Constraining the age of a magnetar possibly associated with FRB 121102

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by  Xiao-Feng Cao

Draftversion April5,2017
PreprinttypesetusingLATEXstyleemulateapjv.12/16/11
CONSTRAINING THE AGE OF A MAGNETAR POSSIBLY ASSOCIATED WITH FRB 121102
Xiao-Feng Cao1, Yun-Wei Yu2,3, and Zi-Gao Dai4,5
Draft version April 5, 2017
ABSTRACT
The similarity of the host galaxy of FRB 121102 with those of long gamma-ray bursts and Type I
7 super-luminous supernovae suggests that this FRB could be associated with a young magnetar. By
1 assuming the FRB emissionto be producedwithin the magnetosphere,we derive a lowerlimit on the
0 ageofthemagnetar,afterwhichGHzemissionisabletoescapefreelyfromthedenserelativisticwind
2 of the magnetar. Another lower limit is obtained by requiring the dispersion measure contributed by
theelectron/positronpairwindtobeconsistentwiththeobservationsofthehostgalaxy. Furthermore,
r
p wealsoderivesomeupperlimitsonthemagnetaragewithdiscussionsonpossibleenergysourcesofthe
A FRB emission and the recently-discoveredpersistent radio counterpart. As a result, some constraints
onmodelparametersare addressedby reconcilingthe lowerlimits withthe possible upper limits that

4 are derived with an assumption of rotational energy source.
Subject headings: radio continuum: general — stars: magnetars — stars: neutron
]
E
H 1. INTRODUCTION shocks due to magnetar flares (Lyubarsky 2014), super-
h. Fast radio bursts (FRBs) are mysterious radio tran- giant pulses from pulsars (Cordes & Wasserman 2016),
encountersofpulsarsandasteroidbelts(Daietal. 2016),
p sients that have been observedto have typical durations
accretion onto a neutron star from its magnetized white
- of a few milliseconds and fluxes of up to a few tens of
o Jansky at 1 GHz (Lorimer et al. 2007; Thornton et dwarfcompanion(Gu etal. 2016),andpulsarssuddenly
r al. 2013; ∼Burke-Spolaor & Bannister 2014; Spitler et “combed” by a nearby strong plasma stream (Zhang
t 2017). In addition, some exotic models have also be
s al. 2014; Ravi et al. 2015; Champion et al. 2016). Al-
a proposed such as oscillations of superconducting cosmic
thoughtheirphysicaloriginisunknown,FRBsarewidely
[ string loops (Cai et al. 2012; Yu et al. 2014).
believed to come from cosmological distances in view
More observational constraints are undoubtedly nec-
3 of their anomalously high dispersion measures (DMs;
v 300 1600pc cm−3) that are difficult to be accounted essary for distinguishing between these models. An im-
2 f∼or by−Galactic high-latitude objects. Thus, the peak portant clue has been provided by the discovery of a re-
peatedFRBintheAreciboPulsarALFASurveyon2012
8 radioluminosityandtotalenergyreleaseofFRBsarees-
4 timated to 1042 1043 erg s−1 and 1039 1040erg, November2(Spitler et al. 2014),whichwassurprisingly
5 respectively∼(e.g. B−era 2016;Cao et al∼. 2017)−. detectedagainon2015May17andJune2(Spitler etal.
0 2016;Scholzetal. 2016). Theselongtimegaps(both572
Accordingtothemillisecondtimescaleandhighenergy
. and23days)makethecatastrophicmodelsdifficulttobe
1 release, on one hand, catastrophic collapses/mergers of
saved. More excitingly, unambiguous multi-wavelength
0 compact star systems are often proposedto be responsi-
counterpartsofFRB121102havebeenrecentlycaptured
7 ble for FRBs, including collapses of supra-massive neu-
andidentifiedby Chatterjeeetal. (2017),Marcoteetal.
