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As a result. a shorter interval to the next state transition would be expected. | As a result, a shorter interval to the next state transition would be expected. |
Unlike the SN'Ts in which the outburst timescale is normally longer than ten vears. 3339.4 has much more frequent state transitions. às seen in e.g. the AASAI data (see Fig. 2)). | Unlike the SXTs in which the outburst timescale is normally longer than ten years, 339–4 has much more frequent state transitions, as seen in e.g. the ASM data (see Fig. \ref{fig:ginga}) ). |
In fact. the 232-d variability seen in the aand data has already hinted at such behaviour. | In fact, the 232-d variability seen in the and data has already hinted at such behaviour. |
The most interesting behaviour of the 1998 LIS is tha he observed. [lat-topped: X-ray. light: curve of 3339.4 »ears Characteristics of Z Cam-type DN in which standstills (i.c. the brightness remains constant) occasionally interrup he recurrent outbursts. | The most interesting behaviour of the 1998 HS is that the observed flat-topped X-ray light curve of 339–4 bears characteristics of Z Cam-type DN in which standstills (i.e. the brightness remains constant) occasionally interrupt the recurrent outbursts. |
Such a scenario can be explainec wv the DIM (Meyer&Alever-Lolmeister1983:WineCan-nizzo1998) in which the accretion rate before standstil is very close to a critical value and a sudden increase in he accretion rate (c.g. bx a starspot or irradiation induce mass overllow) triggers the standstill state. | Such a scenario can be explained by the DIM \cite{meyer83,king98} in which the accretion rate before standstill is very close to a critical value and a sudden increase in the accretion rate (e.g. by a starspot or irradiation induced mass overflow) triggers the standstill state. |
Phe light curve of 33394 is even more similar to the model proposed by Ixing&Cannizzo(1998) in which the stancdstill is at a higher luminosity than the normal outbursts. | The light curve of 339–4 is even more similar to the model proposed by \scite{king98} in which the standstill is at a higher luminosity than the normal outbursts. |
For Z Cam systems. the intensity of the standstills is lower than the maximum of the outbursts (Alever&Mever-Llofmeister 1983)). but we should note that the models for explaining the Z Cam systems are used to account for optical light curves. | For Z Cam systems, the intensity of the standstills is lower than the maximum of the outbursts \pcite{meyer83}) ), but we should note that the models for explaining the Z Cam systems are used to account for optical light curves. |
The second. similarity between 3339.4 and Z Com systems concerns the recurrence of outbursts between stancdstills. | The second similarity between 339–4 and Z Cam systems concerns the recurrence of outbursts between standstills. |
From our earlier period. analysis of the archival 33394 data over 30 vears. we find a variability timescale of 7190240 d and we suggest this to be the recurrent outbursting behaviour seen in Z Cam systems. | From our earlier period analysis of the archival 339–4 data over 30 years, we find a variability timescale of $\sim$ 190–240 d and we suggest this to be the recurrent outbursting behaviour seen in Z Cam systems. |
As pointed out by Dubus et al. ( | As pointed out by Dubus et al. ( |
in preparation). the accretion rate can vary on timescales of 10.100 cavs when a viscously unstable disce is irracliatecl by a constant N-rav. flux. | in preparation), the accretion rate can vary on timescales of 10–100 days when a viscously unstable disc is irradiated by a constant X-ray flux. |
HU irradiation is crucial for determining the acerction rate of 3339.4. this mav explain the variability of the "mini-outbursts observed during the LS. | If irradiation is crucial for determining the accretion rate of 339–4, this may explain the variability of the `mini'-outbursts observed during the LS. |
Phere is also an indication that the accretion rate of 33394 was building up before the LES in 1998 as 1ο average intensity level of both AASM and BATSE data has a slight. increasing trend especially after ALJD 50000 (see Fig. 3)). | There is also an indication that the accretion rate of 339–4 was building up before the HS in 1998 as the average intensity level of both ASM and BATSE data has a slight increasing trend especially after MJD 50000 (see Fig. \ref{fig:batse}) ). |
In addition. the LS occurring at NLJD 48550 and 49450 (see refsec:ibatse)) were just after a strong X-ray outburst in the LS (see Fig. 3)). | In addition, the HS occurring at MJD 48550 and 49450 (see \\ref{sec:batse}) ) were just after a strong X-ray outburst in the LS (see Fig. \ref{fig:batse}) ). |
Perhaps this takes the accretion rate in the LS to be close to the critical value and finally triggers the state transition. | Perhaps this takes the accretion rate in the LS to be close to the critical value and finally triggers the state transition. |
