#  BeppoSAX observations of the X–ray pulsar MAXI J1409$-$619 in low state:
discovery of cyclotron resonance features

Mauro Orlandini11affiliation: INAF/Istituto di Astrofisica Spaziale e Fisica
Cosmica (IASF) Bologna, via Gobetti 101, 40129 Bologna, Italy , Filippo
Frontera11affiliation: INAF/Istituto di Astrofisica Spaziale e Fisica Cosmica
(IASF) Bologna, via Gobetti 101, 40129 Bologna, Italy 22affiliation:
Dipartimento di Fisica, Università degli Studi di Ferrara, via Saragat 1,
44122 Ferrara, Italy , Nicola Masetti11affiliation: INAF/Istituto di
Astrofisica Spaziale e Fisica Cosmica (IASF) Bologna, via Gobetti 101, 40129
Bologna, Italy , Vito Sguera11affiliation: INAF/Istituto di Astrofisica
Spaziale e Fisica Cosmica (IASF) Bologna, via Gobetti 101, 40129 Bologna,
Italy , Lara Sidoli33affiliation: INAF/Istituto di Astrofisica Spaziale e
Fisica Cosmica (IASF) Milano, via Bassini 15, 20133 Milano, Italy

###### Abstract

The transient 500 s X–ray pulsar MAXI J1409$-$619 was discovered by the slit
cameras aboard MAXI on October 17, 2010, and soon after accurately localized
by Swift. We found that the source position was serendipitously observed in
2000 during BeppoSAX observations of the Galactic plane. Two sources are
clearly detected in the MECS: one is consistent with the position of IGR
J14043$-$6148 and the other one with that of MAXI J1409$-$619\. We report on
the analysis of this archival BeppoSAX/MECS observation integrated with newly
analyzed observation from ASCA and a set of high-energy observations obtained
from the offset fields of the BeppoSAX/PDS instrument. For the ON-source
observation, the 1.8–100 keV spectrum is fit by an absorbed power law with a
photon index $\Gamma=0.87_{-0.19}^{+0.29}$, corresponding to 2–10 and 15–100
keV unabsorbed fluxes of $2.7\times 10^{-12}$ and $4\times 10^{-11}$ erg cm-2
s-1, respectively, and a 2–10 keV luminosity of $7\\!\times\\!10^{34}$ erg s-1
for a 15 kpc distance. For a PDS offset field observation, performed about one
year later and showing a 15–100 keV flux of $7\times 10^{-11}$ erg cm-2 s-1,
we clearly pinpoint three spectral absorption features at 44, 73, and 128 keV,
resolved both in the spectral fit and in the Crab ratio. We interpret these
not harmonically spaced features as due to cyclotron resonances. The
fundamental energy of 44$\pm$3 keV corresponds to a magnetic field strength at
the neutron star surface of $3.8\times 10^{12}(1+z)$ G, where $z$ is the
gravitational redshift. We discuss the nature of the source in the light of
its possible counterpart.

###### Subject headings:

X–rays: binaries — Stars: individual: MAXI J1409$-$619 — Stars: individual:
IGR J14043$-$6148

††slugcomment: The Astrophysical Journal, 747:, 2012 March 1

##  1\. Introduction

On October 17, 2010 Yamaoka et al. (2010b) reported on an outburst from a new
source, named MAXI J1409$-$619, revealed by the Gas Slit Camera (GSC; Mihara
et al. 2002) of the Monitor of All-sky X-ray Image (MAXI; Matsuoka et al.
2009) experiment aboard the International Space Station. The 4–10 keV flux
decreased from its initial value of $\sim$40 mCrab on October 17 to $\sim$30
mCrab on the following day.

On October 20 Swift (Gehrels et al. 2004) began a target of opportunity
observation of the MAXI J1409$-$619 error circle (0$\fdg$2 radius). Kennea et
al. (2010b) used the X–Ray Telescope (XRT; Burrows et al. 2005) to pinpoint
the position of the new source with an estimated 90% confidence level
uncertainty of 1$\farcs$9 and coordinates RA(J2000): 14h 08m 02$\fs$56,
Dec(J2000): $-$61° 59′ 00$\farcs$3\. Its spectrum is fit by an absorbed power
law, with photon index $\Gamma=-0.5^{+0.1}_{-0.6}$. A follow-up XRT
observation on November 30 found the source $\sim$7 times brighter than on
October 20, with an average 0.3–10 keV flux of $7\times 10^{-10}$ erg cm-2 s-1
(Kennea et al. 2010a). Only in this latter observation a $\sim$500 s pulsation
was detected, with a 42% sinusoidal rms modulation. The source was detected
since October 18 also in the 15–50 keV energy band by the Swift Burst Alert
Telescope (BAT; Barthelmy et al. 2005) at the level of $\sim$30 mCrab (Kennea
et al. 2011).

Neither catalogued radio nor X–ray sources were present in the Swift error
circle, while a 2MASS IR star, the likely IR counterpart of the X–ray
transient, lays 2$\farcs$1 from the XRT position (Kennea et al. 2010b).

The source was also observed on October 22 by the Proportional Counter Array
(PCA; Jahoda et al. 1996) aboard RXTE (Bradt et al. 1993). The X–ray spectrum
shows a strong 6.5 keV Iron line (E.W. 200 eV), and a continuum modeled by
reflection or partially covering absorption (Yamaoka et al. 2010a). The 2–10
keV flux was about $10^{-10}$ erg cm-2 s-1, and was strongly variable on time-
scales of hundreds of seconds. An observation performed on December 4 found
MAXI J1409$-$619 at a flux level 6–7 times higher than the October observation
(Yamamoto et al. 2010), and confirmed the 500 s pulse period observed by
Swift.

From observations performed on December 2 and 3 by the GLAST Burst Monitor
(GBM; Meegan et al. 2007) aboard the Fermi satellite, the presence of a 500 s
double peaked pulsating signal was confirmed (Camero-Arranz et al. 2010).

Following the MAXI J1409$-$619 localization by Swift, we searched for BeppoSAX
(Boella et al. 1997a) archival observations with the transient position within
the Narrow Field Instruments (NFIs) Field of View (FoV). We found that an
observation performed in the framework of a Galactic plane survey contained
the MAXI J1409$-$619 position (Sguera et al. 2010).

In this Paper we report on the spectral analysis of this BeppoSAX observation,
integrated with high-energy data obtained from the offset fields of the PDS
(Frontera et al. 1997) instrument. We further present the analysis of a newly
analyzed ASCA observation performed during a Galactic plane survey.

##  2\. Observations

###  2.1. BeppoSAX/PDS Observations

We searched our BeppoSAX archive for observations covering the MAXI
J1409$-$619 position. From our local archive we are able to extract not only
products from the NFI nominal pointings (also available from ASDC – the ASI
Scientific data Center111http://www.asdc.asi.it/bepposax), but also net
spectra from the PDS offset fields.

Table 1List of PDS pointings covering the MAXI J1409$-$619 error box

Seq | OPn | Date | Duration | Source Name | Dist | RA | DEC | Type | PDS rate | S/N
---|---|---|---|---|---|---|---|---|---|---
|  |  | (ks) |  | $(\arcmin)$ | (J2000) | (J2000) |  | (15–100 keV c/s) |
1 | 08317 | 29/01/2000 | 42 | Gal Plane Sur 2 | 19.6 | 211.3474 |  $-$61.88437 | ON | $0.247\pm 0.069$ | 3.6
2 | 10480 | 07/01/2001 | 220 | Circinus Galaxy | 27.3 | 212.8952 |  $-$61.80250 | MOFF | $0.450\pm 0.032$ | 14.1
3 | 08386 | 07/02/2000 | 52 | Gal Plane Sur 16 | 49.5 | 210.2841 |  $-$61.83828 | ON | $0.086\pm 0.049$ | 1.7
4 | 01652 | 23/02/1997 | 90 | Alpha Cen | 53.6 | 213.5273 |  $-$62.53019 | POFF | $0.086\pm 0.052$ | 1.6
5 | 08484 | 19/02/2000 | 47 | Gal Plane Sur 23 | 70.0 | 213.5936 |  $-$61.09281 | POFF | $0.144\pm 0.072$ | 2.0

Note. — The “Source Name” refers, in case of offset fields, to the nominal ON
pointing. The distance listed in column 6 refers to the angular distance of
the PDS pointing from the MAXI J1409$-$619 position.

Indeed, because of the rocking technique used to derive the PDS background,
two positions offset by 3$\fdg$5 with respect to the NFI direction were
alternatively observed every 96 s. For fields free of sources, we expect that
differences between the count spectra in the “plus OFF” (thereafter POFF) and
“minus OFF” (thereafter MOFF) positions will be consistent with zero. If, on
the other hand, a contaminating source is present in one of the two offset
fields, then the spectral difference corresponds to a net, background
subtracted PDS spectrum of the offset field.

We found three sets of observations containing the MAXI J1409$-$619 position
in their FoVs. In Figure 1 we show a finding chart centered on the MAXI
J1409$-$619 position and showing both the ON-source and OFF-source pointings
(the PDS FoV is 1$\fdg$3 FWHM). In Table 1 we detail the whole set of BeppoSAX
observations used in our analysis. All errors are given at the 90% confidence
level for a single parameter.

Figure 1.— Region centered on the MAXI J1409$-$619 position (the small black
circle) with the PDS pointings: ON pointings (red), POFF pointings (green),
and MOFF pointings (cyan). The PDS FoV is 1$\fdg$3 (FWHM). Note the two PDS
observations (one ON and one MOFF) fully covering the source position, and
three (one ON and two POFF) partially covering the position (with the term
“partially” we intend that the source is beyond the collimator FWHM but before
its FWZI). The bold dotted line corresponds to the Galactic plane.

####  2.1.1 ON-source observations (OP08317 and OP08386)

The first ON-source BeppoSAX pointing with MAXI J1409$-$619 in the NFI FoV was
performed on January 29, 2000 within a program aimed at performing a
systematic study of part of the Galactic plane (OP08317 – Galactic Plane
Survey Field 02). Another ON-source observation (OP08386 – Galactic Plane
Survey Field 16), was performed a week later. This observation partially
overlaps the OP08317 field (see Fig. 1).

The Galactic plane Field 02 net exposures of the two imaging instruments Low-
Energy Concentrator Spectrometer (LECS; 0.1–10 keV; 37′ FoV; Parmar et al.
1997) and Medium-Energy Concentrator Spectrometer (MECS; 1.8–10 keV; 56′ FoV;
Boella et al. 1997b) are 6812 and 22060 s, respectively. The difference is due
to the constraint that the LECS can be operated only when in the Earth shadow.
Good data were selected when the instrument configurations were nominal, and
with an elevation angle above the Earth limb $>$4°.

