text: et al. (2014b), we deﬁne that the evidence ratio should TABLE 4 Solutions for HD26965. Final set of orbital and be instrumental parameters. 1σ errors. ln B − > 9.2 (8) Parameter HD26965 b k,k 1 ± P (days) 42.364 0.015 TABLE 5 − ± K (m s 1) 1.59 0.15 Logarithm of Bayes factors comparing a k=0, k=1 and k=2 ± Keplerian models with and without activity correlations e 0.017 0.046 ± ω (rad) 0.31 1.93 ± Bayes Factor Activity No Activity Ma (0A(Ura)d) 04.2.9125 ± 10..90208 ln Bk,k−1 Correlations Correlations msini (M⊕) 6.91 ± 0.79 ln B1,0 43.38 35.44 γPFS (m s−1−) 0.13 ±± 0.80 ln B2,1 0.61 2.96 γHIRES (m s 1−) 0.50 ± 0.57 γCHIRON (m s 1−) 0.43 ± 0.79 γ (m s 1) 0.17 0.50 HARPS,old − ± γHARP−S,new (m− s 1) 0.45 ±0.78 γ˙ (m s 1 year 1) -0.031 0.037 which translates posterior odds of 10,000:1 that the k − ± σPFS (m s 1−) 1.54 ± 0.20 model is selected over the k-1 model, in order to satisfy σHIRES (m s 1−) 2.38 ± 0.15 our detection criteria. Table 5 shows the Bayes factors σCHIRON (m s 1−) 1.78 ± 0.15 for k = 0, 1, 2-planet models with and without activity σHARPS,old (m s −1) 1.11 ± 0.05 correlation terms. σHARPS,new (m s 1) 0.69 ± 0.07 Figure 5 shows the minimum mass detection thresh- φPFS 0.82 ± 0.10 olds for additional planets orbiting around HD26965. φHIRES 0.61 ± 0.07 The green-ﬁlled area represents the liquid-water hab- φCHIRON 0.62 ± 0.06 itable zone estimated according to Kopparapu et al. φ 0.81 0.04 HARPS,old ± (2013a,b). The thresholds are calculated following the φHARPS,new− 0.90± 0.08 cSPFS (m s 1−) 61.1±14.0 methods in Tuomi et al. (2014b). From this ﬁgure we cSHIRES (m s 1) 53.1 ± 15.0 can say that planets with minimum masses in excess of c 0.086 0.046 Neptune in the habitable zone can be ruled out meaning BISHARPS,old ± c 1.8 4.5 if there are HZ planets orbiting HD26965, they would FWHMHARPS,old− ± cSHARPS,old (m s 1−) 1.6 ±2.6 likely be super-Earths or smaller. The red circle rep- cHα HARPS,old (m s−1) -12.1± 8.9 resents the planet candidate, barely, but signiﬁcantly cHeIHARPS,old (m s 1) -76± 34 above the detection threshold. cBISHARPS,new 0.27 ± 0.11 5.3. Signal injection cFWHMHARPS,new− 0.089 ±0.018 cSHARPS,new (m s 1−) 105± 36 As an additional test to investigate if the signal was cHα HARPS,new (m s−1) 76 ± 125 supported for each instrument we performed a signal in- cHeIHARPS,new (m s 1) 23 141 jection on the individual datasets. We use the best-ﬁt parameters (i.e. P, K, ω, M , e) of the putative signal 0 by using a Keplerian function described in equation 2. bility of a given model by using the corresponding value The hypothesis is the following: if the 42-day signal is of the Bayesian Information Criterion (BIC). A complete injected in a given dataset and we run our usual Bayesian and detailed discussion can be found in Tuomi & Jones analysis we should, in principle, easily detect it. If the (2012) and Feng et al. (2016). To compare the Mk model signal is indeed present in the dataset we should recover with a previous Mk−1 model, we simply compute the ∼twice the best-ﬁt radial velocity semi-amplitude as it logarithm of the Bayes factor, ln Bk,k−1, deﬁned via has been boosted by the injection. On the other hand, if the recovered velocity semi-amplitude is signiﬁcantly ln Bk,k−1 = ln P(y|Mk) − ln P(y|Mk−1) (6) lower than our best-ﬁt values, that would suggest the ac- tual data is not supported by the instrument, or in other words, the precision of the instrument plus the current Furthermore, the model containing the best-ﬁt param- number of observations do not allow the signal to be eters that support the signal has to fulﬁll the detection detected. When boosting our signal, we recover the can- criteria described in Tuomi (2012). It must hold that didate period for HARPS OLD, HIRES and CHIRON datasets. For CHIRON, however, the expected peak at P(y|M ) = sP(y|M − ) (7) k k 1 42-days in the GLS is not unique, although it is above the 0.1% signiﬁcance level, as can be seen from the pe- where s > 104. Hence using the Bayes factor deﬁned in riodograms shown in Figure 6. In the case of PFS data, equation 6, we require that the M model describing the we did not recover the candidate period. Instead, we k k-keplerian signal has to be more statistically probable found a strong power at ∼5 days. This could be caused Draft version May 30, 2020 than the M − model associated with the k−1-keplerian by the sparse sampling and lower number of observations Preprint typkese1t using LATEX style emulateapj v. 12/16/11 signal. Following the conservative threshold from Tuomi available from this instrument. THE TEST CASE OF HD26965: DIFFICULTIES DISENTANGLING WEAK DOPPLER SIGNALS FROM ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: et

text: al.

