cosmological parameters by a factor of at least $\sim$ 2. |
Given that we have cousicered the most optimistic parameter yield (rom CMB experiments (83). he use of known astrophysics can only. become more valuable for re:istic experimental results. | Given that we have considered the most optimistic parameter yield from CMB experiments 3), the use of known astrophysics can only become more valuable for realistic experimental results. |
This is of particular value for oy (or A) auc. given their implicatious for structure formation aud or theoretical models of the origin of the seeds of structure iu the early universe. | This is of particular value for $\sigma_8$ (or $A$ ) and $n$, given their implications for structure formation and for theoretical models of the origin of the seeds of structure in the early universe. |
The converse situatiou- using a projected exquisite determination of a cosmological parameter ο coustrain astroplivsical reionizatiou parameters- does uot vield quite as interesting results with eniperature data from current experiments or [rom ΑΝ. even though we mace the most optimistic assumptions: the l-o errors for [εις or f, are larger than what are already kuown to be 'easonable. | The converse situation– using a projected exquisite determination of a cosmological parameter to constrain astrophysical reionization parameters– does not yield quite as interesting results with temperature data from current experiments or from $PLANCK$ , even though we made the most optimistic assumptions; the $\sigma$ errors for $f_{esc}$ or $f_\star$ are larger than what are already known to be reasonable. |
When tlie projected polarization data (rom οΝΟN is iucluded. we fouud that fy iu yarticular may be coustrained to [ar greater accuracy than dits current astroplivsical uucertainty: in practice. however. this may prove difficult to achieve. given the ellects of foregrounds aud instrumental noise which we have neglected here. | When the projected polarization data from $PLANCK$ is included, we found that $f_\star$ in particular may be constrained to far greater accuracy than its current astrophysical uncertainty; in practice, however, this may prove difficult to achieve, given the effects of foregrounds and instrumental noise which we have neglected here. |
Iu sumauiary. one may take away that the astroplivsical details of reiouization cau strengthen he limits ou the cosmology of our universe. beyond even the projected parameter vield [rom future CMB data. and that there is more potential to a measurement of 7 thau the determination ofa single uunber out of a large parameter space describing adiabatic CDM moclels. | In summary, one may take away that the astrophysical details of reionization can strengthen the limits on the cosmology of our universe, beyond even the projected parameter yield from future CMB data, and that there is more potential to a measurement of $\tau$ than the determination of a single number out of a large parameter space describing adiabatic CDM models. |
These broad couclusions are naturally subject to the assumptions mace in this analysis. | These broad conclusions are naturally subject to the assumptions made in this analysis. |
The sizes of joint coulideuce regions derived from the CMB data for any. 2-parameter subspace is determined by the full covariauce matrix. whose elements values are depeudent ou the dimension of the chosen parameter space aud the selected parameters. | The sizes of joint confidence regions derived from the CMB data for any 2-parameter subspace is determined by the full covariance matrix, whose elements' values are dependent on the dimension of the chosen parameter space and the selected parameters. |
The inclusion of more parameters has the geueric result of increasing the sizes of the error ellipses: therelore. the primary results of this paper can only. be streugtliened when parameter spaces larger than that aualvzed here are cousiderect. | The inclusion of more parameters has the generic result of increasing the sizes of the error ellipses; therefore, the primary results of this paper can only be strengthened when parameter spaces larger than that analyzed here are considered. |
Iu the sCDMI cosmology assumed here. the values of 7 in our staudard model were relatively low ( 0.060). | In the sCDM cosmology assumed here, the values of $\tau$ in our standard model were relatively low $\sim 0.06$ ). |