1 tron stars to black holes at several thousand to million
(2017), and Tendulkar et al. (2017), including a per-
: years old (Falcke & Rezzolla 2014) or at birth (Zhang
v sistent radio source and a low-metallicity, star-forming
2014),inspiralormergersofdoubleneutronstars(Totani
i dwarfgalaxy. Thedetectionofthehostgalaxyhelpedto
X 2013;Wang et al. 2016),mergersof binary white dwarfs
determinetheredshiftofFRB121102toz =0.19273(8),
r (Kashiyama et al. 2013), mergers of charged black holes which corresponds to a luminosity distance of 972 Mpc
a (Zhang 2016), and collisions of asteroids/comets with
and undoubtedly confirms the cosmologicaloriginof the
neutronstars(Geng&Huang2015). Ontheotherhand,
FRB (i.e. only FRB 121102 has been confirmed, not
in view of the coherent emission property, FRBs are
FRBs in general).
often connected with some energetic activities of pul-
The host galaxy of FRB 121102 was found to share
sars (more specifically, magnetars), such as giant flares
many common properties with those of long gamma-ray
of soft gamma-ray repeaters (Popov & Postnov 2010),
bursts (GRBs) and Type I super-luminous supernovae
synchrotronmaseremissionfromrelativistic,magnetized
(SLSNe). Therefore, the possibility that FRB 121102
is associated with a young magnetar is enhanced sig-
1School of Physics and Electronics Information, Hubei Uni- nificantly, since both GRBs and SLSNe are widely con-
versityofEducation,Wuhan430205, China
sidered to be powered by a newly-born rapidly rotat-
2InstituteofAstrophysics,CentralChinaNormalUniversity,
Wuhan430079, China,[email protected] ing magnetar (Usov 1992; Dai & Lu 1998a,b; Zhang &
3Key Laboratory of Quark and Lepton Physics (Central Meszaros2001;Woosley2010;Kasenetal. 2010). Sucha
China Normal University), Ministry of Education, Wuhan magnetarispossiblyembeddedwithinayoungsupernova
430079, China
remnant, which can trap radio emission at early times
4SchoolofAstronomyandSpaceScience,NanjingUniversity,
Nanjing210093, China and result in a significant contribution to DM (Kulkarni
5Key Laboratory of Modern Astronomy and Astrophysics et al. 2015; Piro 2016; Murase et al. 2016). Following
(NanjingUniversity),MinistryofEducation, China
2
this idea, the age of the magnetar of FRB 121102 was energy flux. Since the total energy flux carried by the
constrained to be about a few decades by Metzger et al. magnetar wind is completely provided by the magnetic
(2017). Nevertheless, such a supernova remnant is not spin-down of the magnetar, i.e., (σ +1)Γ N˙ m c2 =
L L w e
indispensable, even for the explanation of the observed L , the Lorentz factor at the light cylinder can be de-
sd
persistent radio emission (Dai et al. 2017). terminedbyΓ (L /N˙ m c2)1/3. Bytakingthespin-
L sd w e
In any case, before the FRB emission encounters the down luminosity∼as usual as L = B2R6Ω4/(6c3), the
supernova ejecta, it probably has to first penetrate a sd p
maximum value of plasma frequencies of the magnetar
denserelativisticwindfromthemagnetar,iftheemission
wind can be written as
is produced within the stellar magnetosphere. As first
suggestedby Yu (2014),this pulsar wind, whichconsists ν (r )=1.5 104µ1/3B2/3P−7/3GHz, (4)
ofanextremelygreatnumberofelectronsandpositrons, p L × ± p,14 −3
can cause a more serious radio trapping and a more sig- outside of the light cylinder. Hereafter the conventional
nificant DM contribution than the supernova ejecta, in notation Qx = Q/10x is adopted in cgs units. In order
particular, by considering of the relativistic boosting ef- to guarantee that GHz radio emission can freely pene-
fect. InthisLetter,therefore,wederiveamorestringent trate the wind, the above plasma frequency should not
constraintontheageofthemagnetarofFRB121102,by be higher than the radio frequency and thus we can ob-
constricting the electron/positronloading of the magne- tain the spin period as
tar wind to be consistent with the implications from the
P >61µ1/7B2/7ms. (5)
host galaxy observations. ± p,14
This lower limit is much longer than the initial spin pe-
2. LOWERLIMITSONSPINPERIODSANDMAGNETAR
riodofafewmillisecondsthatcanbeinferredfromSLSN
AGES
or GRB observations. Therefore the corresponding age
By considering of a magnetar of an angularfrequency,
of the magnetar can be constrained to
Ω, a surface polar strength of magnetic field, B , and
p
a radius, R = 106 cm, the electron-positron distribu- t >24µ2/7B−10/7yr, (6)
age ± p,14
tion of the magnetar magnetosphere can be described
which is calculated with the expression of spin-down
as usual by the Goldreich & Julian (GJ) particle den-
sity n (r) (ΩB /2πce)(r/R)−3 (Goldreich & Julian timescaleastsd =3Ic3/(Bp2R6Ω2),whereI =1045gcm2
GJ p
≈ is the moment of inertial of the magnetar. Correspond-
1969),theangle-dependenceofwhichisignoredforasim-
ingly, the age constraint due to the radio trapping of
ple order-of-magnitude analysis. Beyond the light cylin-
supernova ejecta can be derived from Equation (9) of
drical radius r = c/Ω, the corotation of the magneto-
L
Metzger et al. (2017) to
sphere can no longer be held. The magnetocentrifugal
force exerting on the plasma and especially the subse-
M 1/3
quent magnetic reconnections will launch a relativistic t >0.7f1/3 ej v−1 yr, (7)
wind. The particle number flux of the wind can be ex- age ion,−1(cid:18)10M⊙(cid:19) ej,9
tphreessspeidnapserNi˙owd.≈T4hπereL2±µ±mnuGltJi(prlLic)icty, wpahrearme ePte=r,µ2±π/,Ωrepis- wmhaesrse,afniodn,thMeevj,elaoncdityveojfatrheetehjeecitoan.izOabtivoinoufsrlayc,ttihoen,cothne-
resentsa ratioof the wind flux to the GJ flux, because a straint presented in Equation (7) can be ignored safely
great number of electrons and positrons could be gener- in comparison with the more stringent constraint given
ated spontaneously as the wind propagation and energy by the wind plasmafrequency asshowninEquation(6).