text: we

text: deﬁne

text: that

text: the

text: evidence

text: ratio

text: should

text: TABLE

text: Solutions

text: for

text: Final

text: set

text: of

text: orbital

text: and

text: be

text: instrumental

text: parameters.

text: 1σ

text: errors.

text: ln

label: equation

text: B

text: Parameter

text: b

text: k,k

text: P

text: (days)

text: TABLE

text: K

text: (m

text: s

text: Logarithm

text: of

text: Bayes

text: factors

text: comparing

text: a

text: and

text: Keplerian

text: models

text: with

text: and

text: without

text: activity

text: correlations

text: e

text: ω

text: (rad)

text: Bayes

text: Factor

text: Activity

text: No

text: Activity

text: Ma

text: ln

text: Correlations

text: Correlations

text: msini

text: (M⊕)

text: ln

text: γPFS

text: (m

text: ln

text: γHIRES

text: (m

text: s

text: γCHIRON

text: (m

text: s

text: γ

text: (m

text: s

text: HARPS,old

text: γHARP−S,new

text: (m−

text: s

text: γ˙

text: (m

text: s

text: year

text: which

text: translates

text: posterior

text: odds

text: of

text: that

text: the

text: k

text: σPFS

text: (m

text: s

text: model

text: is

text: selected

text: over

text: the

text: model,

text: in

text: order

text: to

text: satisfy

text: σHIRES

text: (m

text: s

text: our

text: detection

text: criteria.

text: Table

text: shows

text: the

text: Bayes

text: factors

text: σCHIRON

text: (m

text: s

text: for

text: k

text: models

text: with

text: and

text: without

text: activity

text: σHARPS,old

text: (m

text: s

text: correlation

text: terms.

text: σHARPS,new

text: (m

text: s

text: Figure

text: shows

text: the

text: minimum

text: mass

text: detection

text: thresh-

text: φPFS

text: olds

text: for

text: additional

text: planets

text: orbiting

text: around

text: φHIRES

text: The

text: green-ﬁlled

text: area

text: represents

text: the

text: liquid-water

text: hab-

text: φCHIRON

text: itable

text: zone

text: estimated

text: according

text: to

text: Kopparapu

text: et

text: al.

text: φ

text: HARPS,old

text: The

text: thresholds

text: are

text: calculated

text: following

text: the

text: φHARPS,new−

text: cSPFS

text: (m

text: s

text: methods

text: in

text: Tuomi

text: et

text: al.

text: From

text: this

text: ﬁgure

text: we

text: cSHIRES

text: (m

text: s

text: can

text: say

text: that

text: planets

text: with

text: minimum

text: masses

text: in

text: excess

text: of

text: c

text: Neptune

text: in

text: the

text: habitable

text: zone

text: can

text: be

text: ruled

text: out

text: meaning

text: BISHARPS,old

text: c

text: if

text: there

text: are

text: HZ

text: planets

text: orbiting

text: they

text: would

text: FWHMHARPS,old−

text: cSHARPS,old

text: (m

text: s

text: likely

text: be

text: super-Earths

text: or

text: smaller.

text: The

text: red

text: circle

text: rep-

text: cHα

text: HARPS,old

text: (m

text: resents

text: the

text: planet

text: candidate,

text: barely,

text: but

text: signiﬁcantly

text: cHeIHARPS,old

text: (m

text: s

text: above

text: the

text: detection

text: threshold.

text: cBISHARPS,new

text: Signal

text: injection

text: cFWHMHARPS,new−

text: cSHARPS,new

text: (m

text: s

text: As

text: an

text: additional

text: test

text: to

text: investigate

text: if

text: the

text: signal

text: was

text: cHα

text: HARPS,new

text: (m

text: supported

text: for

text: each

text: instrument

text: we

text: performed

text: a

text: signal

text: in-

text: cHeIHARPS,new

text: (m

text: s

text: jection

text: on

text: the

text: individual

text: datasets.

text: We

text: use

text: the

text: best-ﬁt

text: parameters

text: (i.e.

text: P,

text: K,

text: ω,

text: M

text: e)

text: of

text: the

text: putative

text: signal

text: by

text: using

text: a

text: Keplerian

text: function

text: described

text: in

text: equation

text: bility

text: of

text: a

text: given

text: model

text: by

text: using

text: the

text: corresponding

text: value

text: The

text: hypothesis

text: is

text: the

text: following:

text: if

text: the

text: signal

text: is

text: of

text: the

text: Bayesian

text: Information

text: Criterion

text: (BIC).

text: A

text: complete

text: injected

text: in

text: a

text: given

text: dataset

text: and

text: we

text: run

text: our

text: usual

text: Bayesian

text: and

text: detailed

text: discussion

text: can

text: be

text: found

text: in

text: Tuomi

text: Jones

text: analysis

text: we

text: should,

text: in

text: principle,

text: easily

text: detect

text: it.

text: If

text: the

text: and

text: Feng

text: et

text: al.