As a result. this LIS may be simply a garong. prolonged outburst in an otherwise LS. | As a result, this HS may be simply a strong, prolonged outburst in an otherwise LS. |
In fact. from 10 period analysis of the2D ancl ddata for which state transitions are included. in the calculation. the timescale of variability resembles that found in the AASM data in the LS. | In fact, from the period analysis of the and data for which state transitions are included in the calculation, the timescale of variability resembles that found in the ASM data in the LS. |
Perhaps the LS outbursts may be excursions towards the LES but which they do not. quite reach. | Perhaps the LS outbursts may be excursions towards the HS but which they do not quite reach. |
Lf this is the case. we would expect similar spectral evolution during the LS outhurst. just like during a state transition. | If this is the case, we would expect similar spectral evolution during the LS outburst, just like during a state transition. |
Lowever. such outbursts are usually short and therefore. observations are clillicult to arrange. | However, such outbursts are usually short and therefore observations are difficult to arrange. |
The most likely observations available are from Wilmsetal.(1999).. in which an ppointecl observation. of 3339.4. was mace near the outburst (AIJD 50710) just. before the LS/IS transition. | The most likely observations available are from \scite{wilms99}, in which an pointed observation of 339–4 was made near the outburst (MJD 50710) just before the LS/HS transition. |
Although the energy spectrum appeared to be slightly softer. it is still consistent with other LS observations within the uncertainties. | Although the energy spectrum appeared to be slightly softer, it is still consistent with other LS observations within the uncertainties. |
Lt is worth noting that XX1 also shows LS/LS transitions occasionally ancl it has Daring activities during the LS. | It is worth noting that X–1 also shows LS/HS transitions occasionally and it has flaring activities during the LS. |
Moreover. the long LIS of XX.1 between 1996 Alay ancl August resembles that seen in 3339.4. but with more Lares during the LIS (Zhangetal.1997). | Moreover, the long HS of X–1 between 1996 May and August resembles that seen in 339–4, but with more flares during the HS \cite{zhang97}. |
. Therefore. it is possible that the state transition of NNl is due to a similar mechanism to that. discussed. here (sce however Zhangetal.1997:Esin1998)). | Therefore, it is possible that the state transition of X--1 is due to a similar mechanism to that discussed here (see however \pcite{zhang97,esin98}) ). |
More recently. XX.3 was also shown to have recurring state transitions but the driving force is very likely to be due to other mechanisms (Boveetal.2000:Wilms2001). | More recently, X–3 was also shown to have recurring state transitions but the driving force is very likely to be due to other mechanisms \cite{boyd00,wilms00}. |
. Given that the accretion rate in the LS is close to the critical value. similar LS/LIS transitions would be expected in the near future and multi-waveleneth observations of such transitions. particularly the correlation between the N-ray. optical and radio emissions will constrain the accretion disc structure and behaviour of the secondary star. | Given that the accretion rate in the LS is close to the critical value, similar LS/HS transitions would be expected in the near future and multi-wavelength observations of such transitions, particularly the correlation between the X-ray, optical and radio emissions will constrain the accretion disc structure and behaviour of the secondary star. |
We are grateful to Colleen Wilson-Llodge and Ken Watanabe [or providing the updated: BATSE data. | We are grateful to Colleen Wilson-Hodge and Ken Watanabe for providing the updated BATSE data. |
We also thank Christine Done for the red-noise generator code. | We also thank Christine Done for the red-noise generator code. |
Alytis is supported by a Hong Kong Oxford Scholarship. | AKHK is supported by a Hong Kong Oxford Scholarship. |
This paper utilizies quick-look results provided. by the ASM/IUCNTIEE team and data obtained. through the WEASARC Online Services of NASA/GSEC. | This paper utilizies quick-look results provided by the ASM/RXTE team and data obtained through the HEASARC Online Services of NASA/GSFC. |
33394 has been observed by several X-ray satellites in the past 30 vears and the results [rom those pointed observations are crucial to determine the ‘state of. the source. | 339–4 has been observed by several X-ray satellites in the past 30 years and the results from those pointed observations are crucial to determine the `state' of the source. |
We compile here a list. of pointed. observations of 33394 from the literature (see Table AL). | We compile here a list of pointed observations of 339–4 from the literature (see Table A1). |