Iu an open universe. or one dominated by a cosmological constant. contribution. we expect larger average values of 7 for a fixed reionization iuodel. as structures [reeze out. earlier. resulting iu a longer line-ol-sight to thelast scattering surface at the reionizatiou epoch. | In an open universe, or one dominated by a cosmological constant contribution, we expect larger average values of $\tau$ for a fixed reionization model, as structures freeze out earlier, resulting in a longer line-of-sight to thelast scattering surface at the reionization epoch. |
Increased | Increased |
As expected, the mass flux associated with the galaxy outflow dominates that from the AGN. | As expected, the mass flux associated with the galaxy outflow dominates that from the AGN. |
The mass flux ratio is Hence M9"s~100 for AGN at the Eddington luminosity andMT &~10%¢7/mp. In order to allow for dust, if a factor 7~! is incorporated into the definition of LsRF, this ratio is seen to be inversely proportional to the square root of the adopted (dust) opacity. | The mass flux ratio is Hence ${\dot M_{out}^{gal} / \dot M_{acc}} \sim 100 $ for AGN at the Eddington luminosity and $\kappa \sim 10^3 \sigma_T/m_p.$ In order to allow for dust, if a factor $\tau^{-1}$ is incorporated into the definition of $L_{SRF},$ this ratio is seen to be inversely proportional to the square root of the adopted (dust) opacity. |
The momentum flux ratio is We see that the momentum ejected from the AGN dominates over that in the global outflow by a factor of a few for AGN near the Eddington luminosity LAcGw~Lsnr. The piston model enables downsizing of AGN and spheroids by coupling their growth. | The momentum flux ratio is We see that the momentum ejected from the AGN dominates over that in the global outflow by a factor of a few for AGN near the Eddington luminosity $L_{AGN}\sim L_{SRF}.$ The piston model enables downsizing of AGN and spheroids by coupling their growth. |
For some fiducial AGN energy conversion efficiency η (0.1), we note that LACN=nc?Mace is a measure of the BH accretion rate. | For some fiducial AGN energy conversion efficiency $\eta$ $(\sim 0.1),$ we note that $L^{AGN}=\eta c^2\dot M_{acc}$ is a measure of the BH accretion rate. |
Since LAC controls the star formation rate and is itself controlled by the black hole accretion rate, we infer that black hole growth and star formation triggering downsize together, provided Q is approximately constant due to AGN triggering of SN. | Since $L^{AGN}$ controls the star formation rate and is itself controlled by the black hole accretion rate, we infer that black hole growth and star formation triggering downsize together, provided $Q$ is approximately constant due to AGN triggering of SN. |
The AGN driving of star formation overcomes the pressure suppression of porosity in the absence of the AGN. | The AGN driving of star formation overcomes the pressure suppression of porosity in the absence of the AGN. |
large porosity also results in a wind. | A large porosity also results in a wind. |
The required turbulentA velocity field controls the accretion rate and might be specified by other physics, such as a merger, or even be due to the AGN itself. | The required turbulent velocity field controls the accretion rate and might be specified by other physics, such as a merger, or even be due to the AGN itself. |
Let us try to make these assertions more quantitative. | Let us try to make these assertions more quantitative. |
The AGN is the ultimate driver of the porosity. | The AGN is the ultimate driver of the porosity. |
We need to connect AGN-induced star formation and outflows to the black hole growth rate via the AGN luminosity. | We need to connect AGN-induced star formation and outflows to the black hole growth rate via the AGN luminosity. |
The Boe outflow rate is By momentum conservation, this must equal the global AGN-boosted outflow rate LCN/(ους). Incorporating the effects of porosity-driven star formation means that the outflows must satisfy The AGN luminosityis controlled by the accretion rate onto the central black hole, Mace. Our next step is to evaluate the black hole growth rate, Macc. This is the key to explaining downsizing. | The global outflow rate is By momentum conservation, this must equal the global AGN-boosted outflow rate $L^{AGN}/(cv_c).$ Incorporating the effects of porosity-driven star formation means that the outflows must satisfy The AGN luminosity is controlled by the accretion rate onto the central black hole, $ \dot M_{acc}. $ Our next step is to evaluate the black hole growth rate, $M_{acc}.$ This is the key to explaining downsizing. |