text: To

text: compare

text: the

text: Mk

text: model

text: signal

text: is

text: indeed

text: present

text: in

text: the

text: dataset

text: we

text: should

text: recover

text: with

text: a

text: previous

text: model,

text: we

text: simply

text: compute

text: the

text: ∼twice

text: the

text: best-ﬁt

text: radial

text: velocity

text: semi-amplitude

text: as

text: it

text: logarithm

text: of

text: the

text: Bayes

text: factor,

text: ln

text: deﬁned

text: via

text: has

text: been

text: boosted

text: by

text: the

text: injection.

text: On

text: the

text: other

text: hand,

text: if

text: the

text: recovered

text: velocity

text: semi-amplitude

text: is

text: signiﬁcantly

text: ln

text: ln

text: P(y|Mk)

text: ln

text: lower

text: than

text: our

text: best-ﬁt

text: values,

text: that

text: would

text: suggest

text: the

text: ac-

text: tual

text: data

text: is

text: not

text: supported

text: by

text: the

text: instrument,

text: or

text: in

text: other

text: words,

text: the

text: precision

text: of

text: the

text: instrument

text: plus

text: the

text: current

text: Furthermore,

text: the

text: model

text: containing

text: the

text: best-ﬁt

text: param-

text: number

text: of

text: observations

text: do

text: not

text: allow

text: the

text: signal

text: to

text: be

text: eters

text: that

text: support

text: the

text: signal

text: has

text: to

text: fulﬁll

text: the

text: detection

text: detected.

text: When

text: boosting

text: our

text: signal,

text: we

text: recover

text: the

text: can-

text: criteria

text: described

text: in

text: Tuomi

text: It

text: must

text: hold

text: that

text: didate

text: period

text: for

text: HARPS

text: OLD,

text: HIRES

text: and

text: CHIRON

text: datasets.

text: For

text: CHIRON,

text: however,

text: the

text: expected

text: peak

text: at

text: P(y|M

text: sP(y|M

text: k

text: k

text: in

text: the

text: GLS

text: is

text: not

text: unique,

text: although

text: it

text: is

text: above

text: the

text: signiﬁcance

text: level,

text: as

text: can

text: be

text: seen

text: from

text: the

text: pe-

text: where

text: s

text: Hence

text: using

text: the

text: Bayes

text: factor

text: deﬁned

text: in

text: riodograms

text: shown

text: in

text: Figure

text: In

text: the

text: case

text: of

text: PFS

text: data,

text: equation

text: we

text: require

text: that

text: the

text: M

text: model

text: describing

text: the

text: we

text: did

text: not

text: recover

text: the

text: candidate

text: period.

text: Instead,

text: we

text: k

text: k-keplerian

text: signal

text: has

text: to

text: be

text: more

text: statistically

text: probable

text: found

text: a

text: strong

text: power

text: at

text: ∼5

text: days.

text: This

text: could

text: be

text: caused

text: Draft

text: version

text: May

text: than

text: the

text: M

text: model

text: associated

text: with

text: the

text: by

text: the

text: sparse

text: sampling

text: and

text: lower

text: number

text: of

text: observations

text: Preprint

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: signal.

text: Following

text: the

text: conservative

text: threshold

text: from

text: Tuomi

text: available

text: from

text: this

text: instrument.

text: THE

text: TEST

text: CASE

text: OF

text: DIFFICULTIES

text: DISENTANGLING

text: WEAK

text: DOPPLER

text: SIGNALS

text: FROM

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 30]. It seems therefore pertinent to discuss the similarities and diﬀerences be- tween Eq. (3) and the Breit-Wigner distribution with energy-dependent width. First, both approaches yield quasi-Lorentzian peaks. However, in the case of Eq. (3), the Lorentzian is distorted by the matrix element of the interaction, whereas in the case of the Breit-Wigner distribution with energy-dependent width the distortion is produced by the energy dependence of the width. Sec- ond, a ﬁt of the experimental decay energy spectrum (after deconvolution of the experimental error) using Eq. (3) would yield quantitatively diﬀerent resonant energies and widths than using the Breit-Wigner distribution with energy-dependent width. Third, and most importantly, an energy-dependent width Γ(E) implies an energy-dependent lifetime τ(E) = ¯h . Thus, although Γ(E) from a data-analysis point of view the Breit-Wigner distributions with energy- dependent width may not seem very diﬀerent from Eq. (3), from a theoretical point of view they imply that resonances have diﬀerent lifetimes for diﬀerent energies. By contrast, when one describes a resonance by a Gamow state and its decay energy spectrum by Eq. (3), the (mean) lifetime is given by τ = ¯h , R ΓR and it is an intrinsic property of the resonance that doesn’t depend on its energy. Four, instead of the pole width Γ being energy dependent, in the R Gamow-state approach what depends on the energy is the diﬀerential decay width dΓ(E)/dE of Eq. (1), and such dependence takes into account that the resonance couples to the continuum with diﬀerent strengths at diﬀerent energies, while at the same time Γ , and therefore τ , are energy independent. R R In principle, it is possible to test experimentally whether Eq. (3) or the Breit- Wigner distribution with energy-dependent width should be used as the true theoretical decay energy distribution of resonances. One could prepare the res- onance at diﬀerent energies, measure the mean lifetime for each energy, and check whether the lifetime changes with energy (as in Breit-Wigner distribu- tions with energy-dependent width) or not (as in Eq. (3)). 5 Conclusions Decay energy spectra of radioactive nuclei are routinely measured, and theoret- ical nuclear models should be able to predict such spectra. Within the limita- tions of a simple two-body model of 25O, we have shown that the Gamow-state description of resonances is able to produce such spectra. We have modeled the 25O nucleus as a valence neutron interacting with an 24O core, and applied the formalism of Ref. [18] to obtain the decay spectra of the ground and ﬁrst- excited states. The resulting spectra have a quasi-Lorentzian peak centered around the resonant energy, and are qualitatively similar to those of simpler models [19], another example of the universality of resonance phenomena. We have also seen that the ﬁts of experimental decay energy spectra that use a Breit-Wigner distribution with energy-dependent width are diﬀerent, both quantitatively and phenomenologically, from ﬁts that use the Gamow-state ap- 12 ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: It