dissipation. Thenthedensityofthewindatradiusr can On the other hand, by taking the relativistic boosting
be expressed as effectintoaccount,theDMcontributedbythemagnetar
N˙ r −2 wind can be calculated by (Yu 2014)
w
nw(r)≈ 4πr2c =µ±nGJ(rL)(cid:18)rL(cid:19) . (1) DMw= 1 2Γ(r) nw(r)dr
(1+z)Z ·
On one hand, with the above density, the plasma fre- rL
quency of the magnetar wind at different radii can be =3ΓLµ±nGJ(rL)rL
calculated by (1+z)
Γ e2 µ±nGJ(rL) 1/2 r −1 =1.5×107µ2±/3Bp4/,134P−−311/3 pc cm−3. (8)
ν (r)= (2)
p
1+z (cid:20)πm Γ (cid:21) (cid:18)r (cid:19) For an upper limit value of DM , the spin period of
e L w,up
the magnetar can be constrained to
for r r , where Γ is the Lorentz factor of the wind
L
at the≥radius. Following Drenkhahn (2002), a reference P >90µ2/11B4/11DM−3/11ms, (9)
± p,14 w,up
dynamicalresultcanbeadoptedasΓ=Γ (r/r )1/3 and
L L
which corresponds to an age of
then we have
ν (r)= µ1±/2Γ1L/2 e2 n (r ) 1/2 r −5/6. (3) tage >53µ4±/11Bp−,1144/11DMw−,6u/p11yr. (10)
p GJ L
1+z (cid:20)πme (cid:21) (cid:18)rL(cid:19) For a comparison, the age constraint from the DM con-
straintonthesupernovaejectacanbewrittenasfollows:
The initial velocity of the wind can be set to the Alfv´en
rveesloencittsytahtetihneitliiaglhrtactyiolinbdeetwreaesnΓPLo∼yn√tiσnLg,flwuhxerteoσmLartetper- t >215f1/2 Mej 1/2v−1DM−1/2 yr, (11)
age ion,−1(cid:18)10M⊙(cid:19) ej,9 ej,up
3
4.5 4.5 4.5
4.0 4.0 4.0
Lyr 3.5 Lyr 3.5 Lyr 3.5
H(cid:144)tage0 3.0 H(cid:144)tage0 3.0 H(cid:144)tage0 3.0
g1 2.5 g1 2.5 g1 2.5
o o o
l 2.0 l 2.0 l 2.0
1.5 1.5 1.5
0 2 4 6 8 0 2 4 6 8 0 2 4 6 8
log Μ log Μ log Μ
10 ± 10 ± 10 ±
Fig.1.—Lower limitsonthe magnetar age that areobtained by constrainingthe windplasmafrequency (dotted greenline), the wind
DMcontribution(solidredline),andtheDMcontributionofsupernovaejecta(dashedblueline),whereaputativeupperlimitontheDM
of the FRBsource istaken as DMsrc,up =1pc cm−3. Thepanels fromlefttorightcorrespond to magnetic fields of 1013 G, 1014 G, and