Table 2Sources detected in the MECS field for OP08317  # | RateaaIn units of $10^{-3}$ Counts/s | pixel |  | R.A. | Dec |  | S/N | Counterpart
---|---|---|---|---|---|---|---|---
| (2–10 keV) | x | y |  | (2000) |  |  |
1 | 45.5$\,\pm\,$ | 3.6 | 373.2 | 317.1 |  | 14 04 31.3 |  $-$61 47 10.8 |  | 12.6 | IGR J14043$-$6148
2 | 10.6$\,\pm\,$ | 1.3 | 188.2 | 233.2 |  | 14 08 00.4 |  $-$61 58 24.1 |  | 8.15 | MAXI J1409$-$619
3 | 3.77$\,\pm\,$ | 0.79 | 293.9 | 250.1 |  | 14 06 00.5 |  $-$61 56 09.6 |  | 4.77 |

Figure 2.— 2–10 keV MECS image of a mosaic of the two, partially overlapping,
ON-source BeppoSAX observations OP08317 and OP08386, smoothed with a Gaussian
filter with a $\sigma$ of 24″. Two sources are clearly detected. The one
present in the MAXI J1409$-$619 error box is listed as #2 in Table 2. The two
“cuts-out” are due to the removal of the calibration source events.

A mosaic of the two, partially overlapping, ON-source OP08317 and OP08386 MECS
images is shown in Figure 2, where two relatively bright sources are clearly
present (see Table 2). Source #2 position is consistent with MAXI J1409$-$619,
while the source #1 position is consistent with the X–ray source IGR
J14043$-$6148 (Bird et al. 2010). The third detection (source #3) corresponds
to extended low-energy emission observed in the LECS image, and may be
associated with a radio source at approximately the same position (Cohen &
Green 2001). The uncertainty in the MECS positions is 30″ (Perri & Capalbi
2002).

We extracted LECS and MECS data from a 4′ (30 pixel) radius circular region
centered on the MAXI J1409$-$619 position. Concerning LECS data, there are not
enough counts to allow a spectral reconstruction. On the other hand, we are
able to extract a 1.8–10 keV MECS spectrum, that we rebinned to a minimum of
30 counts per bin to allow the use of $\chi^{2}$ statistics. We used a
response matrix appropriate for the off-axis position of the source. Because
the position of MAXI J1409$-$619 is close to the Galactic plane, we checked
for possible contamination from the Galactic ridge emission. To this end, from
the MECS image we extracted background spectra from two source-free regions,
offset by about the same angle as the source (to take into account
vignetting), and with the same extraction radii. These backgrounds are
consistent with the mean background used routinely for MECS observations. For
this reason the mean background was used in the spectral analysis.

Given that the PDS is not an imaging instrument, before performing joint
MECS/PDS spectral fits it is necessary to address the problem of determining
whether the high energy emission observed by the PDS is due to MAXI
J1409$-$619 or to the INTEGRAL source. Bear in mind that, in computing fluxes,
it is necessary to take into account the triangular angular response of the
PDS collimators (see Fig. 2 in Frontera et al. 2007).

First, we analyzed the ON-source OP08386 MECS observation, that contains IGR
J14043$-$6148 (detected at $>$15$\sigma$) but not MAXI J1409$-$619\. For this
observation we have only a marginal PDS detection (see Table 1), therefore it
seems plausible to associate the high-energy emission observed in the OP08317
observation with MAXI J1409$-$619\. To further confirm this association, we
verified that the PDS spectrum extrapolated at lower energies is consistent
with the MAXI J1409$-$619 MECS spectrum.

Figure 3.— 1.8–100 keV joint MECS/PDS count rate spectrum (plus signs) and the
power law best fit model (histogram) of MAXI J1409$-$619 in the OP08317 field.
The fit residuals are shown in the bottom panel.

In Figure 3 we show the joint MECS/PDS 1.8–100 keV MAXI J1409$-$619 count rate
spectrum, together with its power law best fit (normalized $\chi^{2}_{\nu}$ of
0.18 for 15 degrees of freedom – dof). It is characterized by a hard spectrum,
with power law photon index $0.87_{-0.19}^{+0.29}$,
$N_{H}=2.8_{-2.2}^{+3.4}\times 10^{22}$ cm-2 (consistent with galactic
absorption, Dickey & Lockman 1990) and unabsorbed 2–10 and 15–100 keV fluxes
of $2.7\times 10^{-12}$ and $4\times 10^{-11}$ erg cm-2 s-1, respectively.

####  2.1.2 MOFF offset field observation (OP10480)

From Figure 1 we can see that we have three sets of PDS offset observations
covering the MAXI J1409$-$619 position. For two of them (OP01652 and OP08484)
we do not have sufficient statistics to perform a useful analysis (see the
observed count rates in Table 1). On the other hand, for the MOFF offset field
of the Circinus Galaxy observation, OP10480, data have enough statistics for a
thorough spectral analysis.

First, we checked for the presence of any catalogued X–ray sources in the MOFF
observation FoV. The only source present is the EGRET (Hartman et al. 1999)
source 3EG J1410$-$6147, that was recently associated with the 50 ms radio
pulsar PSR J1410$-$6132 (O’Brien et al. 2008). No significant X–ray emission
has been detected around the EGRET source (Doherty et al. 2003); therefore we
associate the PDS detection with MAXI J1409$-$619.

We started by extracting a 15–100 keV coarser binned PDS spectrum of the MOFF
field (assuming as its background the corresponding POFF spectrum). The fit
with a power law gives a photon index of $2.2\pm 0.3$, with a normalized
$\chi^{2}_{\nu}$ of 1.31 for 11 dof. From the shape of the residuals and from
the quite different photon index with respect to the ON-source observation, we
suspected the presence of a change of slope in the spectrum between 20 and 30
keV. This break is not detected in the ON-source observation because of its
poor statistics. We tried both a broken and a cutoff power law. By fixing the
cutoff energy at 25 keV (we were only able to constrain the break/cutoff
energy between 20 and 40 keV), a cutoff power law fit gives a power law index
$\Gamma=0.9_{-0.3}^{+0.4}$, with a normalized $\chi^{2}_{\nu}$ of 1.06 for 11
dof. While from a statistical point of view the two fits are equivalent (an
F-test gives a 36% probability of chance improvement (PCI) of the $\chi^{2}$ –
see Appendix A for the discussion on the correctness of the use of the
F-test), the cutoff fit nicely matches the spectral index observed at lower
energies. The corresponding 15–100 keV flux, corrected for the triangular
response of the collimator, is $7.0\times 10^{-11}$ erg cm-2 s-1.

Intrigued by the shape of the residuals, we tried a smaller rebinning factor
and a wider (15–200 keV) energy range. We were surprised to find the spectrum
shown in Figure 4. Absorption features at $\sim$40, $\sim$70, and $\sim$120
keV are quite evident. We also noted that the points that trail these features
at $\sim$50, $\sim$90, and $\sim$150 keV seems to be in agreement with a hard
continuum up to 200 keV. To test this hypothesis, we performed a fit to the
data below 30 keV together with these trailing points. No cutoff (the break
energy is not constrained) or broken power law (the two photon indices are
equal) model are able to fit the data, while a power law with photon index
$1.0\pm 0.2$ does ($\chi^{2}_{\nu}=0.56$ for 20 dof). Therefore the change of
slope in 20–40 keV derived from the analysis of the coarser binned energy
spectrum is really due to the presence of the absorption features. The first
absorption feature drives the position of the break, while the higher energy
ones are spread by the rebinning, resulting in low-count, high energy bins
that are fit by a change of slope.

Figure 4.— 15–200 keV MOFF PDS count rate spectrum (plus signs) and the best
fit power law plus 3 Gaussians in absorption model (histogram). The fit
residuals are shown in the bottom panel. The Gaussian energies are left free
and are $44\pm 3$, $73_{-5}^{+4}$, and $128_{-8}^{+5}$ keV. The line widths
were constrained to 4, 4, and 7 keV for the three lines.

To further support this interpretation, we performed a joint fit with the MECS
spectrum (this analysis is detailed in Appendix B). We find that a cutoff or a
broken power law models are not able to fit this broad band spectrum with the
constraint $\Gamma\sim 0.9$ below 10 keV. This implies that the origin of the
change of slope is not due to the presence of a cutoff but to something else,
likely absorption features. For this reasons as the continuum we will use a
power law with $\Gamma$ fixed at $0.87$, as derived by the analysis on the
broad band spectrum of the first observation.

As throughly discussed in Appendix A, the assessment of the statistical
significance of these absorption features has to be done with great care. Our
strategy was to evaluate the significance of each feature one at a time. We
started with a power law fit in the energy range 15–55 keV, and we obtained an
unacceptable $\chi^{2}_{\nu}$ of 1.4 for 25 dof. The inclusion of a Gaussian
in absorption (gabs in Xspec) with centroid energy $E_{\rm cyc}=44\pm 3$ keV,
$\tau=16_{-7}^{+14}$, and width fixed at 4 keV significantly improved the fit
($\chi^{2}_{\nu}$ equal to 0.48 for 23 dof).

Two comments on this measurement are in order: first, the $\sim$44 keV line is
statistically significant (PCI is $6\times 10^{-3}$). Second, the difficulties
in determining the other line parameters (the line width had to be fixed,
while the line normalization is affected by large errors) are due to the fact
that the lines are seen as “holes” in the continuum: we have not enough
statistics to clearly reconstruct the line profile. The line width is
therefore computed from the “hole” width, while for the line depth we can
infer only an upper limit (this will be more evident for the lines at higher
energies).

Following our detection strategy, we performed a second fit on the 15–80 keV
energy range. The fit with only one Gaussian in absorption was not acceptable:
$\chi^{2}_{\nu}$ was 3.6 for 32 dof. By adding another Gaussian in absorption
we obtained a significantly better fit, with $\chi^{2}_{\nu}$ of 0.48 for 30
dof (PCI is $2\times 10^{-7}$). The second line parameters are $E_{\rm
cyc}=73_{-5}^{+4}$ keV, $\tau=91_{-42}^{+\infty}$, while the the line width
was fixed at 7 keV. Note how the two line energies are not in an harmonic
ratio.

Then we extended the energy range to the whole 15–200 keV band. Again the fit
with two lines was not acceptable ($\chi^{2}_{\nu}$ of 4.5 for 46 dof), so we
added another absorption line, the width of which was fixed at 7 keV. The fit
significantly improved ($\chi^{2}_{\nu}$ of 0.58 for 44 dof, PCI of
$10^{-10}$), with the best fit $E_{\rm cyc}=128_{-8}^{+5}$ keV. $\tau$ was not
constrained (this was not a surprise because, as discussed above, we have no
signal in the “hole”, and therefore we are not able to put a limit to the line
normalization).