text: seems

text: therefore

text: pertinent

text: to

text: discuss

text: the

text: similarities

text: and

text: diﬀerences

text: be-

text: tween

text: Eq.

text: and

text: the

text: Breit-Wigner

text: distribution

text: with

text: energy-dependent

text: width.

text: First,

text: both

text: approaches

text: yield

text: quasi-Lorentzian

text: peaks.

text: However,

text: in

text: the

text: case

text: of

text: Eq.

text: the

text: Lorentzian

text: is

text: distorted

text: by

text: the

text: matrix

text: element

text: of

text: the

text: interaction,

text: whereas

text: in

text: the

text: case

text: of

text: the

text: Breit-Wigner

text: distribution

text: with

text: energy-dependent

text: width

text: the

text: distortion

text: is

text: produced

text: by

text: the

text: energy

text: dependence

text: of

text: the

text: width.

text: Sec-

text: ond,

text: a

text: ﬁt

text: of

text: the

text: experimental

text: decay

text: energy

text: spectrum

text: (after

text: deconvolution

text: of

text: the

text: experimental

text: error)

text: using

text: Eq.

text: would

text: yield

text: quantitatively

text: diﬀerent

text: resonant

text: energies

text: and

text: widths

text: than

text: using

text: the

text: Breit-Wigner

text: distribution

text: with

text: energy-dependent

text: width.

text: Third,

text: and

text: most

text: importantly,

text: an

text: energy-dependent

text: width

text: Γ(E)

text: implies

text: an

text: energy-dependent

text: lifetime

text: τ(E)

text: ¯h

text: Thus,

text: although

text: Γ(E)

text: from

text: a

text: data-analysis

text: point

text: of

text: view

text: the

text: Breit-Wigner

text: distributions

text: with

text: energy-

text: dependent

text: width

text: may

text: not

text: seem

text: very

text: diﬀerent

text: from

text: Eq.

text: from

text: a

text: theoretical

text: point

text: of

text: view

text: they

text: imply

text: that

text: resonances

text: have

text: diﬀerent

text: lifetimes

text: for

text: diﬀerent

text: energies.

text: By

text: contrast,

text: when

text: one

text: describes

text: a

text: resonance

text: by

text: a

text: Gamow

text: state

text: and

text: its

text: decay

text: energy

text: spectrum

text: by

text: Eq.

text: the

text: (mean)

text: lifetime

text: is

text: given

text: by

text: τ

text: ¯h

text: R

text: ΓR

text: and

text: it

text: is

text: an

text: intrinsic

text: property

text: of

text: the

text: resonance

text: that

text: doesn’t

text: depend

text: on

text: its

text: energy.

text: Four,

text: instead

text: of

text: the

text: pole

text: width

text: Γ

text: being

text: energy

text: dependent,

text: in

text: the

text: R

text: Gamow-state

text: approach

text: what

text: depends

text: on

text: the

text: energy

text: is

text: the

text: diﬀerential

text: decay

text: width

text: dΓ(E)/dE

text: of

text: Eq.

text: and

text: such

text: dependence

text: takes

text: into

text: account

text: that

text: the

text: resonance

text: couples

text: to

text: the

text: continuum

text: with

text: diﬀerent

text: strengths

text: at

text: diﬀerent

text: energies,

text: while

text: at

text: the

text: same

text: time

text: Γ

text: and

text: therefore

text: τ

text: are

text: energy

text: independent.

text: R

text: R

text: In

text: principle,

text: it

text: is

text: possible

text: to

text: test

text: experimentally

text: whether

text: Eq.

text: or

text: the

text: Breit-

text: Wigner

text: distribution

text: with

text: energy-dependent

text: width

text: should

text: be

text: used

text: as

text: the

text: true

text: theoretical

text: decay

text: energy

text: distribution

text: of

text: resonances.

text: One

text: could

text: prepare

text: the

text: res-

text: onance

text: at

text: diﬀerent

text: energies,

text: measure

text: the

text: mean

text: lifetime

text: for

text: each

text: energy,

text: and

text: check

text: whether

text: the

text: lifetime

text: changes

text: with

text: energy

text: (as

text: in

text: Breit-Wigner

text: distribu-

text: tions

text: with

text: energy-dependent

text: width)

text: or

text: not

text: (as

text: in

text: Eq.

text: Conclusions

text: Decay

text: energy

text: spectra

text: of

text: radioactive

text: nuclei

text: are

text: routinely

text: measured,

text: and

text: theoret-

text: ical

text: nuclear

text: models

text: should

text: be

text: able

text: to

text: predict

text: such

text: spectra.