1015 G,respectively.
whereDM istheupperlimitontheDMoftheejecta. al. 2011), which indicates a ratio of this energy to the
ej,up
ThisexpressionisderivedfromEquation(12)ofMetzger totalstellarrotationalenergyas∆E/E <5 10−22. If
rot
etal. (2017),whichwashowevernotaddressedtherebe- asame emissionmechanismis assumedfor FR×B 121102,
cause they did not separatedthe DM of the FRB source thenanabsolutelyimpossiblerotationalenergyof 1060
from that of the host galaxy. By comparing Equation ergwouldbe requiredtoexplainthe isotropic-equ∼ivalent
(10)with (11), the ageconstraintgivenbythe magnetar FRBenergyof∆E E =2 1039 erg. This difficulty
iso
wind can be more stringent than the supernova ejecta was also recently po∼inted out b×y Lyutikov (2017).
constraint as long as µ4/11B−14/11 >5. As an alternative scenario, we here propose that the
± p,14
energy release of an FRB is connected with a glitch-like
3. DISCUSSIONSONFRB121102 process,althoughthephysicsofthisprocessiscompletely
unknown. By denoting the sudden change of spin fre-
For FRB 121102, its total DM was measured to be
DM =558pccm−3(Spitleretal. 2016;Chatterjeeet quency by ∆Ω, the energy release can be calculated by
total
al. 2017),whichis contributedjointly bythe Milky Way 1 1
and its halo, the intergalactic medium, the host galaxy, ∆E = IΩ2 I(Ω ∆Ω)2 ≈IΩ∆Ω. (12)
2 − 2 −
and the FRB source itself. As analyzed by Tendulkar et
al. (2017), the sum of the last two contributions can be Observations of Galactic pulsars usually found ∆Ω/Ω
constrained to 55 . (DM +DM ) . 225pc cm−3. 10−9 10−6 for their glitches and the currentmaximum∼
Moreover, according to Ehqousattion 6sorcf Tendulkar et al. value−can be as large as 10−5 (Yuan et al. 2010;Manch-
(2017), the value of DM is probably not much lower ester & Hobbs 2011). Therefore,for a releasedenergy of
host
than 100pc cm−3, if FRB 121102does not offsetvery 2 1039 erg, the total rotational energy of the mag-
much∼from its host galaxy. Therefore,the DM contribu- n∼etar×can be constrained to be Erot (105 109)Eiso,
≫ −
tion leaving to the FRB source including the magnetar wherethesymbol“ ”isusedbecauseofthe repeatabil-
≫
windandthesupernovaremnantisverylimited,whichis ity of FRB 121102. As a result, the spin period and age
consistent with the small fluctuation of the DM of FRB of the magnetar can be derived to be P (0.1 10) s
121102 during the past few years (Spitler et al. 2016). andt (67 6.7 105)B−2 yr. Suchu≪pperlim−itson
For a putative DM upper limit of DM = 1pc cm−3 the maaggen≪etara−ge can×in prinpc,i1p4le be consistentwith the
src,up
andthreetypicalmagneticfields,thedifferentlowerlim- obtained lower limits, if FRBs can indeed be associated
its on the magnetar age are presented in Figure 1, as with the most giant glitches.
functions of the uncertain e± multiplicity. According to In any case, besides the rotational energy, some other
observations of pulsar wind nebulae, µ± is usually con- energy sources could still be available to power FRBs.
sideredtobeveryhigh,sinceagreatnumberofelectrons The most popular choice could be the magnetic en-
and positrons are needed to produce the observed wind ergy within a magnetar, which is of the order E
B
emission and to determine a typical Lorentz factor of 3 1049B2 ergforaninternalmagneticfieldstrengt∼h
104 105 of the wind (e.g. Yu et al. 2014). of×B int1,10616 G. However, the disadvantage of this
∼ − int
As a general result, the lower limit on the magnetar ∼
model is that no bright radio pulse was detected from
agecanbefoundtobe,atleast,aboutafewhundredsto
the giant flare of SGR 1806-20 (Tendulkar et al. 2017).
thousands years. Then, a question could arise: whether
As another possible solution, Dai et al. (2016) proposed
or not the rotational energy of the magnetar can power
that the FRB energy could be provided by the gravi-
the FRB emission and also the persistent radio coun-
tational energy of an asteroid as it is captured by and
terpart. First of all, according to some observations of
collides with the magnetar. Such a process can repeat
Galactic pulsars, it has been suggested that FRBs could
naturally if an asteroid belt is around the magnetar. In
be analogical to giant pulses that are powered by the
anycase,bothoftheabovealternativescenarioscansur-
spin-down of a magnetar. In this case, however, the
vive from the constraints on the magnetar age, with at
magnetar age of FRB 121102 would be constrained to
least a few hundreds to thousands years.