Note that the “state? quoted is determined. by spectral and temporal analysis. | Note that the `state' quoted is determined by spectral and temporal analysis. |
Although the X-ray llux level can more or less reflect the ‘state’ of the source. on several occasions the ray [lux in the LS is actually higher than the LIS (e.g. 1981 May ancl 1984 Mas). | Although the X-ray flux level can more or less reflect the `state' of the source, on several occasions the X-ray flux in the LS is actually higher than the HS (e.g. 1981 May and 1984 May). |
Hence the X-ray intensity itself is not an accurate indicator of X-ray ‘state’. | Hence the X-ray intensity itself is not an accurate indicator of X-ray `state'. |
distributions. while data allow us to probe a regionlarger than these radii. which permits us to investigate the core radius phenomenon at large clistances ancl consequently to constrain the size of the core radii (note. however. the case of NGC 7339 discussed in Appendix A). | distributions, while data allow us to probe a region than these radii, which permits us to investigate the core radius phenomenon at large distances and consequently to constrain the size of the core radii (note, however, the case of NGC 7339 discussed in Appendix A). |
The rotation curves used. here consist of the Lla data for the inner parts and the data for the outer parts. | The rotation curves used here consist of the $\alpha$ data for the inner parts and the data for the outer parts. |
The claim of unreliability of observed kinematics put forward by lthee.IXIvpin&Valenzuela(2003). does not apply here: a) the derivation of the rotation curve was performed in a more thorough and. reliable way than the standard. tiltecd-ring analvsis on the velocity field: b) the galaxies studied here were at an inclination optimal to minimise projection elfects: c) in the regions were they coexist. the Ho and data (emerging from cillerent. physical processes) do agree. iniplving a high quality of thedatat. | The claim of unreliability of observed kinematics put forward by \citet*{Rh:03} does not apply here: a) the derivation of the rotation curve was performed in a more thorough and reliable way than the standard tilted-ring analysis on the velocity field; b) the galaxies studied here were at an inclination optimal to minimise projection effects; c) in the regions were they coexist, the $\alpha$ and data (emerging from different physical processes) do agree, implying a high quality of the. |
. The small internal scatter inside the cach radial bin indicates that. for the galaxies of our sample. the ellects of non-circular motions are negligible (even though a more thorough investigation should involve 2-dimensional Ho cata). cillerently from the case shown by Swatersetal.(2003b). | The small internal scatter inside the each radial bin indicates that, for the galaxies of our sample, the effects of non-circular motions are negligible (even though a more thorough investigation should involve 2-dimensional $\alpha$ data), differently from the case shown by \citet{Sw:03}. |
. Moreover. it can be shown (see e.g. deBlok.MeGaugh&Dosma 2003)) that the bias towards lower velocity gracients in the rotation curve due to a misalignment between the slit with the kinematical axis is likely to be small. | Moreover, it can be shown (see e.g. \citealt{dB:03}) ) that the bias towards lower velocity gradients in the rotation curve due to a misalignment between the slit with the kinematical axis is likely to be small. |
The five galaxies of this sample were chosen from the sample of 967 galaxies with optical rotation curves presented by Persic&Salucei(1995).. according to the following criteria: a) they belong to the "exeellent subsample (i.e. the two sides are svmmetric. the cata are extended out to at. least ro and the number of data points is z 30): b) they have a reasonably high total flux: ο) they have a relatively low I-band luminosity (AZ;7 21.8). d) they have a large angular size. and e) they have an inclination / suitable forLIL studies (50°r<7< 857). | The five galaxies of this sample were chosen from the sample of 967 galaxies with optical rotation curves presented by \citet{PS:95}, according to the following criteria: a) they belong to the “excellent” subsample (i.e. the two sides are symmetric, the data are extended out to at least $r_{opt}$ and the number of data points is $\geq$ 30); b) they have a reasonably high total flux; c) they have a relatively low I-band luminosity $M_{\rm I} > -21.8$ ), d) they have a large angular size, and e) they have an inclination $i$ suitable for studies $50^{\circ} < i < 85 ^{\circ}$ ). |
For each galaxy. the raw Lla data were binned in groups of 4 to 6 (Persic&Salucei1995).. and the given error is the uncertainty on the average value inside the radial bin. | For each galaxy, the raw $\alpha$ data were binned in groups of 4 to 6 \citep{PS:95}, and the given error is the uncertainty on the average value inside the radial bin. |