text: Within

text: the

text: limita-

text: tions

text: of

text: a

text: simple

text: two-body

text: model

text: of

text: we

text: have

text: shown

text: that

text: the

text: Gamow-state

text: description

text: of

text: resonances

text: is

text: able

text: to

text: produce

text: such

text: spectra.

text: We

text: have

text: modeled

text: the

text: nucleus

text: as

text: a

text: valence

text: neutron

text: interacting

text: with

text: an

text: core,

text: and

text: applied

text: the

text: formalism

text: of

text: Ref.

text: to

text: obtain

text: the

text: decay

text: spectra

text: of

text: the

text: ground

text: and

text: ﬁrst-

text: excited

text: states.

text: The

text: resulting

text: spectra

text: have

text: a

text: quasi-Lorentzian

text: peak

text: centered

text: around

text: the

text: resonant

text: energy,

text: and

text: are

text: qualitatively

text: similar

text: to

text: those

text: of

text: simpler

text: models

text: another

text: example

text: of

text: the

text: universality

text: of

text: resonance

text: phenomena.

text: We

text: have

text: also

text: seen

text: that

text: the

text: ﬁts

text: of

text: experimental

text: decay

text: energy

text: spectra

text: that

text: use

text: a

text: Breit-Wigner

text: distribution

text: with

text: energy-dependent

text: width

text: are

text: diﬀerent,

text: both

text: quantitatively

text: and

text: phenomenologically,

text: from

text: ﬁts

text: that

text: use

text: the

text: Gamow-state

text: ap-

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [28] M.D. Jones, et al.; arXiv:1710.04706. [29] E. Lunderberg, Phys. Rev. Lett. 108 (2012) 142503. [30] Z. Kohley, et al., Phys. Rev. C 87 (2013) 011304(R). [31] G. Gamow, Z. Phys. 51 (1928) 204. [32] J.R. Taylor, Scattering Theory, John Wiley & Sons (1972). [33] T. Vertse, K.F. Pal, Z. Balogh, Comput. Phys. Commun. 27 (1982) 309. [34] L.Gr. Ixaru, M. Rizea, T. Vertse, Comput. Phys. Commun. 85 (1995) 217. [35] R.J. Liotta, E. Maglione, N. Sandulescu, T. Vertse, Phys. Lett. B 367 (1996) 1. [36] https://www.wolfram.com/mathematica/ [37] Numerial Recipes, http://apps.nrbook.com/fortran/index.html. [38] A. Volya, V. Zelevinsky, Phys. Rev. C 74 (2006) 064314. [39] Evaluated nuclear structure data ﬁle, http://www.nndc.bnl.gov. [40] H. Frauenfelder, E.M. Henley, Subatomic Physics, Prentice Hall (1991). [41] The LHCb Collaboration, arXiv:1711.05490. [42] S. Rakityansky, J. Phys.: Conf. Ser. 915 (2017) 012008. [43] L. Schwartz, Mathematics for the Physical Sciences, Hermann (1966). 17

text: M.D.

text: Jones,

text: et

text: al.;

text: E.

text: Lunderberg,

text: Phys.

text: Rev.

text: Lett.

text: Z.

text: Kohley,

text: et

text: al.,

text: Phys.

text: Rev.

text: C

text: G.

text: Gamow,

text: Z.

text: Phys.

text: J.R.

text: Taylor,

text: Scattering

text: Theory,

text: John

text: Wiley

text: Sons

text: T.

text: Vertse,

text: K.F.

text: Pal,

text: Z.

text: Balogh,

text: Comput.

text: Phys.

text: Commun.

text: L.Gr.

text: Ixaru,

text: M.

text: Rizea,

text: T.

text: Vertse,

text: Comput.

text: Phys.

text: Commun.

text: R.J.

text: Liotta,

text: E.

text: Maglione,

text: N.

text: Sandulescu,

text: T.

text: Vertse,

text: Phys.

text: Lett.

text: B

text: Numerial

text: Recipes,

text: A.

text: Volya,

text: V.

text: Zelevinsky,

text: Phys.

text: Rev.

text: C

text: Evaluated

text: nuclear

text: structure

text: data

text: ﬁle,

text: H.

text: Frauenfelder,

text: E.M.

text: Henley,

text: Subatomic

text: Physics,

text: Prentice

text: Hall

text: The

text: LHCb

text: Collaboration,

text: S.

text: Rakityansky,

text: J.

text: Phys.:

text: Conf.

text: Ser.

text: L.