To reproduce the downsizing phenomenon, observed for AGN (Hasingeretal.2005) and their massive host galaxies (Krieketal.|/2007) to occur almost coevally, we need to understand why massive SMBH and spheroids form before their less massive counterparts. | To reproduce the downsizing phenomenon, observed for AGN \citep{has05} and their massive host galaxies \citep{krie07} to occur almost coevally, we need to understand why massive SMBH and spheroids form before their less massive counterparts. |
The required scaling for LAGN oy Macer is reminiscent of the scaling found for proto-stellar jets. | The required scaling for $L^{AGN}$ or $M_{accr}$ is reminiscent of the scaling found for proto-stellar jets. |
The magnetically-regulated disk phenomenon plausibly obeys a universal scaling law, that could equally apply to jets and outflows from disks around SMBH. | The magnetically-regulated disk phenomenon plausibly obeys a universal scaling law, that could equally apply to jets and outflows from disks around SMBH. |
TheM? proto-stellar[Allenetal scaling is X,ος (2006) find that for the black holes that power the AGNs in massive ellipticals, the Bondi accretion rate is approximately proportional to the jet power. | The proto-stellar scaling is \citep{moh05} $\dot M_{acc}\propto M^2.$ \citet{alle06} find that for the black holes that power the AGNs in massive ellipticals, the Bondi accretion rate is approximately proportional to the jet power. |
The connection with outflows and jets that are magnetically guided by the wound-up field in the accretion disk proposed by (2006) is a generic scaling in their study of proto-stellar jets, =—fwMace, with fy~0.1, for outflows associated withMAGN central objects that range from brown dwarfs to super-massive black holes. | The connection with outflows and jets that are magnetically guided by the wound-up field in the accretion disk proposed by \citet{ban06}
is a generic scaling in their study of proto-stellar jets, $\dot M_{wind}^{AGN} =f_w \dot M_{acc},$ with $f_w\sim 0.1, $ for outflows associated with central objects that range from brown dwarfs to super-massive black holes. |
The Bondi accretion formula therefore regulates SMBH growth and, implicitly, outflow. | The Bondi accretion formula therefore regulates SMBH growth and, implicitly, outflow. |
It yields Mey=7G?Μηη”. A simple interpretation of this scaling(p,/o°) is that for Bondi accretion, in combination with adiabatic so that pocc?. For the AGN case, we writecompression, Mace=aM2g with axG?p/a?. Rewriting this we see that Therefore if Q reaches a self-regulating constant and fr and f, are also constant then the black hole and galaxy growth move together on a fixed trajectory in the Magorrian plane as discussed below. | It yields $ {\dot M_{BH}} = \pi G^2\left({p_g / \sigma^5}\right) {M_{BH}}^2 .$ A simple interpretation of this scaling is that for Bondi accretion, in combination with adiabatic compression, so that $\rho\propto \sigma^3.$ For the AGN case, we write $\dot M_{acc}=\alpha M_{BH}^2$ with $\alpha \propto {G^2\rho/\sigma^3}.$ Rewriting this we see that Therefore if $Q$ reaches a self-regulating constant and $f_L$ and $f_w$ are also constant then the black hole and galaxy growth move together on a fixed trajectory in the Magorrian plane as discussed below. |
Substituting further the relevant quantities for the case of SN energy input we find for the case without AGN feedback where Thus the critical parameter determining the logarithmic slope in the Magorrian plane is ἐς. | Substituting further the relevant quantities for the case of SN energy input we find for the case without AGN feedback where Thus the critical parameter determining the logarithmic slope in the Magorrian plane is $ {\bar t_S}$. |
There is a critical density perit= above which black hole growth dominates and (Gfgts)~?below which star formation dominates. | There is a critical density $\rho_{crit} = (G f_g {\bar t_S})^{-2}$ above which black hole growth dominates and below which star formation dominates. |
This can be rewritten in terms of a critical velocity dispersion if one takes ρε to be the density at the edge of the Bondi accretion sphere (the sphere of influence of the black hole, Razz) namely p~(Mfr?)(Meu/reu®)(G-0°Mgu”) giving then an equivalent σοι which is a satisfying combination of black hole-galaxy and ISM properties. | This can be rewritten in terms of a critical velocity dispersion if one takes $\rho_{crit}$ to be the density at the edge of the Bondi accretion sphere (the sphere of influence of the black hole, $R_{BH}$ ) namely $\rho \sim (M/r^3)\sim (M_{BH} / {r_{BH}}^3) \sim (G^{-3} \sigma^6 {M_{BH}}^{-2})$ giving then an equivalent $\sigma_{crit}$ which is a satisfying combination of black hole-galaxy and ISM properties. |