betage <9(LFRB/1041erg s−1)−1Bp−,114 yr(Metzgeretal. For the steady radio emission associated with FRB
2017), which is probably in conflict with the lower lim- 121102, it was currently considered to be produced by
its given above. Moreover, observationally, giant pulses synchrotronemissionofpulsarwindnebulae(Kashiyama
from the Crab pulsar, statistically, only have an en- & Murase 2017; Metzger et al. 2017; Dai et al. 2017).
ergy release ∆E 1028erg per giant pulse (Majid et By considering that the luminosity of the wind emis-
∼
4
sionis ultimately determinedby the spin-downluminos- sistent radio counterpart. In this case, the value of µ±
ity of the magnetar, it is convenient to simply require at emitting radii is required to be very high (see Dai et
the spin-down luminosity to be higher than the lumi- al. 2017 for an estimate of the number of emitting elec-
nosity of the steady radio emission, i.e., L > L = trons/positrons), which is very much higher than that
sd radio
3 1038erg s−1. This gives a very stringent constraint presented in Equation (17) for the light cylinder radius.
×
of Itisthereforeindicatedthattheleptonloadofmagnetar
wind could significantly evolve at large radii.
P <134B1/2 ms (13)
p,14
4. SUMMARY
and
TherecentdiscoveryofthehostgalaxyofFRB121102
t <116B−1 yr. (14) implies a possible connection between FRBs and long
age p,14 GRBs/SLSNe. CombiningwiththerepeatabilityofFRB
In any case, if this steady radio emission is not powered 121102,it was suggested that this FRB could be associ-
ated with a young magnetar and originate from some
by the spin-down of magnetar, the above constraint can
activities of the magnetar. In order to test this possi-
be removed but then some alternative scenarios should
besuggested. BycomparingEquation(14)with(10)and bility, we investigated the important influences on FRB
emission from the wind of magnetar, if the FRB emis-
(11), some relationships between the model parameters
sion is produced in the inner magnetosphere. Specifi-
can be derived as
cally, by evaluating the radio trapping and the DM con-
B >0.06µ4/3DM−2 (15) tribution by wind electrons/positrons, we derived some
p,14 ± w,up
lower limits on the spin periods and ages of magnetars
and visible as FRBs and applied these results to the case
of FRB 121102. Meanwhile, some possible upper limits
Bp,14 <0.5DM1ej/,2up (16) are also discussed by considering that FRB 121102 and
moreover its persistent radio emission could be powered
inordertomakethelowerandupperlimitsofthemagne- bytherotationalenergyofmagnetar. Byreconcilingthe
tar age consistent with each other. As shown, relatively
lower and upper limits, some constraints on the model
low magnetic fields are favored, which indicates FRBs
parameters were revealed. For example, for a putative
more probably associated with SLSNe than long GRBs. DMsrc,up 5 pc cm−3, µ± 100, and a relatively low
The coexistence of Equations (15) and (16) can further ∼ ∼
magnetic field of B 1014 G, all of the limits on the
give p ∼
age can reach a consensus at the age of about 100
∼
yr. According to the most allowable values of magnetic
µ± <5.5DM3w/,2upDM3ej/,8up. (17) fieldstrengths,FRBs are suggestedto be moreprobably
associated with SLSNe than long GRBs.
The value of µ± at radii much far away from the light
cylinder can in principle been inferred from the emis-
sion property of the wind. So far no hard X-ray/soft We thank Kohta Murase and Bing Zhang for their
gamma-raycounterpartshavebeendetectedwithin 1◦ comments and X. Zhou for her discussions on pulsar
∼
ofFRB 121102’sposition(Scholz et al. 2016;see Petroff glitches. This work was supported by the National Ba-
et al. 2015 for other FRBs), which could be a nature sic Research Program (“973” Program) of China (grant
result of the small value of µ±. However, as suggested No. 2014CB845800) and the National Natural Sci-
by Kashiyama & Murase (2017), Metzger et al. (2017), ence Foundation of China (grant Nos. 11473008 and
andDaietal. (2017),differentfromthetypicalCrab-like 11573014). XFC was supported by the National Nat-
nebulae, the wind emission of FRB 121102 could actu- ural Science Foundation of China (grant Nos. 11303010
ally be mainly in the radio band exhibiting as the per- and 11673008).
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