text: Schwartz,

text: Mathematics

text: for

text: the

text: Physical

text: Sciences,

text: Hermann


text: where Γ is the total decay width, (cid:32) (cid:33) (cid:90) (cid:90) ∞ ∞ dΓ |(cid:104) | | (cid:105)| Γ = dE = 2π L(E) E V z 2dE . (4) dE R 0 0 We will identify the theoretical spectrum of Eq. (3) with the experimental decay energy spectrum. It should be noted that Γ is in general diﬀerent from Γ , and therefore Γ is R not related to the lifetime of the resonance. Physically, we can interpret Γ as a measure of the overall strength of the interaction between the resonance and the continuum. This is why in Sec. 1 we will use Γ to quantify the relative strength of the spectra of diﬀerent resonances. It should also be noted that Eqs. (3) and (4) represent, in a way, an extension of Fermi’s Golden Rule to the case where the broadening of the decay energy spectrum is taken into account. Such broadening is provided by the Breit- Wigner distribution. 2.2 The Schr¨odinger equation of the valence neutron Because 24O is a doubly magic nucleus [22], we are going to neglect the many- body nature of 25O and treat it as an 24O core plus a single neutron that is subject to a mean-ﬁeld potential created by the 24O core. As it is customary, we will model such mean-ﬁeld potential by the Woods-Saxon potential and a spin-orbit interaction, − 1 df(r) V (r) = V (r) + V (r) = V f(r) + V ξ , (5) WS SO 0 so r dr l,j where V > 0 represents the potential well depth, and V > 0 represents the 0 so strength of the spin-orbit interaction. The function f(r) is given by (cid:16) (cid:17) 1 f(r) = , (6) − 1 + exp r R a where a is the diﬀuseness parameter (or surface thickness), and R = r A1/3 is 0 the nuclear radius, A being the mass number. The function ξ is given by l,j 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: Γ

text: is

text: the

text: total

text: decay

text: width,

text: dΓ

text: Γ

text: dE

text: 2π

text: L(E)

text: E

text: V

text: z

text: dE

text: R

text: We

text: will

text: identify

text: the

text: theoretical

text: spectrum

text: of

text: Eq.

text: with

text: the

text: experimental

text: decay

text: energy

text: spectrum.

text: It

text: should

text: be

text: noted

text: that

text: Γ

text: is

text: in

text: general

text: diﬀerent

text: from

text: Γ

text: and

text: therefore

text: Γ

text: is

text: R

text: not

text: related

text: to

text: the

text: lifetime

text: of

text: the

text: resonance.

text: Physically,

text: we

text: can

text: interpret

text: Γ

text: as

text: a

text: measure

text: of

text: the

text: overall

text: strength

text: of

text: the

text: interaction

text: between

text: the

text: resonance

text: and

text: the

text: continuum.

text: This

text: is

text: why

text: in

text: Sec.

text: we

text: will

text: use

text: Γ

text: to

text: quantify

text: the

text: relative

text: strength

text: of

text: the

text: spectra

text: of

text: diﬀerent

text: resonances.

text: It

text: should

text: also

text: be

text: noted

text: that

text: Eqs.

text: and

text: represent,

text: in

text: a

text: way,

text: an

text: extension

text: of

text: Fermi’s

text: Golden

text: Rule

text: to

text: the

text: case

text: where

text: the

text: broadening

text: of

text: the

text: decay

text: energy

text: spectrum

text: is

text: taken

text: into

text: account.

text: Such

text: broadening

text: is

text: provided

text: by

text: the

text: Breit-

text: Wigner

text: distribution.

text: The

text: Schr¨odinger

text: equation

text: of

text: the

text: valence

text: neutron

text: Because

text: is

text: a

text: doubly

text: magic

text: nucleus

text: we

text: are

text: going

text: to

text: neglect

text: the

text: many-

text: body

text: nature

text: of

text: and

text: treat

text: it

text: as

text: an

text: core

text: plus

text: a

text: single

text: neutron

text: that

text: is

text: subject

text: to

text: a

text: mean-ﬁeld

text: potential

text: created

text: by

text: the

text: core.

text: As

text: it

text: is

text: customary,

text: we

text: will

text: model

text: such

text: mean-ﬁeld

text: potential

text: by

text: the

text: Woods-Saxon

text: potential

text: and

text: a

text: spin-orbit

text: interaction,

text: df(r)

text: V

text: (r)

text: V

text: (r)

text: V

text: (r)

text: V

text: f(r)

text: V

text: ξ

text: WS

text: SO

text: so

text: r

text: dr

text: l,j

text: where

text: V

text: represents

text: the

text: potential

text: well

text: depth,

text: and

text: V

text: represents

text: the

text: so

text: strength

text: of

text: the

text: spin-orbit

text: interaction.

text: The

text: function

text: f(r)

text: is

text: given

text: by

text: f(r)

text: exp

text: r

text: R

text: a

text: where

text: a

text: is

text: the

text: diﬀuseness

text: parameter

text: (or

text: surface

text: thickness),

text: and

text: R

text: r

text: is

text: the

text: nuclear

text: radius,

text: A

text: being

text: the

text: mass

text: number.