Continuing to the case with AGN feedback, we find | Continuing to the case with AGN feedback, we find |
The accuracy of this fitting formula is demonstrated in Figures and [D]. | The accuracy of this fitting formula is demonstrated in Figures \ref{fig:sstar} and \ref{fig:sfall}. |
For very low values of Duw<0.01 the fit is [6]not very accurate. | For very low values of $\D\lesssim0.01$ the fit is not very accurate. |
This is most likely due to the limited volume of our simulations: at such low dust-to-gas ratios the atomic-to-molecular transition shifts to extremely high gas densities, my~10?cm, and our simulations lack 500pc sized regions that would be dominated by such dense gas. | This is most likely due to the limited volume of our simulations: at such low dust-to-gas ratios the atomic-to-molecular transition shifts to extremely high gas densities, $n_\Ht\sim10^3\dim{cm}^{-3}$, and our simulations lack $500\dim{pc}$ sized regions that would be dominated by such dense gas. |
Large volume simulations containing substantially more massive galaxies will be need to test the accuracy of the fitting formula (14) in this regime. | Large volume simulations containing substantially more massive galaxies will be need to test the accuracy of the fitting formula \ref{eq:sstar}) ) in this regime. |
Finally, we have checked that our results are not particularly sensitive to the specific choice of the fiducial parameters esr and risp. | Finally, we have checked that our results are not particularly sensitive to the specific choice of the fiducial parameters $\epsilon_{\rm SF}$ and $n_{\rm SF}$. |
While the fiducial values provide the best fit to the median values of THINGS measurements(?),, a substantial variation in the adopted values for these parameters has only mild effecton our results, as we demonstrate in Figure[7. | While the fiducial values provide the best fit to the median values of THINGS measurements, a substantial variation in the adopted values for these parameters has only mild effecton our results, as we demonstrate in Figure \ref{fig:sflpars1}. . |
Cepheids against their pulsation period. P. | Cepheids against their pulsation period $P$. |
Fundamental nmiode pulsators ancl overtone pulsators are displayed: with different symbols. | Fundamental mode pulsators and overtone pulsators are displayed with different symbols. |
V440 Per is plotted with an open circle. | V440 Per is plotted with an open circle. |
It is immediately. obvious. that it is located far apart from the fundamental mode progression. | It is immediately obvious, that it is located far apart from the fundamental mode progression. |
This notion can be put on à quantitative basis. | This notion can be put on a quantitative basis. |
We selected a sample of 23 fundamental mode Cepheids with periods 2 in the range of Ztx:1.1 day. where £4)=7.5121 day is the period of V440 Per. | We selected a sample of 23 fundamental mode Cepheids with periods $P$ in the range of $P_0\pm1.1$ day, where $P_0=7.5721$ day is the period of V440 Per. |
To this sample we fitted a straight line σι}=atc?1)| 6. | To this sample we fitted a straight line $\phi_{21}(P) =
a(P-P_0)+b$ . |
With this procedure. we find that at the period of V440 Per the expected: o», of the fundamental moce Copheicl is rad. with the average scatter of individual values of συ=0.026 rad. | With this procedure, we find that at the period of V440 Per the expected $\phi_{21}$ of the fundamental mode Cepheid is rad, with the average scatter of individual values of $\sigma_0=0.026$ rad. |
This estimate of the intrinsic scaller is conservative. as the nominal ó», measurement errors would account for at least half of it. | This estimate of the intrinsic scatter is conservative, as the nominal $\phi_{21}$ measurement errors would account for at least half of it. |
The G2, value measured for V440 Per is rack. with an error of σι= .IlTrad. | The $\phi_{21}$ value measured for V440 Per is rad, with an error of $\sigma_1=0.117$ rad. |
Phe probability. distribution of the o», olfset. εν, is obtained by convolution of two normal clistributions. Αθσα) anc IN(GNσι). | The probability distribution of the $\phi_{21}$ offset, $\Delta$, is obtained by convolution of two normal distributions, $N(0,\sigma_0)$ and $N(\Delta,\sigma_1)$. |
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