We took a "minimum error equal to half the average error in order to avoid data points. with unrealistically low errors that could bias the rotation curves decompositions: for the same reason. when a data point was Clearly inconsistent with the two neighbouring points and the general trend of the rotation curve. the value of its error was increased. | We took a “minimum error” equal to half the average error in order to avoid data points with unrealistically low errors that could bias the rotation curves decompositions; for the same reason, when a data point was clearly inconsistent with the two neighbouring points and the general trend of the rotation curve, the value of its error was increased. |
The complete description. of the optical observations (spectroscopic and photometric) can be found in Mathewson.Ford&Buchhorn(1992) and Persic&Salucci (1995). | The complete description of the optical observations (spectroscopic and photometric) can be found in \citet*{Ma:92} and \citet{PS:95}. |
. We observed ESO 116-G12. ESO 287-G13 and ESO 79-Ci14 with the Australia “Telescope Compact Array in the 750 m. and 1.5 km configurations: the resulting baselines range from 31 mto 1500 m. The galaxies were observed for 12 hrs in cach configuration. | We observed ESO 116-G12, ESO 287-G13 and ESO 79-G14 with the Australia Telescope Compact Array in the 750 m and 1.5 km configurations; the resulting baselines range from 31 m to 1500 m. The galaxies were observed for 12 hrs in each configuration. |
The correlator setup vielded 512 channels of | The correlator setup yielded 512 channels of |
Lt is welbestablished that rotation is a function of object mass (see?.forareview)... | It is well-established that rotation is a function of object mass \citep[see][for a review]{2007prpl.conf..297H}. |
Specifically. there is accumulating evidence that in the very low mass regime the average period drops steadily with decreasing mass. | Specifically, there is accumulating evidence that in the very low mass regime the average period drops steadily with decreasing mass. |
This positive period-mass correlation at masses <0.8O44. has been found in the ONC (?).. ©OOrt (7).. and the Pleiacles (?).. clusters with ages between 1 and MMwyr. | This positive period-mass correlation at masses $<0.3-0.4\,M_{\odot}$ has been found in the ONC \citep{2001ApJ...554L.197H}, $\epsilon$ Ori \citep{2005A&A...429.1007S}, and the Pleiades \citep{2004A&A...421..259S}, clusters with ages between 1 and Myr. |
Similarly. in the clusters Νάς25160 ancl M34 (22)... which roughly mark the ZAMS for very low mass objects at ages of 150 and 200Myr. there seems to be a general decline of the upper envelopeM of the periods with decreasing mass (7.seetheirFig.17).. | Similarly, in the clusters NGC2516 and M34 \citep{2006MNRAS.370..954I,2007MNRAS.tmp..276I}, which roughly mark the ZAMS for very low mass objects at ages of 150 and Myr, there seems to be a general decline of the upper envelope of the periods with decreasing mass \citep[][see their Fig. 17]{2007MNRAS.tmp..276I}. |
]t has been pointed. out that this trend. is consistent. with constant angular momentum for all object masses (e.g.?).. and thus might just reflect the drop in stellar radius. | It has been pointed out that this trend is consistent with constant angular momentum for all object masses \citep[e.g.][]{2001ApJ...554L.197H}, and thus might just reflect the drop in stellar radius. |
Since it is already. established at very voung ages. it may be related to the initial distribution of angular momentum. | Since it is already established at very young ages, it may be related to the initial distribution of angular momentum. |
Llere we probe if our small period sample in. Praesepe older than all other clusters with rotation periods in this mass range allows us to see a similar trend. | Here we probe if our small period sample in Praesepe – older than all other clusters with rotation periods in this mass range – allows us to see a similar trend. |
We derived masses for the five objects in Table 2. by comparing the available photometry in the Lhand from the literature with evolutionary tracks from ? for an age of GiCivr. | We derived masses for the five objects in Table \ref{periods} by comparing the available photometry in the I-band from the literature with evolutionary tracks from \citet{1998A&A...337..403B}
for an age of Gyr. |
According to these estimates. all five objects are in the very low mass regime with masses 0.4AL... | According to these estimates, all five objects are in the very low mass regime with masses $\le 0.4\,M_{\odot}$. |
Due to age uncertainties. photometric band inconsistencies. ancl mocel shortcomüngs. the uncertainties in the derived. masses. are considerable (probably on the order of )). | Due to age uncertainties, photometric band inconsistencies, and model shortcomings, the uncertainties in the derived masses are considerable (probably on the order of ). |