Finally, we constrained the line energies to be in an harmonic ratio. In this
case we obtain a $\chi^{2}_{\nu}$ of 1.22 for 46 dof. The best fit parameters
of the fundamental222Thereafter we will indicate the fundamental resonance as
(1:1), meaning with this the ordinal number of the resonance with respect to
the fundamental. With this notation the first harmonic will be indicated as
(1:2), the third (1:3), and so on. were $E_{\rm cyc}=41\pm 1$ keV,
$\tau=11_{-7}^{+9}$. All the line widths were fixed, as in the previous case
with the line energies left free, but we were not able to constrain the (1:2)
and (1:3) line normalizations.

In order to verify that the observed features were not of instrumental origin,
and at the same time to better characterize the cyclotron resonance features
(thereafter CRFs), we performed a normalized Crab ratio analysis (Orlandini
2004) on the MOFF spectrum, as we successfully employed for the detection of
CRFs in numerous X–ray pulsars (see, e.g. Orlandini et al. 1998). As it is
evident from the top panel of Figure 5, the ratio between the MAXI J1409$-$619
and the Crab count rate spectra shows a “hole” in the 38–44 keV range, due to
the fundamental CRF. To enhance the feature, we multiplied the MAXI
J1409$-$619/Crab ratio by the functional form of the Crab spectrum, i.e. a
simple power law with photon index equal to 2.1. The result is shown in the
second panel of Figure 5. Finally, in the lower panel, we show the ratio
between the previous function and the MAXI J1409$-$619 best fit continuum
power law model, together with a Gaussian fit to the fundamental CRF. The
Gaussian width was fixed at 4 keV, while the Gaussian centroid energy is
$43_{-1}^{+2}$ keV. Please note that this value is a lower limit, because it
does not take into account the intrinsic energy resolution of the PDS
instrument.

Figure 5.— Top: Ratio between MAXI J1409$-$619 MOFF and Crab count rate
spectra. Middle: the MAXI J1409$-$619/Crab ratio multiplied by E-2.1, the
functional form of the Crab spectrum. The dotted line corresponds to the best
fit power law continuum. Bottom: ratio between the former expression and the
best fit power law continuum, together with a Gaussian fit to the $\sim$43 keV
CRF. The statistical assessment of the CRF has been evaluated with a run test,
finding that we can reject the line being random at the 99% confidence level
(see text and Appendix A for details).

In order to give a quantitative evaluation of the CRF in the normalized Crab
Ratio we performed a run test (see detailed discussion in Appendix A). Up to
$\sim$34 keV the residuals are consistent with random fluctuations
($N_{+}=12$; $N_{-}=9$; $N_{r}=13$; consistent with random fluctuation at the
84%). On the other hand, in the 34–50 keV band we see a clear structure in the
residuals. In this case from a run test with $N_{+}=2$, $N_{-}=14$, $N_{r}=2$
we we can reject the null hypothesis of randomness at the 99% confidence
level.

###  2.2. ASCA observation

The region around MAXI J1409$-$619 was observed by ASCA on March 2, 1998 for a
net exposure time of about 18 ks. The source is clearly detected, and its
spectrum is confirmed to be quite hard, with a best fit power law photon index
$\Gamma=0.1_{-0.6}^{+0.8}$ and an unabsorbed 1–10 keV flux of $3\times
10^{-12}$ erg cm-2 s-1. The absorption in the direction of the source is not
well constrained by the fit, with an upper limit of $3\times 10^{22}$ cm-2,
consistent with the Swift and BeppoSAX/MECS results.

##  3\. A very red(dened) counterpart for MAXI J1409$-$619

As stressed by Kennea et al. (2010b), a NIR source belonging to the 2MASS
catalogue (Skrutskie et al. 2006) is found within 2$\farcs$1 of the MAXI
J1409$-$619 position as determined by the XRT. According to that catalogue,
this source (labeled as 2MASS J14080271$-$6159020) has NIR magnitudes $J$ =
15.874$\pm$0.086, $H$ = 13.620$\pm$0.022 and $K$ = 12.560$\pm$0.021. No
catalogued optical counterpart is present at the position of this NIR source,
and inspection of the DSS-II-Red digitized archival
plates333http://archive.eso.org/dss/dss does not show any evident optical
object at that location. Following Monet et al. (2003) we can thus place a
conservative upper limit, $R\\!\\!>$20, to the $R$-band magnitude of the
optical counterpart of this source. The above magnitudes thus indicate that
this is an extremely red object.

Indeed, assuming the Milky Way extinction law (Cardelli et al. 1989), we find
that the NIR color indices of this source are consistent with those of a late
O/early B-type star (Wegner 1994) with a reddening of
$A_{V}\\!\\!\approx\\!\\!20$ mag. This, using the formula of Predehl & Schmitt
(1995), implies a column density of NH $\sim$ 3.6$\times$1022 cm-2, which is
consistent with that inferred from the Swift and BeppoSAX spectral analysis
results. This large extinction is also supported by the non-detection of the
optical counterpart of the source. All this points to a likely HMXB nature of
this source, hosting a heavily absorbed early-type star, similar to several
cases of INTEGRAL hard X–ray sources identified as HMXBs at optical and/or NIR
wavelengths (see, e.g., Masetti et al. 2010).

Assuming then $A_{V}\\!\\!\approx\\!\\!20$ mag along the MAXI J1409$-$619 line
of sight and a B0 spectral type for the companion star in this system, we can
infer its distance, depending on the luminosity class (main sequence, giant or
supergiant) of the star. Using the tabulated absolute magnitudes for this type
of star (Lang 1992) we find, for these three cases, respective distances of
$\sim$4.6, $\sim$7.9 and $\sim$14.5 kpc. Given that the large absorption found
along the line of sight of this source at both NIR and X–ray wavelengths is
consistent with the Galactic one, we consider most likely that the correct
distance is the largest one. Thus, MAXI J1409$-$619 is quite likely located in
the far side of the Galaxy, i.e. in the farthest parts of the Sagittarius-
Carina arm, and its mass donor star is likely an early-type supergiant.

##  4\. Discussion

We found five sets of observations containing the position of MAXI J1409$-$619
in our BeppoSAX/PDS archive, performed in 1997, 2000, and 2001, and one in the
ASCA archive (March 1998). In all our observations the source was in a low
state, with 15–100 keV fluxes in the range $\sim$2–8 mCrab, and no spectral
variability during the observations. For comparison, an integrated exposure
(over 5 years) of 2.4 Ms by INTEGRAL/IBIS provides 2$\sigma$ upper limits on
the persistent quiescent emission of 0.2 and 0.4 mCrab in the 20–40 and 40–100
keV energy bands, respectively (Sguera et al. 2010). When assuming the source
fluxes in outburst as measured by Swift (Kennea et al. 2010b, 2011) and RXTE
(Yamamoto et al. 2010), from our low state measurement we can infer a dynamic
range of 400 in 15–50 keV, and of 300 in 2–10 keV.

The discovery of CRFs in the low state spectrum of MAXI J1409$-$619, together
with its $\sim$500 s pulsations, unambiguously identify the source as an
accreting X–ray binary pulsar. Only few sources show multiple CRFs, and only
two show resonances above the (1:2). Three CRFs were observed during the
2004–2005 outburst of the X–ray pulsar V0332+53 (Coburn et al. 2005;
Kreykenbohm et al. 2005; Tsygankov et al. 2006), and five (possible six) CRFs
were discovered in the spectrum of 4U 0115+63 during its 1999 giant outburst
(Santangelo et al. 1999; Heindl et al. 1999; Ferrigno et al. 2009). In both
cases, deviations from a pure harmonic ratio among the CRFs were observed, and
were explained in terms of departures from a classical dipolar structure of
the magnetic field in the line-forming region (Nishimura 2005, 2008).

The magnetic field strength at the neutron star surface corresponding to
$E_{\rm cyc}\\!=\\!44$ keV is $3.8\times 10^{12}(1+z)$ G (Canuto & Ventura
1977), where $z$, the gravitational redshift, for a typical neutron star of
mass 1.4 M⊙ and radius 10 Km, is about 0.3. When left as free parameters in
the fit, we found that the best fit CRF line energies do not follow the
harmonic relation $E_{n}=n\,E_{1}$. A slightly non-harmonicity is expected
when relativistic effects are taken into account (see, e.g., Meszaros 1992)

$E_{n}=m_{e}c^{2}\,\frac{\displaystyle\sqrt{1+2n(B/B_{\rm
crit})\sin^{2}\theta}-1}{\sin^{2}\theta}\,\frac{1}{1+z}$ (1)

where $m_{e}$ is the electron rest mass, $c$ the speed of light, $\theta$ the
angle between the photon and the magnetic field direction, and $B_{\rm
crit}=4.414\times 10^{13}$ G is the critical magnetic field strength where the
cyclotron energy equals the electron rest mass.

The observed (1:2) and (1:3) ratios of the line energies with respect to the
fundamental, 1.7$\pm$0.2 and 2.9$\pm$0.3, cannot be explained in terms of Eq.
(1), as it is evident from Figure 6. A fit to the harmonic relation, shown as
the dashed line, gives $E_{1}=41\pm 3$ keV (in agreement with the best fit
value $E_{\rm cyc}=41\pm 1$ keV found when imposing the harmonic relation in
the spectral fit), but with poor significance ($\chi^{2}_{\nu}=2.27$ for 2
dof). In the same figure we also show the harmonic relation from Eq. (1) for
different values of the magnetic field and the angle $\theta$. Taking into
account relativistic effects, we found that the magnetic field responsible for
the (1:1) and (1:3) CRFs is about 20% higher than that responsible for the
(1:2) CRF.

Figure 6.— Harmonicity of the MAXI J1409$-$619 CRF line energies. The dashed
black line corresponds to a linear fit (that is, $E_{n}=n\,E_{1}$) to the
data, while the colored strips take into account relativistic effects, as
detailed by Eq. (1). Each strip corresponds to a fixed value of the magnetic
field strength $B_{12}=B\times 10^{12}$ G, and to a range 0.0001–1 for
$\sin^{2}\theta$, where $\theta$ is the angle between the photon direction and
that of the magnetic field. While $E_{1}$ and $E_{3}$ are consistent with
$B_{12}\sim 3.8$, $E_{2}$ is consistent with a magnetic field about 20% lower.

Two points are worth noticing: it is always the (1:2) resonance that shows the
larger disagreement with the harmonic relation, and this could be due to the
fact that this CRF is due to pure absorption (Nishimura 2003), while for the
other resonances other effects, like multiple scattering and photon spawning,
enter into play (see, e.g., Schönherr et al. 2007, and references therein).
Unfortunately, because we are not able to reconstruct the CRF profiles, we
cannot extract more information, like the electron temperature and the
geometry of the emitting region.