text: The

text: function

text: ξ

text: is

text: given

text: by

text: l,j

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: a is very small, V (r) becomes, for practical purposes, the delta-shell potential a centered at r = R. Since the potential V (r) becomes very singular when a is small, care must be a taken in obtaining the resonant energies when a tends to zero. In our case, we used a two-step process to obtain the resonant energies. In the ﬁrst step, we obtained the resonant energies for decreasing values of a; in the second → step, we extrapolated [37] the result to a 0. In this way, we ﬁrst calculated the l = 0 ground (bound) and ﬁrst excited (unbound) states for the case that − − R = 6.466 fm and 2µ = 0.047892 MeV 1-fm 2 (which correspond to 133Sn), ¯h2 and for decreasing values of a up to a = 0.04 fm. Afterward, we extrapolated − the results using four order algebraic extrapolation [37] up to a = 10 5 fm. In order to compare our results with those of Ref. [19], we need to recall that the results of Ref. [19] were given in terms of the dimensionless coupling constant λ = 2µRg, Hence, with our choice of R and µ (R = 6.466 fm and ¯h2 − − 2µ = 0.047892 MeV 1-fm 2), the strength g can be written in terms of λ as ¯h2 g = 3.229 λ MeV-fm, and the energies and widths of Ref. [19] are given in units of h¯2/2µR2 = 0.4994 MeV. In Table 1, we compare the calculated ground state and ﬁrst-excited state − − − energies with those of Ref. [19] for λ = 0.5, 10, and 100. As can be seen in Table 1, our numerical results are in fairly good agreement with the exact ones. Table 1 Comparison of the calculated ground state energy E and ﬁrst excited state energy gs z with those of Ref. [19]. R Strength E (MeV) z (MeV) gs R λ g (MeV) Exact Present work Exact Present work − − − − − − 0.5 1.6146 0.19711 0.20554 5.935 i 5.180 5.895 i 5.204 − − − − − − 10 32.29 12.484 12.475 5.897 i 0.356 5.823 i 0.335 − − − − − − 100 322.9 1248.5 1139.0 5.029 i 0.00320 5.030 i 0.00298 The calculation of the decay width of Eq. (4) involves the resonant states and the free scattering states, and therefore constitutes a more demanding test. In Table 2, we compare the calculated decay width with the exact one for the − − − ﬁrst excited state when λ = 0.5, 10, and 100. As can be seen in Table 2, our calculated decay width agrees well with the − − exact one for strong couplings (λ = 10, 100) but not for weak couplings 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: a

text: is

text: very

text: small,

text: V

text: (r)

text: becomes,

text: for

text: practical

text: purposes,

text: the

text: delta-shell

text: potential

text: a

text: centered

text: at

text: r

text: R.

text: Since

text: the

text: potential

text: V

text: (r)

text: becomes

text: very

text: singular

text: when

text: a

text: is

text: small,

text: care

text: must

text: be

text: a

text: taken

text: in

text: obtaining

text: the

text: resonant

text: energies

text: when

text: a

text: tends

text: to

text: zero.

text: In

text: our

text: case,

text: we

text: used

text: a

text: two-step

text: process

text: to

text: obtain

text: the

text: resonant

text: energies.

text: In

text: the

text: ﬁrst

text: step,

text: we

text: obtained

text: the

text: resonant

text: energies

text: for

text: decreasing

text: values

text: of

text: a;

text: in

text: the

text: second

text: step,

text: we

text: extrapolated

text: the

text: result

text: to

text: a

text: In

text: this

text: way,

text: we

text: ﬁrst

text: calculated

text: the

text: l

text: ground

text: (bound)

text: and

text: ﬁrst

text: excited

text: (unbound)

text: states

text: for

text: the

text: case

text: that

text: R

text: fm

text: and

text: 2µ

text: MeV

text: (which

text: correspond

text: to

text: and

text: for

text: decreasing

text: values

text: of

text: a

text: up

text: to

text: a

text: fm.

text: Afterward,

text: we

text: extrapolated

text: the

text: results

text: using

text: four

text: order

text: algebraic

text: extrapolation

text: up

text: to

text: a

text: fm.

text: In

text: order

text: to

text: compare

text: our

text: results

text: with

text: those

text: of

text: Ref.

text: we

text: need

text: to

text: recall

text: that

text: the

text: results

text: of

text: Ref.

text: were

text: given

text: in

text: terms

text: of

text: the

text: dimensionless

text: coupling

text: constant

text: λ

text: Hence,

text: with

text: our

text: choice

text: of

text: R

text: and

text: µ

text: (R

text: fm

text: and

text: 2µ

text: MeV

text: the

text: strength

text: g

text: can

text: be

text: written

text: in

text: terms

text: of

text: λ

text: as

text: g

text: λ

text: MeV-fm,

text: and

text: the

text: energies

text: and

text: widths

text: of

text: Ref.

text: are

text: given

text: in

text: units

text: of

text: MeV.

text: In

text: Table

text: we

text: compare

text: the

text: calculated

text: ground

text: state

text: and

text: ﬁrst-excited

text: state

text: energies

text: with

text: those

text: of

text: Ref.

text: for

text: λ

text: and

text: As

text: can

text: be

text: seen

text: in

text: Table

text: our

text: numerical

text: results

text: are

text: in

text: fairly

text: good

text: agreement

text: with

text: the

text: exact

text: ones.

text: Table

text: Comparison

text: of

text: the

text: calculated

text: ground

text: state

text: energy

text: E

text: and

text: ﬁrst

text: excited

text: state

text: energy

text: gs

text: z

text: with

text: those

text: of

text: Ref.

text: R

text: Strength

text: E

text: (MeV)

text: z

text: (MeV)

text: gs

text: R

text: λ

text: g

text: (MeV)

text: Exact

text: Present

text: work

text: Exact

text: Present

text: work

text: i

text: i

text: i

text: i

text: i

text: i

text: The

text: calculation

text: of

text: the

text: decay

text: width

text: of

text: Eq.

text: involves

text: the

text: resonant

text: states

text: and

text: the

text: free

text: scattering

text: states,

text: and

text: therefore

text: constitutes

text: a

text: more

text: demanding

text: test.