However. most of this uncertainty is svstematic. thus we expect that in a relative sense our masses are realistic. | However, most of this uncertainty is systematic, thus we expect that in a relative sense our masses are realistic. |
We caution against comparing these values with masses derived using a dilferent approach. | We caution against comparing these values with masses derived using a different approach. |
Using the same model isochrone. we determined ellective temperatures for our five targets. | Using the same model isochrone, we determined effective temperatures for our five targets. |
Comparing with the Zi scale from ? gives an estimate for the spectral types (sce “Table 2)). | Comparing with the $T_{\mathrm{eff}}$ scale from \citet{2003ApJ...593.1093L} gives an estimate for the spectral types (see Table \ref{periods}) ). |
The uncertainties in these “photometric? spectral tvpes are probably +12 subclasses. | The uncertainties in these 'photometric' spectral types are probably $\pm 1-2$ subclasses. |
In Fig. 2.. | In Fig. \ref{f2}, |
upper panel. we plot periods vs. masses for 16 five objects in Table 2.. | upper panel, we plot periods vs. masses for the five objects in Table \ref{periods}. |
Lt is immediately obvious that 10 periods appear to increase with mass in a roughly linear wav. | It is immediately obvious that the periods appear to increase with mass in a roughly linear way. |
A linear least-square fit (shown as dotted line) gives: P=(24040)(ΑΛ(21x10)h. | A linear least-square fit (shown as dotted line) gives: $P = (240 \pm 40)\,(M/M_{\odot}) - (21\pm10)\,h$. |
The correlation is significant with a false alarm probability of4. | The correlation is significant with a false alarm probability of. |
. The slope in the relationship is clearly steeper than in the Pleiades (105€GLCAZI/AL.). 2)). maybe indicating that rotational braking on the main-sequence is a function of object mass. | The slope in the relationship is clearly steeper than in the Pleiades $105 \pm 61\,(M/M_{\odot})$, \citet{2004A&A...421..259S}) ), maybe indicating that rotational braking on the main-sequence is a function of object mass. |
Based on only five datapoints. however. this finding is of somewhat limited value. | Based on only five datapoints, however, this finding is of somewhat limited value. |
Clearly. more datapoints are needed to solidifv the main-sequence P-M correlation in the VEM Lt is more instructive to look on the VLM periods in comparison with rotation measurements obtained for more massive stars. | Clearly, more datapoints are needed to solidify the main-sequence P-M correlation in the VLM It is more instructive to look on the VLM periods in comparison with rotation measurements obtained for more massive stars. |
To our knowledge. such data is not available [or Praesepe. but for the ELEvades. with 600Myr a roughly coeval cluster (22)... | To our knowledge, such data is not available for Praesepe, but for the Hyades, with Myr a roughly coeval cluster \citep{1981A&A....97..235M,1998A&A...331...81P}. |
For this comparison. we prefer to use spectral types instead. of masses. in order not to be biased » model inconsistencies in the mass estimates. which are illieult to avoid when covering a mass range of more than one order of magnitude. | For this comparison, we prefer to use spectral types instead of masses, in order not to be biased by model inconsistencies in the mass estimates, which are difficult to avoid when covering a mass range of more than one order of magnitude. |
Since we use only coeval objects. Ίο spectral type is a valid. indicator for stellar mass. | Since we use only coeval objects, the spectral type is a valid indicator for stellar mass. |
We ‘olleetecl a sample of 25 periods for Hades members from 16 photometric monitoring campaigns published by 7? and S | We collected a sample of 25 periods for Hyades members from the photometric monitoring campaigns published by \citet{1987ApJ...321..459R} and \citet{1995PASP..107..211P}. |
pectral types for. these objects have originally been rublished by 2? and. ον. | Spectral types for these objects have originally been published by \cite{1952BAN....11..385V} and \citet{1969AJ.....74....2V}. |
Combined. with our sample. the spectral range from late Foto late M is covered. roughly corresponding to a mass range from 0.1 to 2AL.. | Combined with our sample, the spectral range from late F to late M is covered, roughly corresponding to a mass range from 0.1 to $M_{\odot}$. |
The period/spectral type relation is shown in Fig. 2.. | The period/spectral type relation is shown in Fig. \ref{f2}, |
lower panel. | lower panel. |
Lt clearly demonstrates that for. E-Ix. stars the periods increase towards later spectral types. | It clearly demonstrates that for F-K stars the periods increase towards later spectral types. |