Second, at variance with the V0332+53 and 4U 0115+63 observations, both
performed during giant outbursts, our BeppoSAX observations were performed
while MAXI J1409$-$619 was in a low state. This did not allow the study of the
dependence of the CRF parameters as a function of luminosity, an important
tool for study of the physical conditions in the line-forming region (Mihara
et al. 2004; Nakajima et al. 2006; Klochkov et al. 2011).

The likely early-type optical counterpart and the 500 s pulsation makes MAXI
J1409$-$619 a HMXB pulsar. According to the nature of the secondary star we
have two possibilities: the source is a supergiant fast X–ray transient (SFXT;
Sguera et al. 2005; Negueruela et al. 2006) or a Be/HMXB. In favor of the
former interpretation is the highly reddened supergiant as possible
counterpart, the typical outburst X–ray luminosity of $2\times 10^{37}$ erg
s-1 (assuming a distance of 14.5 kpc), and the dynamic range of more than two
orders of magnitude ($\sim$300) that is typical of the so-called
“intermediate” SFXT (Sguera et al. 2007; Clark et al. 2010). Against we have
that the source active phase, about two months long (Ueno et al. 2010; Kennea
et al. 2011), is significantly longer that that typical of SFXT (see, e.g.,
Sidoli 2009). If this were the case, then our magnetic field measurement would
rule out the magnetar nature for SFXTs (Bozzo et al. 2008). The observed
properties of MAXI J1409$-$619 are also in agreement with those observed in
other Be/HMXBs, like 1A 1118$-$615, a 400 s X–ray pulsar with a hard X–ray
spectrum ($\Gamma\\!\sim\\!1$), a CRF at $\sim$55 keV, and long (tens of
years) periods of quiescence interrupted by giant (Type-II) outbursts lasting
weeks to months, in which the X–ray luminosity increases by a factor $\sim$200
(see, e.g., Rutledge et al. 2007). This would put the source a factor
$\sim$2–3 closer.

Figure 7.— Error box of AGL J1410$-$6143 (big red circle) superimposed on the
20–100 keV INTEGRAL/IBIS deep mosaic image ($\sim$2.3 Ms exposure). The
BeppoSAX/MECS position of MAXI J1409$-$619 is marked by the green square. The
two small yellow ellipses represent the Fermi $\gamma$–ray sources 2FGL
J1409.9$-$6129 and 2FGL J1413.4$-$6204, respectively. White contours (from 50%
to 99%) refer to the EGRET source 3EG J1410$-$6147.

Finally, we note that MAXI J1409$-$619 is located within the $0\fdg 5$ error
box of the unidentified transient MeV source AGL J1410$-$6143 (see Fig. 7)
discovered on February 21, 2008 by the $\gamma$–ray satellite AGILE (Tavani et
al. 2009a) during a bright MeV flare lasting only about one day (Pittori et
al. 2008; Orlandini et al. 2008). The AGILE large position uncertainty makes
very difficult the identification of its lower energy counterpart responsible
for the $\gamma$–ray emission. Despite this drawback, it is intriguing to note
that the flaring source MAXI J1409$-$619 is the only catalogued hard X–ray
source above 20 keV (20–100 keV) to be located inside the AGILE error box,
according to all the available catalogs in the HEASARC database. This spatial
correlation is further supported by a similar transient nature for both MAXI
J1409$-$619 and AGL J1410$-$6143\. We also point out that this is not a unique
case. To date, a few HMXBs have been unambiguously detected as flaring MeV
sources lasting only a few days (Sabatini et al. 2010; Tavani et al. 2009b;
Abdo et al. 2009). In addition, there are several other HMXBs proposed as best
candidate counterparts of unidentified transient MeV sources located on the
Galactic plane (Sguera et al. 2009, 2011; Sguera 2009). For the sake of
completeness, we note that within the large AGILE error box there are a number
of high energy MeV sources (see Fig. 7): i) 2FGL J1409.9$-$6129 and 2FGL
J1413.4$-$6204 have been reported in the second Fermi source catalog (Abdo et
al. 2011) as firmly identified $\gamma$–ray pulsars and this unambiguously
excludes their association with the transient AGL J1410$-$6143, ii) 3EG
J1410$-$6147 is still unidentified although it has been likely associated with
the $\gamma$–ray pulsar 2FGL J1409.9$-$6129 (O’Brien et al. 2008). However,
Wallace et al. (2000) reported a possible MeV flare from 3EG J1410$-$6147
lasting a few days on November 1991\. This behavior is at variance with the
proposed association with the $\gamma$–ray pulsar 2FGL J1409.9$-$6129 while it
is more compatible with the flaring nature of AGL J1410$-$6143\. Further
multi-wavelength studies (radio, NIR, X–ray and $\gamma$–ray) of the sky
region are strongly needed and encouraged to shed more light on the nature of
such high energy emitters.

## Acknowledgements

We thank an anonymous referee for helpful comments and suggestions that
greatly improved the paper. The BeppoSAX satellite was a joint Italian-Dutch
programme. Part of this work is based on archival data, software or online
services provided by the ASI Scientific Data center (ASDC), and the High
Energy Astrophysics Science Archive Research Center (HEASARC), provided by
NASA’s Goddard Space Flight Center. This publication makes use of data
products from the 2MASS archive. The authors acknowledge support from ASI/INAF
grants I/088/06/0, I/009/10/0 and I/033/10/0 and grant PRIN INAF/2009.

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##  Appendix A On the statistical significance of absorption features

The analysis of the significance of spectral absorption features, and in
particular CRFs, has historically been fraught with serious misunderstanding,
that we attribute to the application of techniques made for emission features
to absorption ones. This has a very profound impact on the statistical
analysis, as detailed in the following.

First, when searching for spectral features, and especially absorption
features, the binning of data is crucial: indeed a narrow feature can be lost
if the data binning is too high (see, e.g., Eadie et al. 1971, page 259), as
we showed in the data analysis detailed in Section 2.1.2. But before searching
for the best binning factor, it has become customary to rebin data in order to
achieve a minimum number of counts in each bin. The reason of this procedure
is that we must avoid bins with few counts if we want to use the $\chi^{2}$
statistics to test for the goodness of our fit (in other words, errors must be
normally distributed). A minimum 20 counts per bin filter is sufficient to
achieve this goal (Cash 1979).

The application of this criterion to background subtracted spectra is correct
for instruments that are not background dominated (that is, they have an
intrinsically low number of counts per bin), like a detector on the focal
plane of X–ray optics. On the other hand, when the net source spectrum is
obtained by subtracting two high counts spectra this correction must not be
applied (instruments like BeppoSAX/PDS or RXTE/HEXTE are background dominated,
with a very high number of counts per energy channel). Although the net source
counts can be very low, they are still Gaussian distributed, because resulting
from the difference of two normally distributed counts.

Second, if features are present in the source spectrum, they (clearly) show up
in the fit residuals, and the standard goodness of fit estimator, the
minimization of the $\chi^{2}$ statistics (Lampton et al. 1976), should
(clearly) indicate that the fit is not “good” and an additional component is
necessary. The inclusion of a new component to the continuum should result in
a reduction of the reduced $\chi^{2}$. In order to assess whether the
improvement of the $\chi^{2}$ is due to chance or it is because the new
component is significant it is customary to use the F-test (see, e.g., Barlow
1989, page 160).

A very important point has to be raised here, because we have two completely
different approaches whether we are dealing with emission or absorption
features. In the former case we are dealing, using a Xspec terminology, with
an additive component, and we test the null hypothesis that the coefficient of
the new term is zero (Bevington 1969, page 200). This is the F-test routine
that is incorporated in the recent versions of Xspec.

On the other hand, when dealing with an absorption feature, the component is
not additive but multiplicative (see routines gabs or cyclabs). It is
therefore obvious that we cannot use the F-test for an additive component,
because we cannot vary to zero the new component coefficient without modifying
the parameters of the original model. The two components (the multiplicative
model and the original continuum) are strongly coupled, and variations in one
will affect also the other. Therefore we must use a different F-test, as
described in Press et al. (2007, page 730), that tests the null hypothesis
that the observed variances comes from the same sample (and this routine is
not incorporated in Xspec). This same test must be used to assess if two
different models are statistically equivalent or not (and indeed we used it in
Section 2.1.2 to compare the power law and the cutoff power law fits). It is
worth noting that the application of the F-test to the statistical assessment
of CRFs is not affected by the concerns raised by its use with emission lines
(Protassov et al. 2002).

Third, CRFs are very shallow features, with their count rates often at the
instrument sensitivity level. We therefore expect errors to be dominant, and
consequently the $\chi^{2}$ could not be the best estimator of the CRF
significance. We need a test that takes into account the structure of the line
(an information which is lost in the $\chi^{2}$, because we compute the square
of the difference between the data and the model). In other words, we need to
compute what is the probability that a particular structure visible in the fit
residuals occurs by chance.

The run test (also known as Wald-Wolfowitz test; Barlow 1989; Eadie et al.
1971) works on the signs of the deviations, that is on the form of the
residuals. To better clarify how the run test works give a look at Figure 8,
adapted from Barlow (1989): when fitting the 12 data points with a straight
line the normalized $\chi^{2}$ is exactly 1 (likely due to error
overestimation). But it is evident by eye that the fit is not good (indeed the
data come from a parabolic model). The reason is that if the fit were good we
should expect that the number of points “above” the fitting line should not
group together, but should be intermixed with points “below” the fitting line
(and this should be more true as the number of data points increases). If, on
the other hand, we observe only small groups of data with the same “sign”
(called runs), this means that our data are not randomly distributed with
respect to the fitting model, but there is an underlying trend.

Figure 8.— Example on how the goodness of fit estimation performed by the
$\chi^{2}$ statistics not always is able to find the best fit model to the
data. In this case the linear fit to the 12 data points yields a normalized
$\chi^{2}$ of 1, but the deviation with respect to the straight line is
evident. A run test shows that these deviations have only 1.3% probability to
be due to random fluctuations (see text for details).

As we can see from Figure 8, we have 12 points, 6 points “above” the fitting
line (let us call them $N_{+}$) and 6 points “below” the fitting line
($N_{-}$). The number of runs $N_{r}$ is only 3, suspiciously small. Indeed
the probability of obtaining $N_{r}\leq 3$ is 1.3%, telling us that the
structure observed in the residuals is not due to random fluctuations but to a
wrong modelization (the linear fit).

From this example, and from the real-case analysis performed on the residuals
shown in Figures 5 and 9, should be clear that the goodness of fit in the case
of data structures in the residuals should not be addressed only with the
$\chi^{2}$ estimator, but it has to be supported by the run test.

In conclusion, this is a sort of cookbook for a correct evaluation of the
statistical significance of absorption (CRFs) features in the spectra of X–ray
sources:

  1. 1.

Be careful with the binning: although CRFs are usually broad features, a too
small binning can hide structures. When dealing with background dominated data
(like, for example, BeppoSAX/PDS or RXTE/HEXTE net spectra) never rebin data
in order to have a minimum number of counts per bin, otherwise we risk to lose
the absorption feature;

  2. 2.