text: In

text: Table

text: we

text: compare

text: the

text: calculated

text: decay

text: width

text: with

text: the

text: exact

text: one

text: for

text: the

text: ﬁrst

text: excited

text: state

text: when

text: λ

text: and

text: As

text: can

text: be

text: seen

text: in

text: Table

text: our

text: calculated

text: decay

text: width

text: agrees

text: well

text: with

text: the

text: exact

text: one

text: for

text: strong

text: couplings

text: (λ

text: but

text: not

text: for

text: weak

text: couplings

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: SparsePak Survey of Nearby Galaxy Mergers 15 0.2 2 2 2 primary secondary 0.0 0.2 3 3 3 0.4 1 1 1 0.6 µ < 1. 85 µ ∼ 2. 5 µ > 4 L L L 6 7 6 0.8 0 50 100 150 200 2500 50 100 150 200 2500 50 100 150 200 250 velocity dispersion σ (km/s) velocity dispersion σ (km/s) velocity dispersion σ (km/s) Figure 11. Plots log ([N II]/H ) vs. velocity dispersion for components in diﬀerent mass ratio bins: equal mass mergers (left) 1 85, non-equal major mergers (middle) 1 8150 α4 0, and minor majors (right) 4 0. The number of systems in the bins is written in lower riµgLht<corn.er of each panel. As we we go from righ.t to<lefµt,Lo<ne c.an see that not only is the fraµcLtio>n o.f points in Groups 2 and 3 increasing, but also the points in Group 1 reveal a generally higher log ([N II]/H ) and velocity dispersion. The components of the primary and secondary galaxies are plotted with black and cyan points, respectively. 10 α In the minor mergers (right panel), on average, the components of the primary galaxies have a signiﬁcantly higher log ([N II]/H ) compared to the secondary 10 α ones. system name light separation previous models 5.1.1 UGC 07593 ratio (kpc) Figure 12 shows slices of the map of scores across three en- UGC 07593 1.81 5.6 — counterparameters,eccentricity,pericentricdistance,andtimesince NGC 5257/8 1.17 33.8 Privon et al. (2013) pericenter. The slices are taken at the best-ﬁt parameters shown by UGC 12914 1.61 14.8 Vollmer et al. (2012) the cyan box. This system is in a relatively late stage at 0 68±0 08 Arp 238 1.51 19.7 — . 0.01 of the time between the ﬁrst and the second passages. The reco.n- Arp 87 1.27 2.5 — structed pericentric distance and eccentricity are 0 50 ±0 16 Table 3. List of the systems we attempted to model using our method. The and 0 8±0 05, respectively. Taking the physical lengt.h and0..v1e6loRcvitiyr lightratiosareobtainedfrom2MASSKsband,exceptforUGC07593which . 0.05 scaling int.o account we derive the encounter parameters with phys- is not resolved in the 2MASS Ks survey, and we used SDSS r-band light ratio instead. ical units (See Mortazavi et al. 2016). The reconstructed time since the ﬁrst passage is 27±30 Myr. The physical pericentric distance is 3 2 5±0 6 kpc. Both galaxies are almost half-way between a prograde an. d a0p..7olar orbit with the inclinations being at about 45◦ ±20◦ from orientations have all resulted in good scores, and consequently, the the orbital plane. reconstructed encounter parameters are not unique. Mortazavi et al. 5.1.2 Unsuccessful Attempts (2016) found good convergence in most of the tests on discs that were almost in a prograde orbit. In contrast, they found poor and UGC 12914: The velocity map of this system, shown in Figure 5, point in the direction of why our method cannot reconstruct the fair convergence when any of the two interacting discs were in a observed features. Fibers with strong H line emission are scattered retrograde or polar orbit. α in the region between the two galaxies, and the velocity of Group Table3showsthesystemsweselectedfordynamicalmodeling. 1 components do not show an smooth rotation. The gas in the Here we exclude NGC 4676, the Mice, which we modeled and bridgecontainsasigniﬁcantamountofshocks.Inaddition,powerful discussed extensively in Mortazavi et al. (2018). The systems in HI, CO, radio continuum, and X-ray emission is detected in the Table 3 are the most equal mass systems based on their 2MASS bridge, suggesting that the gas originally in the two discs have been K-band magnitude (Skrutskie et al. 2006), which is used as a crude stripped oﬀ as a result of a virtually head on encounter (Condon estimate of stellar mass. We select the galaxies with a light ratio et al. 1993; Braine et al. 2003; Gao et al. 2003; Vollmer et al. less that 1.85, and make an attempt to model them with equal 2012; Appleton et al. 2015). It does not accompany the rest of the mass identikit models. For two of these systems we found prior baryons, particularly the collisionless stars, anymore and it can not dynamical models in the literature, which are mentioned in Table 3. be reconstructed by identikit test particles. 5.1 Results NGC 5257/8: NGC 5257 has a prominent tail in HI that is We found a ﬁrst ever model for UGC 07593, though the dynam- absent in the optical image. This tail played an important role in the ical modeling method we used did not well converge for the other modeling by Privon et al. (2013). Without the information in the systems. Here we present the encounter parameters we found for extended tail we do not have the same constraints to pin down the UGC 07593 and discuss brieﬂy the possible reasons for the failure encounter parameters. Furthermore, the model presented by Privon of our method for some of the other systems. et al. (2013) indicates that NGC 5258 is in a nearly polar orbit. 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