According to ?.. these periods can be explained in terms of the correlation between magnetic activity and the inverse Rossby number Ro. the ratio between rotation period and convective turnover timescale το (e.g.οι. | According to \citet{1987ApJ...321..459R}, these periods can be explained in terms of the correlation between magnetic activity and the inverse Rossby number $Ro$, the ratio between rotation period and convective turnover timescale $\tau_C$ \citep[e.g.][]{1984ApJ...279..763N}. |
This relation basically implies that the magnetic field amplification mainly depencls on convection properties and rotation supporting the idea of an aw dyvnamo operating in F-Ix stars. | This relation basically implies that the magnetic field amplification mainly depends on convection properties and rotation – supporting the idea of an $\alpha\omega$ dynamo operating in F-K stars. |
We note that the Ilvades periods for E-Ix stars represent what ο called the | We note that the Hyades periods for F-K stars represent what \cite{2003ApJ...586..464B} called the |
with Az=x—rg and Ay=y—yo. | with $\Dx \equiv x - x_0$ and $\Dy \equiv y - y_0$. |
Note forp=0 (uncorrelated x and y), the x? looks familiar. | Note for $\rho = 0$ (uncorrelated $x$ and $y$ ), the $\chi^2$ formula looks familiar. |
For correlated x and y (p> 0), x? is formulareduced. | For correlated $x$ and $y$ $\rho > 0$ ), $\chi^2$ is reduced. |
Consider a Fisher matrix provided by the DETF (Table for optimistic Stage IV BAO observations for the []))following variables: Qa, Ωμ), where Wm=Qh? and OQ,--Q4Q0;=1. | Consider a Fisher matrix provided by the DETF (Table \ref{tab:fish}) ) for optimistic Stage IV BAO observations for the following variables: $\om, \OL, \Ok$ ), where $\om \equiv \Om h^2$ and $\Om + \OL + \Ok = 1$. |
The (Wm,covariance matrix (inverse of the Fisher matrix) is given in Table | The covariance matrix (inverse of the Fisher matrix) is given in Table \ref{tab:cov}. |
For example, the top- element tells us that Aw,,5”Bl.0.00566z3.20E—5. | For example, the top-left element tells us that $\Delta \om \approx 0.00566 \approx \sqrt{3.20E-5}$. |
When quoting these uncertainties on Ww», the other variables (Q4, Ωκ) have automatically been marginalized over. | When quoting these uncertainties on $\om$, the other variables $\OL, \Ok$ ) have automatically been marginalized over. |
That is, their probabilities have been integrated over: they have been set free to hold any values while we calculate the range of acceptable wy. | That is, their probabilities have been integrated over: they have been set free to hold any values while we calculate the range of acceptable $\om$. |
To calculate a new Fisher matrix marginalized over any variable, simply remove that variable's row and column from the covariance matrix, and take the inverse of that to yield the new Fisher matrix. | To calculate a new Fisher matrix marginalized over any variable, simply remove that variable's row and column from the covariance matrix, and take the inverse of that to yield the new Fisher matrix. |
Suppose instead want the opposite: perfect knowledge of a parameter. | Suppose instead want the opposite: perfect knowledge of a parameter. |
For example, we want to consider a flat universe with a fixed value of Ωμ=0. | For example, we want to consider a flat universe with a fixed value of $\Ok = 0$. |
To do this, simply remove Ωμ from the Fisher matrix )). | To do this, simply remove $\Ok$ from the Fisher matrix (Table \ref{tab:fishflat}) ). |
The new covariance matrix and parameter uncertainties(Table are calculated from the revised Fisher matrix. | The new covariance matrix and parameter uncertainties are calculated from the revised Fisher matrix. |
Alternatively, the on-diagonal element corresponding to that parameter can be set to a very large value. | Alternatively, the on-diagonal element corresponding to that parameter can be set to a very large value. |
For example, if we set the bottom-right element in Table to 1013, that would correspond to a 1079 uncertainty in Wm, or nearly fixed. | For example, if we set the bottom-right element in Table \ref{tab:fish} to $10^{12}$, that would correspond to a $10^{-6}$ uncertainty in $\om$, or nearly fixed. |
Note that higher values in the Fisher matrix correspond to higher certainty. | Note that higher values in the Fisher matrix correspond to higher certainty. |
Rather than fixing a parameter to an exact value,(1- we may want to place a prior such as AQ,=0.01 e). | Rather than fixing a parameter to an exact value, we may want to place a prior such as $\Delta \Ok = 0.01$ $\sigma$ ). |
In this case,corresponding simply add 1/σ3=103 to the on-diagonal element to that variable (in this case, the bottom left element). | In this case, simply add $1 / \sigma^2 = 10^4$ to the on-diagonal element corresponding to that variable (in this case, the bottom left element). |
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