In order to assess the significance of a CRF the F-test is perfectly suitable,
but we must use the correct routine: the F-test routine included in Xspec is
applicable only to an additive component (in particular, an emission line),
while both gabs and cyclabs are multiplicative;

  3. 3.

The evaluation of the statistical significance of a CRF must be supported by
other tests, especially in the presence of structures in the fit residuals:
the run test is able to discriminate whether the structure is due to random
fluctuations or not.

##  Appendix B The continuum spectral model of MAXI J1409$-$619

The analysis performed on the broad band spectrum of the pointed MAXI
J1409$-$619 observation yields as best fit continuum a power law with index
$\sim$0.9 (see section 2.1.1). On the other hand, the analysis performed on a
coarser binned spectrum of the offset observation indicates the presence of a
change of slope around 20–40 keV (see section 2.1.2). Because a finer
rebinning of this second observation revealed the presence of features that
can be explained as CRSFs, it is important to understand whether this change
of slope is due to a cutoff in the continuum or is a combined effect of the
binning and the presence of the absorption features.

From the pointed observation we know that data below 10 keV must be described
by a power law with $\Gamma\sim 0.9$. To take into account this constraint we
performed a joint fit of the high energy PDS spectrum from the second
observation with the low energy MECS spectrum from the first observation.

Although the two spectra come from different observations, X–ray pulsars are
known to not show any spectral variability besides that at low energy due to
reprocessing of X–rays by the circumstellar material (see, for example, the
case of Vela X–1 or GX301–2). Because our source does not show any intrinsic
absorption or emission lines, signatures of reprocessing, we are confident
that our MECS spectrum did not change between the two observations.

First we tried a fit with a power law, modified at low energy by photoelectric
absorption. All the parameters are left free but the $N_{H}$ (fixed at the
$2.8\times 10^{22}$ cm-2 value obtained from the ON-source observation), and
the result is shown in panel a. of Figure 9. The fit parameters are listed in
Table 3.

Figure 9.— 1.8–200 keV joint MECS (from the pointed observation) / PDS (from
the offset observation) spectrum. a. Fit with a power law continuum. b. Fit
with a cutoff continuum. c. Fit with a broken power law continuum. d. Fit with
a power law continuum and three CRSFs. The only model able to fit
simultaneously the low (below 10 keV) and high energy data is the power law
plus three CRSFs. The best fit parameters are listed in Table 3.

The fit is not statistically acceptable, and the slope is driven by the low
energy part of the spectrum. To fit also the PDS data we need something that
is able to change the slope at about 40 keV.

Now let us test whether we can explain this change of slope in terms of a
cutoff in the continuum. We first tried a cutoff power law (see panel b. in
Figure 9): the fit is statistically acceptable but the power law index does
not match with that of the low energy part, below 10 keV. The systematic
deviations from the model (information lost by the $\chi^{2}$ test, as
discussed in Appendix A) can be quantified by means of the run test on the
MECS residuals ($N_{+}=8$, $N_{-}=6$, $N_{r}=3$). The probability of obtaining
$N_{r}\leq 3$ is 0.5%, therefore the structure observed in the residuals is
not due to random fluctuations but to the wrong modelization of the continuum.
The same result is obtained with the broken power law fit (see panel c. in
Figure 9): there is no match to the MECS data.

These results demonstrate that the change of slope is not in the continuum but
is due to something else: we made the hypothesis that its origin are
absorption features. Therefore the true continuum model is a power law
together with CRSF features (see panel d. in Figure 9). Because we are not
able to disentangle the continuum and the CRSFs, due to the broadness of the
features and their low statistics, we need to fix the power law index to the
value obtained by the broad band fit.

Just as a test, we performed all the analysis on the CRSFs by fixing the power
law index at 1.0, 1.1, and 1.2, corresponding to the $1.0\pm 0.2$ value
obtained from the fit to the PDS data below 30 keV and the points trailing the
CRSFs. The CRSF parameters are all consistent with each other, demonstrating
that they do not depend on the particular value of the power law index.

Table 3Best fit parameters for the joint MECS/PDS spectral fits  | Power Law | Cutoff | Broken PL | Power Law
---|---|---|---|---
|  |  |  | \+ 3 gabs
$\Gamma$ | $0.87\pm 0.07$ | $-0.75_{-0.5}^{+0.3}$ | $0.06_{-0.2}^{+0.2}$ | $1.0_{-0.2}^{+0.3}$
$E_{\rm cutoff}$ |  $\cdots$ | $14\pm 4$ | $27_{-4}^{+7}$ |  $\cdots$
$\Gamma_{2}$ |  $\cdots$ |  $\cdots$ | $2.93_{-0.6}^{+1.9}$ |  $\cdots$
$\chi^{2}_{\nu}$ (dof) | 2.48 (66) | 0.91 (65) | 0.78 (64) | 0.54 (59)


# Dynamics of Polymer Ejection from a Nano-sphere

Farzaneh Moazemi  Institute of Nanoscience and Nanotechnology, University of
Kashan, Kashan 87317-53153, Iran  Samaneh Ghanbari-Kashan  Institute of
Nanoscience and Nanotechnology, University of Kashan, Kashan 87317-53153, Iran
Narges Nikoofard<EMAIL_ADDRESS>Institute of Nanoscience and
Nanotechnology, University of Kashan, Kashan 87317-53153, Iran

###### Abstract

Polymer ejection from nano-confinement has been of interest due to its
relation to various fundamental sciences and applications. However, the
ejection dynamics of a polymer with different persistence lengths from
confinement through a nanopore is still poorly understood. In this manuscript,
a theory is developed for the ejection dynamics of a polymer with the total
length $L_{0}$ and persistence length $l$ from a sphere of diameter $D$. These
length-scales specify different regimes, which determine the polymer dynamics
and its ejection rate. It is seen that the polymer undergoes between two to
three confinement regimes, in some cases. The total ejection time $\tau$
depends on the polymer dynamics in various relevant regimes that the polymer
experiences. Dependence of the ejection time on the system parameters is
discussed according to the theory. The theory predicts that $\alpha$ in
$\tau\sim L_{0}^{\alpha}$ changes between 1 and 1.7, $\beta$ in $\tau\sim
D^{\beta}$ changes between 3 and 5, and $\gamma$ in $\tau\sim l^{\gamma}$ is
often smaller than 1, in the studied range of the parameters.

###### pacs:

87.15.A-, 36.20.Ey, 87.15.H-

##  I Introduction

Advances in nano-technology have enabled confinement of polymers in various
nano-structures [1, 2, 3, 4]. This has new efforts to improve theories for
describing the behavior of confined polymers [5, 6, 7, 8]. On the other hand,
polymer confinement is a ubiquitous phenomenon in nature. In a eukaryotic
cell, biopolymers are often confined in different organelles, including the
nucleus [14]. The genetic material is also confined in bacterias [15] and
viruses [16]. Confinement of polymers has also notably important applications,
such as long DNA sequencing [9], data storage [10] and polymer separation
[11].

Previous studies show that the polymers have different properties when they
are confined in nano-scale geometries [17, 18, 19]. The effect of confinement
is more complicated when the confined polymer has a persistence length
comparable to the size of the confining geometry [22]. This is due to the
existence of several length scales in the system. Among different natural and
synthetic polymers, double-stranded DNA has the largest persistence length.

Compaction and ejection of a polymer from the closed geometry of a sphere is
related to natural phenomena, such as viral genome packaging and ejection [20,
21]. Ejection of a confined polymer from a sphere has been elucidated
theoretically, by using computer simulations and experimentally [23, 24, 25].
These studies consider how the system parameters such as the size of the
confining geometry, the polymer length, the packaging history or the
triggering force affect the ejection process. The effect of persistence length
and DNA cholesteric interactions are considered using computer simulations
[26, 27]. However, a theory that describes dependence of the ejection process
on the persistence length of the polymer is lacking, to the authors’
knowledge.

Initial theories for polymer ejection from confinement assumed that the
ejection time is proportional to the inverse of the free energy of the polymer
in confinement [28]. Sakaue and Yoshinaga presented another theory, in which
the rate of change of the free energy of confinement is balanced with the rate
of energy dissipation in the system [29]. Later, this theory was improved by
revising the final stage of the polymer ejection, when there is no free energy
of confinement [23].

In this manuscript, the theory is revisited to describe the ejection dynamics
of a polymer, which has an arbitrary persistence length. It is shown that
considering the persistence length changes the theory of polymer ejection,
significantly. Predictions of the theory for dependence of the ejection time
on the total length of the polymer, size of the confining geometry and the
persistence length are discussed.

$\begin{array}[]{cc}\includegraphics[width=227.62204pt,height=170.71652pt]{fig1a.eps}&\includegraphics[width=284.52756pt,height=227.62204pt]{fig1b.eps}\end{array}$

Figure 1: Left: Phase diagram shows the different confinement and the
attachment regimes for a polymer of length $L_{0}$ and Kuhn length $l$
confined in a nano-sphere of size $D$. During ejection, the polymer length
inside the sphere decreases and the system moves along the x-axis in the phase
diagram. The arrows show some typical paths in which the polymer experiences
several regimes, during its ejection. Right: The polymer starts ejection, from
the regime III with an ordered state. Then, it experiences the regimes II and
afterwards the regime IV, before complete ejection from the sphere. The
polymer in the sphere is shown in the regimes III, II and IV, from left to
right, respectively.

##  II Free Energy of a Semi-flexible Polymer in a nano-sphere

In this section, the theory of a semi-flexible polymer confined inside a
sphere is reviewed briefly, as an integral part of the study. A semi-flexible
polymer is effectively described by cylindrical Kuhn monomers. For a polymer
with persistence length $P$, the Kuhn monomers have length $l=2P$ and width
$b$. Number of the Kuhn monomers is equal to $N=\frac{L}{l}$, where $L$ is
contour length of the polymer. Radius of gyration of a free polymer is
described by $R_{g}\approx v^{\frac{1}{5}}l^{\frac{2}{5}}N^{\frac{3}{5}}$; in
which $v=l^{2}b$ is the second virial coefficient [30]. A polymer with
persistence length $P$ and contour length $L$ confined in a closed cavity of
size $D$ ($<R_{g}$) is below described in five different regimes (Fig. 1)
[22].

###  II.1 Fluctuating Semi-dilute Regime

In this regime, the polymer is divided into confinement blobs. Inside each
blob, statistics of the monomers is not perturbed by the confinement (Fig.
2(a)). Thus, dependence of the size of the blobs $\xi$ on the contour length
inside each blob $L_{b}$ is

$\xi\approx l^{\frac{1}{5}}b^{\frac{1}{5}}L_{b}^{\frac{3}{5}}.$ (1)

The blobs are closely packed inside the sphere;
$\frac{L_{b}}{\xi^{3}}\approx\frac{L}{D^{3}}$. The two former equations give

$L_{b}\approx\frac{D^{\frac{15}{4}}}{l^{\frac{3}{4}}b^{\frac{3}{4}}L^{\frac{5}{4}}}.$
(2)

The free energy of confinement is equal to thermal energy times the number of
confinement blobs [22];

$F_{c}\approx k_{B}T\frac{L}{L_{b}}\approx
k_{B}T\frac{l^{\frac{3}{4}}b^{\frac{3}{4}}L^{\frac{9}{4}}}{D^{\frac{15}{4}}}.$
(3)

The sphere size should be smaller than the radius of gyration of the polymer
in the bulk, $D<R_{g}$. Besides, the confinement blob should be larger than
thermal blob, $\xi>\xi_{T}$. Thermal blob, $\xi_{T}$ is a length scale in the
system. On smaller length scales, the excluded volume interaction is smaller
than the thermal energy. Indeed, the excluded volume interactions between the
confinement blobs are assumed in the derivation of Eq. 2. In the above
conditions, substituting $R_{g}$, $\xi$ and $\xi_{T}\approx\frac{l^{2}}{b}$
results that the regime I is true within certain boundaries [22]

$l\left(\frac{L}{b}\right)^{\frac{1}{3}}<D<l^{\frac{1}{5}}b^{\frac{1}{5}}L^{\frac{3}{5}}.$
(4)

Figure 2: Configurations of a semi-flexible polymer inside a nano-sphere. (a)
In the regime I, the polymer is divided into confinement blobs. The polymer
length inside each blob does not feel confinement. (b) In the regime II, the
polymer density inside the sphere increases. This results in an ideal behavior
for the polymer, in this regime. (c) The density further increases in the
regime III. Thus, entropic forces between different parts of the polymer cause
an ordered state in the polymer configuration. (d) In the regime IV, the
radius of the sphere is smaller. The system is no longer uniform, unlike a
polymer in the bulk. (e) In the regime V, bending energy governs the polymer
behavior and results in another ordered state.

###  II.2 Mean-field Semi-dilute Regime

As the density of the polymer in the sphere increases, the blob size becomes
smaller than the size of the thermal blob, $\xi<\xi_{T}$, and there is a
crossover to the regime II. In this regime, the polymer behavior is Gaussian
at all length scales (Fig. 2(b)). Besides, the mean-field-approximation is
valid and the free energy is calculated from the second term of the virial
expansion [22]

$F_{c}\approx vc^{2}\approx k_{B}T\frac{L^{2}b}{D^{3}}.$ (5)

Here, $c=\frac{L}{lD^{3}}$ is concentration of the Kuhn monomers inside the
sphere. The length scale at which the concentration fluctuations can be
ignored is called the correlation length, and is obtained from the random
phase approximation;
$\xi\approx\left(\frac{l^{2}}{12cv}\right)^{\frac{1}{2}}\approx\left(\frac{D^{3}l}{Lb}\right)^{\frac{1}{2}}$
[22]. The contour length related to the correlation length is obtained using
$\xi\approx\left(lL_{b}\right)^{\frac{1}{2}}$;

$L_{b}\approx\frac{D^{3}}{Lb}.$ (6)

Each correlation length should contain several monomers and be smaller than
the size of the sphere; $l<\xi<D$. These conditions besides the initial
equation for this regime, $\xi<\xi_{T}$, give the boundaries for the regime II
[22]

$\left(Lbl\right)^{\frac{1}{3}}<D<\min\\{l\left(\frac{L}{b}\right)^{\frac{1}{3}},\frac{Lb}{l}\\}.$
(7)

###  II.3 Liquid Crystalline Regime

As the density increases, the correlation length becomes comparable to the
Kuhn length, $\xi<l$. The system responds by a transition and the coexistence
of isotropic and nematic phases (Fig. 2(c)). The free energy of confinement is
due to the loss in the orientational entropy, and is estimated by the number
of Kuhn segments [22];

$F_{c}\approx k_{B}T\frac{L}{l}.$ (8)

In this regime, the Kuhn segments have arranged inside the sphere, and the
bending energy of the Kuhn segments is not relevant yet. Thus, the size of the
sphere should be larger than the Kuhn length, $D>l$. Obviously, the volume of
the sphere should be larger than the total volume of the monomers,
$D^{3}>Lb^{2}$. Overall, the boundaries for the regime III are [22]

$\max\\{\left(Lb^{2}\right)^{\frac{1}{3}},l\\}<D<\left(Lbl\right)^{\frac{1}{3}}.$
(9)

###  II.4 Ideal Chain Regime

When the correlation length of the regime II becomes larger than the size of
the system, $\xi\sim\left(\frac{D^{3}l}{Lb}\right)^{\frac{1}{2}}>D$, the
polymer enters the regime IV [22]. The correlation length gives a description
of fluctuations in the polymer configuration (Fig. 2(d)). Also, in this
regime, the polymer behavior is Gaussian at all length scales. The polymer is
divided into blobs, inside which the confinement is not felt by the monomers;
$\xi\approx\left(lL_{b}\right)^{\frac{1}{2}}$. These blobs completely overlap
each other and their size is equal to the size of the sphere; $\xi\approx D$.
The contour length of the polymer inside the blobs is

$L_{b}\approx\frac{D^{2}}{l}.$ (10)

The free energy of confinement results from the loss of the degrees of freedom
for each blob;

$F_{c}\approx k_{B}T\frac{L}{L_{b}}\approx k_{B}T\frac{lL}{D^{2}}.$ (11)

In this regime, the sphere size is smaller than the correlation length,
however, the bending energy is not determining yet. Thus, the sphere size
should be larger than the Kuhn length, $D>l$. Besides, the size of the sphere
should also be smaller than the radius of gyration of a free _Gaussian_
polymer, in order to have confinement effect. These conditions result in [22]

$\max\\{\frac{Lb}{l},l\\}<D<\left(lL\right)^{\frac{1}{2}}.$ (12)

###  II.5 Bending Regime

When the size of the sphere becomes smaller than the Kuhn length $D<l$, the
behavior of the confined polymer is dominated by bending energy (Fig. 2(e))
[22]. In this regime, the Odijk length, $\lambda$ is the contour length of the
polymer between two deflections from the sphere surface [31];

$\lambda^{3}\approx D^{2}l.$ (13)

The free energy is obtained from the bending energy of a polymer of length $L$
and bending modulus $\kappa=k_{B}Tl$ with a radius of curvature of order $D$;

$F_{c}\approx\kappa\frac{L}{D^{2}}\approx k_{B}T\frac{lL}{D^{2}}.$ (14)

This approximation is valid at low volume fractions, $\phi<0.2$. At higher
volume fractions, interaction between the Kuhn segments becomes determining.
In this regime, the volume of the sphere should be larger than the total
volume of the monomers. The boundaries for the regime V are [22]

$\left(Lb^{2}\right)^{\frac{1}{3}}<D<l.$ (15)

Figure 3: The process of polymer ejection, from the nano-sphere. $\dot{F}_{c}$
and $\dot{F}_{d}$ are confinement and drag forces. They are parallel to the
polymer contour and are applied on the monomer which is inside the nano-pore.
(a) Top: polymer starts ejection, in the regime I. Middle: There has remained
one blob inside the sphere. Bottom: The nano-sphere has no confinement effect
on the polymer. Attachment to the sphere governs the ejection. (b) Top:
polymer starts ejection from regime II. Middle: Polymer experiences the regime
I, another confinement regime. Bottom: Attachment of the polymer to the sphere
governs the ejection. (c) Top: polymer starts ejection, in the regime IV.
Middle: There has remained one blob inside the sphere. The blob is different
from the blob in the regime I, since the sphere is smaller. Bottom: Attachment
to the sphere governs the ejection. (d) Top: polymer starts ejection, in the
regime V. Middle: The polymer that is remained inside the sphere is equal to
the radius of the sphere. Bottom: There is no bending energy. Attachment to
the sphere governs the ejection.

##  III Ejection Dynamics of a Semi-flexible Polymer from a nano-sphere

In this section, we develop a theory for the ejection dynamics of a semi-
flexible polymer from a sphere. The contour-length of the polymer remained
inside the sphere at time $t$ is shown by $L(t)$. The rate of change of the
free energy is balanced by the rate of energy dissipation [29],

$\dot{F}(t)\approx-\eta l\left[\dot{L}(t)\right]^{2}.$ (16)

The right hand side is the rate of energy dissipation; where $\eta$ is the
viscosity of the solvent. Similar to previous studies [22, 23], it is assumed
that one Kuhn length $l$ near the nano-pore contributes to the dissipation.
This is because nearly a Kuhn length of the polymer beside the nano-pore is
moving and the rest of the polymer does not feel the ejection.

At first, the ejection is driven by the confinement free energy. As the length
of the polymer inside the sphere decreases, the free energy of confinement
ceases. Then, the free energy of attachment of the polymer to the sphere
becomes dominant. The free energy of attachment to a surface results from the
reduction in the possible configurations of the chain besides the surface;
$k_{B}T(1-\lambda)\ln(\frac{L}{l})$. Here, $\lambda$ is a constant of order
unity and $\frac{L}{l}$ is the number of Kuhn segments in the attached chain.
For a polymer escaping through a nano-pore between two compartments, the free
energy of attachment has two terms arising from the two sections of the chain
on both sides; $F_{a}(t)\sim
k_{B}T\left[(1-\lambda_{i})\ln(\frac{L(t)}{l})+(1-\lambda_{o})\ln(\frac{L_{0}-L(t)}{l})\right]$
[23]. Using the rate balance (Eq. 16) gives
$l\dot{L}(t)\sim-\frac{b^{3}}{\tau_{0}}\left[\frac{1-\lambda_{i}}{L(t)}-\frac{1-\lambda_{o}}{L_{0}-L(t)}\right]$,
where $\tau_{0}=\frac{\eta b^{3}}{k_{B}T}$ and $L_{0}$ is the total length of
the polymer. The second term in the right side is negligible, because
$L_{0}\gg L(t)$. Solving the resulting differential equation
$\dot{L}(t)\sim-\frac{b^{3}}{\tau_{0}lL(t)}$ gives

$L(t)\sim
L_{\tau}\left(1-\frac{t-\tau}{\tau_{a}}\right)^{\frac{1}{2}},\qquad\qquad{\tau}_{a}\sim\tau_{0}\frac{L_{\tau}^{2}l}{b^{3}}.$
(17)

Here, $\tau$ is the crossover time between the confinement and the attachment
stages. $L_{\tau}$ is the length of the polymer inside the sphere at time
$\tau$ [23].

In the following, the polymer dynamics is described in the different regimes
of confinement. The polymer behavior is simple in the regimes I, IV and V.
However, the polymer experiences a combination of different confinement
regimes during its ejection, in the regimes II and III. Accordingly, we first
discuss the simple regimes, then, the more complicated ones.

###  III.1 Regime I

At the beginning, the polymer dynamics is determined by the confinement free
energy (Fig. 3(a)). Using the rate balance (Eq. 16) and the confinement free
energy in the regime I (Eq. 3) gives the polymer velocity;
$\dot{L}(t)\sim-\frac{1}{\tau_{0}}l^{-\frac{1}{4}}b^{\frac{15}{4}}D^{-\frac{15}{4}}L(t)^{\frac{5}{4}}$.
By integrating this equation, the time evolution of the number of monomers
inside the sphere becomes

$L(t)\approx
L_{0}\left(1+\frac{t}{{\tau}_{1,I}}\right)^{-4},\qquad\qquad{\tau}_{1,I}\sim{\tau}_{0}\left(\frac{lD^{15}}{b^{15}L_{0}}\right)^{\frac{1}{4}}.$
(18)

This equation is valid until $t<\tau_{2,I}$, that the sphere has no
confinement effect on the polymer. At this time, there has remained one blob
inside the sphere (Fig. 3(a)). Using Eq. 2 to find the contour length inside a
blob at this time, one has $L(\tau_{2,I})\approx L_{b}(\tau_{2,I})\approx
D^{\frac{5}{3}}\left(lb\right)^{-\frac{1}{3}}$. Substituting in Eq. 18 gives
$\tau_{2,I}+\tau_{1,I}\sim{\tau}_{0}\left(\frac{D^{10}l}{b^{11}}\right)^{\frac{1}{3}}$.

The polymer continues its ejection, while it is attached to the sphere (Fig.
3(a)). The polymer dynamics at this stage is obtained by substituting
$L_{\tau}=L(\tau_{2,I})$ in Eq. 17; $L(t)\approx
D^{\frac{5}{3}}\left(lb\right)^{-\frac{1}{3}}\left(1-\frac{t-\tau_{2,I}}{\tau_{a,I}}\right)^{\frac{1}{2}}$,
where
${\tau}_{a,I}\sim\tau_{0}\left(\frac{D^{10}l}{b^{11}}\right)^{\frac{1}{3}}$.
The polymer completely leaves the sphere, $L(\tau_{I})=0$, at the time
$\tau_{I}\approx\tau_{a,I}+\tau_{2,I}\approx{\tau}_{0}\left[2\left(\frac{lD^{10}}{b^{11}}\right)^{\frac{1}{3}}-\left(\frac{lD^{15}}{b^{15}L_{0}}\right)^{\frac{1}{4}}\right]$.
The obtained equations are in relative agreement with Ref. [23]. There are
subtle differences, because the dissipation rate in that study depends on the
total length of the polymer.

###  III.2 Regime IV

At the beginning of the ejection, the confinement free energy determines the
polymer dynamics (Fig. 3(c)). Balancing the energy (Eq. 11) and the
dissipation terms gives the polymer velocity;
$\dot{L}(t)\approx-\frac{1}{\tau_{0}}\frac{b^{3}}{D^{2}}$. Accordingly, one
obtains the instant length of the polymer

$L(t)\approx
L_{0}\left(1-\frac{t}{\tau_{1,IV}}\right),\qquad\qquad\tau_{1,IV}\approx\tau_{0}\frac{L_{0}D^{2}}{b^{3}}.$
(19)

Again, this equation is valid until $t<\tau_{2,IV}$, when the polymer length
inside the sphere becomes so small that it does not feel the confinement (Fig.
3(c)). At this time, there has remained one blob inside the sphere;
$L(\tau_{2,IV})\approx L_{b}\approx\frac{D^{2}}{l}$. Substituting this
condition in Eq. 19 gives
$\tau_{1,IV}-\tau_{2,IV}\approx\tau_{0}\frac{D^{4}}{lb^{3}}$.

The rest of the process continues under the attachment free energy (Fig.
3(c)). Substituting $L(\tau_{2,IV})\approx\frac{D^{2}}{l}$ in Eq. 17 gives the
polymer dynamics;
$L(t)\approx\frac{D^{2}}{l}\left(1-\frac{t-\tau_{2,IV}}{\tau_{a,IV}}\right)^{\frac{1}{2}}$,
where ${\tau}_{a,IV}\sim\tau_{0}\frac{D^{4}}{b^{3}l}$. The total ejection time
of the polymer from the sphere is the time at which the polymer length becomes
zero, in the above equation; $L(\tau_{IV})=0$. The result is
$\tau_{IV}\approx\tau_{a,IV}+\tau_{2,IV}\approx\tau_{0}\left(\frac{D^{2}L_{0}}{b^{3}}\right)$.
It should be noted that the dynamics is different in the attachment stages of
the regimes I and IV. This is because the polymer lengths at the crossover
times from the confinement to the attachment stage, $L(\tau_{2,I})$ and
$L(\tau_{2,IV})$, are different.

###  III.3 Regime V

Substituting the derivative of the confinement free energy (Eq. 14) in the
rate balance (Eg. 16) gives the velocity of polymer ejection from the sphere;
$\dot{L}(t)\approx-\frac{1}{\tau_{0}}\frac{b^{3}}{D^{2}}$. Thus, the polymer
dynamics is described by

$L(t)=L_{0}\left(1-\frac{t}{\tau_{1,V}}\right),\qquad\qquad\tau_{1,V}\approx\tau_{0}\frac{L_{0}D^{2}}{b^{3}}.$
(20)

Indeed, the ejection is dominated by the bending energy of the polymer from
the sphere boundaries. Thus, the equation is valid until $t<\tau_{2,V}$, when
the polymer length inside the sphere becomes equal to the sphere diameter
(Fig. 3(d)). Substituting this condition, $L(\tau_{2,V})\approx D$, into Eq.
20 gives $\tau_{1,V}-\tau_{2,V}\approx\tau_{0}\frac{D^{3}}{b^{3}}$. The above
equations for the regime V are only applicable in small volume fractions.

The rest of the polymer ejection continues in the attachment regime (Fig.
3(d)). Using $L(\tau_{2,V})\sim D$ in Eq. 17 gives the polymer dynamics;
$L(t)\approx D\left(1-\frac{t-\tau_{2,V}}{\tau_{a,V}}\right)^{\frac{1}{2}}$,
where ${\tau}_{a,V}\sim\tau_{0}\frac{D^{2}l}{b^{3}}$. The total ejection time
from the sphere is obtained using $L(\tau_{V})=0$ in the above equation;
$\tau_{V}\approx\tau_{2,V}+\tau_{a,V}\approx\tau_{0}\left(\frac{D^{2}l}{b^{3}}+\frac{D^{2}L_{0}}{b^{3}}-\frac{D^{3}}{b^{3}}\right)$.

Figure 4: Predictions of the theory for the ejection process versus the
polymer length, for different persistence lengths. The volume fraction of the
polymer inside the nano-sphere is fixed, $\phi=0.1$. (a) The initial regime of
the polymer. (b) Log-log plot of the ejection time versus the polymer length.
A linear behavior is observed. (c) The exponent $\alpha$ in the relation
$\tau\approx L_{0}^{\alpha}$, which describes dependence of the ejection time
on the persistence length. It is seen that the exponent changes between 1 and
1.7, for all parameters.

###  III.4 Regime II

When the ejection starts, the polymer dynamics is governed by the confinement
free energy of the regime II; Balancing the rate of change of the free energy
(the time derivative of Eq. 5) with the dissipation rate gives
$\dot{L}(t)\sim-{\frac{1}{\tau_{0}}}{\frac{b^{4}}{lD^{3}}}L(t)$. Solving this
differential equation results in the polymer dynamics;

$L(t)\sim
L_{0}\exp(-\frac{t}{\tau_{1,II}}),\qquad\qquad\tau_{1,II}\sim\tau_{0}{\frac{lD^{3}}{b^{4}}}.$
(21)

This equation is valid until $t<\tau_{2,II}$, when the density of the polymer
in the sphere decreases so much that the polymer enters one of the regimes I
and IV. In other word, one of the upper limits for the regime II (Eq. 7) is
violated, at this time.

Case 1, $\frac{Db}{l^{2}}>1$: When $L(\tau_{2,II})\sim
b\left(\frac{D}{l}\right)^{3}$, the driven process in the regime II ceases and
the system enters the regime I (Fig. 3(b)). Substituting this condition into
Eq. 21, one has
$\tau_{2,II}\sim\tau_{0}{\frac{lD^{3}}{b^{4}}}\log({\frac{L_{0}}{b}}(\frac{l}{D})^{3})$.
The polymer dynamics in the regime I is obtained by substituting
$L(\tau_{2,II})$ for $L_{0}$ in Eqs. 18; $L(t)\sim
b(\frac{D}{l})^{3}(1+\frac{t-\tau_{2,II}}{\tau_{1,I}})^{-4}$, where
$\tau_{1,I}\sim\tau_{0}\frac{lD^{3}}{b^{4}}$. The relation is valid in the
time interval $\tau_{2,II}<t<\tau_{3,II}$, in which $\tau_{3,II}$ is the time
that the polymer ejection is no longer driven by the confinement energy. The
required condition is that there has remained one blob inside the sphere,
$L(\tau_{3,II})\approx D^{\frac{5}{3}}\left(lb\right)^{-\frac{1}{3}}$. This
relation with the latter equation for polymer dynamics in the regime I gives
$\tau_{1,I}+\tau_{3,II}-\tau_{2,II}\sim\tau_{0}(\frac{lD^{10}}{b^{11}})^{\frac{1}{3}}$
.

The ejection continues under the dominance of the attachment free energy,
similar to the regime I (Fig. 3(b)). Thus, the polymer dynamics is
$L(t)\sim\frac{D^{\frac{5}{3}}}{(lb)^{\frac{1}{3}}}\left(1-\frac{t-\tau_{3,II}}{\tau_{a,I}}\right)^{\frac{1}{2}}$,
in which
$\tau_{a,I}\sim\tau_{0}\left(\frac{D^{10}l}{b^{11}}\right)^{\frac{1}{3}}$.
Using $L(\tau_{II})=0$ in this equation, the total ejection time from the
sphere becomes $\tau_{II}\sim\tau_{3,II}+\tau_{a,I}\sim
2\tau_{0}(\frac{lD^{10}}{b^{11}})^{\frac{1}{3}}+\tau_{0}{\frac{lD^{3}}{b^{4}}}{\log({\frac{L_{0}}{b}}(\frac{l}{D})^{3})}-\tau_{0}{\frac{lD^{3}}{b^{4}}}$.

Case 2, $\frac{Db}{l^{2}}<1$: When $L(\tau_{2,II}){\frac{b}{l}}\sim D$, the
driven process in the regime II ceases and the system enters the regime IV.
Substituting this condition into Eq. 21 gives the crossover time
$\tau_{2,II}\sim\tau_{0}{\frac{lD^{3}}{b^{4}}}{\log(\frac{L_{0}b}{Dl})}$.
Using $L(\tau_{2,II})$ instead of $L_{0}$ in Eqs. 19 gives the polymer
dynamics in the regime IV;
$L(t)\sim{\frac{Dl}{b}}(1-{\frac{t-\tau_{2,II}}{\tau_{1,IV}}})$, with
$\tau_{1,IV}\sim\tau_{0}{\frac{lD^{3}}{b^{4}}}$. This relation is valid in the
time interval $\tau_{2,II}<t<\tau_{3,II}$. At time $\tau_{3,II}$, the driven
process ceases, because there has remained one blob inside the sphere;
$L(\tau_{3,II})\approx\frac{D^{2}}{l}$. Using this condition in the latter
equation for polymer dynamics gives
$\tau_{1,IV}-\tau_{3,II}+\tau_{2,II}\sim\tau_{0}{\frac{D^{4}}{lb^{3}}}$. The
rest of the ejection is dominated by the attachment free energy, and the
dynamics is similar to that of the regime IV. After minor calculations, the
total ejection time from the sphere becomes
$\tau_{II}\sim\tau_{3,II}+\tau_{a,IV}\sim{\tau_{0}\frac{D^{3}l}{b^{4}}}\left(1+\log\frac{L_{0}b}{Dl}\right)$.

###  III.5 Regime III

The rate of change of the free energy is obtained by taking the derivative of
Eq. 8. Balancing the energy and the dissipation rates gives the polymer
velocity $\dot{L}(t)\sim-\frac{1}{\tau_{0}}\frac{b^{3}}{l^{2}}$. From this
relation, the number of remaining monomers inside the sphere at time $t$ is

$L(t)\approx
L_{0}\left(1-\frac{t}{\tau_{1,III}}\right),\qquad\qquad\tau_{1,III}\approx\tau_{0}\frac{l^{2}L_{0}}{b^{3}}.$
(22)

This relation is valid until $t<\tau_{2,III}$, when the polymer density inside
the sphere decreases so much that the system experiences the regime II.
Indeed, as the contour length of the polymer inside the sphere decreases, the
polymer first enters the regime II and then one of the regimes I or IV (Fig.
1).

Entering the regime II occurs when the upper boundary for $D$ (Eq. 9) is
violated; $L(\tau_{2,III})\sim\frac{D^{3}}{lb}$. This condition with Eqs. 22
gives $\tau_{1,III}-\tau_{2,III}\approx\tau_{0}\frac{D^{3}l}{b^{4}}$.
Substituting $L(\tau_{2,III})$ for $L_{0}$ into Eqs. 21 for the regime II
gives $L(t)\sim\frac{D^{3}}{lb}\exp(-\frac{{t-\tau_{2,III}}}{\tau_{1,II}})$,
in which $\tau_{1,II}\sim\tau_{0}{\frac{l{D^{3}}}{b^{4}}}$. This relation is
valid in the time interval $\tau_{2,III}<t<\tau_{3,III}$, when the polymer
enters one of the regimes I or IV.

Case 1, $\frac{Db}{l^{2}}>1$: The polymer enters the regime I and then
experiences the attachment stage, until it completely leaves the sphere. The
polymer dynamics and the crossover times at these two final stages are similar
to those of the case 1 of the regime II. The total ejection time becomes
$\tau_{III}\sim{2\tau_{0}(\frac{l{D^{10}}}{b^{11}})^{\frac{1}{3}}}+2\tau_{0}{\frac{lD^{3}}{b^{4}}}\left(-1+\log{\frac{l}{b}}\right)+\tau_{0}{\frac{{L_{0}}{l^{2}}}{b^{3}}}$.

Case 2, $\frac{Db}{l^{2}}<1$: The polymer enters the regime IV and then
experiences the attachment stage, until its ejection ends. The polymer
dynamics and the crossover times in the regime IV and the attachment stage are
similar to those of the case 2 of the regime II. The total ejection time
becomes $\tau_{III}\sim
2\tau_{0}{\frac{l{D^{3}}}{b^{4}}}\log{\frac{D}{l}}+\tau_{0}{\frac{L_{0}{l^{2}}}{b^{3}}}$.

##  IV Predictions of the theory

In the previous section, the theory was obtained by using scaling
calculations. Besides, the ejection time for each regime contains several
terms, so, it requires many fitting parameters. To the authors’ knowledge, the
effect of persistence length on the ejection time of a polymer confined inside
a sphere has been investigated in one previous study, by using computer
simulations [27]. Unfortunately, the simulation data is not enough to find all
the fitting parameters, to validate the theory. However, predictions of the
theory are given here, for comparison with future simulations and experiments.
In the following, the volume fraction of the polymer inside the sphere
$\phi=\frac{L_{0}b^{2}}{D^{3}}=0.1$ is kept fixed, according to the related
simulations [27].

Figure 5: Dependence of the ejection time on the sphere diameter and
persistence length, for volume fraction $\phi=0.1$. (a) The exponent $\beta$
that describes dependence of the ejection time on the sphere diameter. It is
seen that $\beta$ takes values between 3 and 5, for different sphere diameters
and persistence lengths. Inset: the relevant regime for the studied
parameters. (b) The exponent $\gamma$ that describes dependence of the
ejection time on the persistence length. It is seen that $\gamma$ takes values
between 0 and 1, for most of the sphere diameters and the persistence lengths.
Inset: the relevant regime for the studied parameters.

The predictions of the theory for dependence of the ejection process on the
system parameters are summarized in Figs. 4 and 5. Persistence length of the
double-stranded DNA is equal to 50nm, while its width is equal to 2.5nm. Thus,
we can take $b=2.5nm$ and $l=20b$. Other synthetic and natural polymers have
smaller persistence lengths [32]. As a result, we study persistence lengths
below 20b, in the following. The largest length of the polymer in this study
is $L_{0}=1000b$. Larger polymers are also tested and the same behavior is
observed.

Figure 4(a) shows initial regime versus the polymer length, for different
persistence lengths. On the plots, the values for cases 1 and 2 of the regime
II (III) are shown by 2 (3) and 2.5 (3.5), respectively. The zero value for
the regime describes the condition in which the sphere has no confining effect
on the polymer. It should be noted that the diameter of the sphere changes
with the total length of the polymer, to keep the volume fraction fixed. Thus,
one cannot deduce the regimes that the polymer experiences during its
ejection, from Fig. 4(a). It is seen that both persistence and polymer lengths
determine the initial regime of the polymer. The initial regime determines
different aspects of the statics and the dynamics of the semi-flexible
polymer.

Figure 4(b) represents the log-log plot of the ejection time versus the
polymer length. The ejection time is an increasing function of the persistence
and the polymer lengths. Dependence of the ejection time on the system
parameters is simple in e.g. the regime IV, by using the related equation.
However, dependence of the the ejection time on the system parameters is not
easily deduced from the equations of Sec. III, in the other regimes. Instead,
it is possible to use local slope of the curves for the ejection time, in Fig.
4(b).

Figure 4(c) shows the exponent $\alpha$ in the relation $\tau\approx
L_{0}^{\alpha}$. It is calculated by using the local slope of the curves in
Fig. 4(b). The value of $\alpha$ changes between 1 and 1.7 for all studied
parameters. It is seen that $\alpha$ is a decreasing function of the polymer
length in the regime I, the regime II case 2 and the regime III case 2. It is
constant with respect to the polymer length in the regime II case 1 and the
regime IV. $\alpha$ increases with the polymer length in the regime V.

Figure 5(a) shows the exponent $\beta$ that describes dependence of the
ejection time on the sphere diameter, $\tau\approx D^{\beta}$. The exponent is
calculated for various persistence lengths and sphere diameters, in which the
polymer experiences different initial regimes (inset of Fig. 5(a)). It is
observed that the exponent $\beta$ takes values between 3 and 5. The range of
values for $\beta$ is different from those of $\alpha$. Thus, dependence of
the ejection time on the sphere diameter is stronger than its dependence on
the polymer length.

Besides, Fig. 5(a) shows that $\beta$ is a decreasing function of the sphere
diameter in the regime I, the regime II case 2 and the regime III case 2. It
is constant with respect to the sphere diameter in the regime II case 1 and
the regime IV. $\beta$ increases with the sphere diameter in the regime V. It
should be noted that the polymer length and the sphere diameter depend on each
other, through the relation $\phi=\frac{L_{0}b^{2}}{D^{3}}$. Thus a similar
behavior is observed for $\alpha$ and $\beta$.

Figure 5(b) shows the exponent $\gamma$ that describes dependence on the
persistence length, $\tau\approx l^{\gamma}$. The relevant initial regimes are
shown in the inset, for different sphere diameters. It is seen that the
polymer falls in the regime II case 1, in a small interval of persistence
lengths. Outside this interval, the exponent $\gamma$ takes values smaller
than 1. So, dependence of the ejection time on the persistence length is
generally weaker than its dependence on the polymer length and the sphere
diameter.

Figure 5(b) shows that the ejection time does not depend on the persistence
length, in the regime IV. It is also seen that the exponent $\gamma$ is
constant in the regime II case 1. It decreases with the persistence length in
the regime II case 2. An increasing dependence of the persistence length is
observed, in the regime V.

##  V Conclusions

Here, the ejection dynamics of a polymer of finite persistence length from a
nano-sphere was studied, theoretically. The dynamics and the ejection time
were obtained for different parameters of the system. In this study, it was
assumed that one Kuhn length beside the nano-pore contributes in the
dissipation. The authors have also investigated the case that one correlation
length in each regime contributes to the dissipation. Assuming one correlation
length for the dissipation, it is observed that the ejection time is not an
increasing function of the persistence length, which is not in agreement with
previous simulations [27].

The theory was obtained in the situation that there are no hydrodynamic
interactions between the monomers, in accordance with the simulations in Refs.
[27, 23]. However, it is straightforward to include hydrodynamics in the
present formalism. In this situation, the dissipation would occur in the range
of a correlation length, not a Kuhn length. Thus, the rate of dissipation
would be $T\dot{S}(t)\approx\eta\xi(t)\left[\dot{L}(t)\right]^{2}$ [22].
Definition of the correlation length is easy in the regimes I, II and IV,
which is equal to the blob size. However, the correlation length in the
regimes III and V needs more consideration. As an estimate, it is possible to
define the correlation length equal to the persistence length and the Odijk
length in the regimes III and V, respectively.

Further simulations are needed to check the formalism, more exactly. On the
other hand, it would be interesting to extend the present formalism to study
the packaging dynamics of a polymer with finite persistence length into a
cavity. This problem is already studied by using computer simulations [33].

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