On the second derivatives of periods and braking indices in radio pulsars
aa r X i v : . [ a s t r o - ph . H E ] A ug On the second derivatives of periods and braking indices in radiopulsars
Malov I.Pushchino Radioastronomical Observatory, 142290 Moscow Region, Pushchino, RussiaSeptember 10, 2018
Abstract
The analysis of some braking mechanisms for neutron stars was carried out to determine the sign ofthe second derivative of the pulsar period. This quantity is the important parameter for calculations ofthe braking index n. It is shown that this derivative can be positive and lead to decreasing of n. It isnecessary to correct the methods of calculations of n used this moment because they are based as a ruleon the suggestion on the constancy of pulsar parameters (magnetic fields, angles between some axes and soon). The estimations of corrections to braking indices are obtained. It is shown that these corrections canbe marked for pulsars with long periods and their small derivatives. keywords braking mechanisms – magnetic fields – pulsars
One of the most important parameters characterizing evolution of pulsars is the so called braking index ndescribing the dependence of the angular rotation frequency on time: d Ω dt = − K Ω n , (1)where K is a constant determined by the mechanism braking the neutron star. The quantity n can becalculated by the following expression: n = Ω d Ω /dt ( d Ω /dt ) (2)It is worth noting that (2) is correct for any braking mechanism if K is constant. Usually the following form n = 2 − P d P/dt ( dP/dt ) (3)is used instead (2). Here P = π Ω is the rotation period. This period and its derivatives can be measuredduring long enough observations., and we can calculate on principle the braking index n and define the brakingmechanism. However measurements of the second derivatives are complicated as a rule and to this momentthey are determined precisely enough for several pulsars only. But there is the second difficulty in calculationsof n connected with the value of K . It is suggested usually that its dependences on magnetic induction, oninclination of magnetic moment to the rotation axis and other parameters do not depend on time. If K evolveswith the pulsar age (2) and (3) demand some corrections. Blandford and Romani (1988) pointed out on thepossible importance of the dependence of K on time many years ago. However the majority of authors believenow that K is constant.Here we discuss the possible corrections for (2) and (3) and signs of the second derivative of the rotationperiod. 1 Braking due to magneto-dipole radiation
The most popular braking mechanism in radio pulsar investigations is connected with the magneto-dipoleradiation of a magnetized spherical neutron star. In this case the rate of losses of the rotation energy is equaledto the radiation power: I Ω d Ω dt = − µ Ω sin β c , (4)where I is the moment of inertia of the neutron star, B s the magnetic induction at the pole of the neutronstar, R ∗ the radius of the neutron star, β the angle between the rotation axis and the magnetic dipole moment µ = B s R ∗ , c the speed of light. It is suggested usually that all quantities besides Ω are constants and that β = 90 ◦ . In this case (4) leads to n = 3. If I = 10 g cm and R ∗ = 10 cm , then we obtain the magneticinduction at the pole: B s = 6 . × (cid:18) P dPdt (cid:19) / G (5)The known catalogs (for example, Manchester et al., 2005) contain inductions at the magnetic equators.Their values are two times less than given by (5).The progress of the magneto-dipole model can be connected with the work of Davis and Goldstein (1970)suggesting the exponential falling of sin β with time:sin β = exp( − t/τ ) , (6)Using the formula (2) we obtain from (6): n = 3 + 2 tan β, (7)This equality has been cited many times in much more late works. In (6) τ is the characteristic time ofdecreasing of the angle β . In the common case the parameter K does not constant and (6) means that K asa function of β must depend on time. The correct form for n is the following one which has been given byBlandford and Romani (1988): n = Ω d Ω /dt (Ω /dt ) − Ω d Ω /dt dK/dtK (8)The model of braking due to magneto-dipole losses gives in the case of the constant magnetic field and theevolution of the angle β by the law (6) n = Ω d Ω /dt (Ω /dt ) + 2Ω( d Ω /dt ) τ (9)Philippov et al. (2014) gave the magneto-hydrodynamical model of the pulsar magnetosphere and showedthat the evolution of the angle β ran slower than the exponential falling (6). For the dependencesin β = (cid:18) tτ (cid:19) − / (10)we obtain n = Ω d Ω /dt (Ω /dt ) + sin βτ Ω( d Ω /dt ) (11)Using the period and its derivative instead of Ω and d Ω dt gives instead of (9) and (11) l the following expres-sions: 2 = 2 − P d P/dt ( P/dt ) − Pτ dP/dt (12)and n = 2 − P d P/dt ( dP/dt ) + P sin βτ dP/dt (13)for the laws of the evolutions (6) and (10), correspondingly.Let us estimate the correction to the value n = 2 .
51, calculated using (3) for the Crab pulsar B0531+21(Lyne et al.,1993). From the catalog of Manchester et al. (2005) we have P = 33 msec and dPdt = 4 . × − ,and from the work of Loginov et al. (2016) τ = 1 . × years . For these values of the parameters weobtain ∆ n = − . × − , i.e. for this pulsar the correction is inessential. However for objects with the largecharacteristic age τ c = P dP/dt (14)such a correction can be noticeableAs follows from (3) and (8) the value of n depends strongly on the sign of the second derivative of the periodand on the value of dKdt .For the magneto-dipole mechanism the sign of d Pdt coincides with the sign of the following polinomial:sin β dBdt + B cos β dβdt − AB sin β P , (15)where A = 8 π R ∗ Ic (16)We will omit the index s of the quantity B meaning that we will deal with the magnetic induction at thesurface. It is evident from (15) that for constant or falling with time values of B and β the derivative d Pdt < n >
2. Since the angle β decreases in this model the positive second derivative is possible for the increasingmagnetic field only.This moment there are no reliable data showing the decay of pulsar magnetic fields. On the other handthere are mechanisms of generation of magnetic fields during the pulsar evolution (see, for example, Blandfordet al., 1983 and Sedrakyan and Movsisyan, 1986). Therefore the suggestion on the increasing field is not absurd.Suggesting similar to Philippov et al. (2014) that the decreasing of the angle β is very slow , omitting thesecond term in (15), and putting sin β = , we obtain: dBdt ≥ AB P (17)If the period P grows linearly the solution of this equation leads to the following inequality:1 B − B ≥ A dP/dt (cid:18) P − P (cid:19) (18)Here index 0 means the values of parameters taken in the initial moment of time. To estimate the necessarygrowth rate of magnetic field we put P = 0 . sec, P = 1 sec, dPdt = 10 − , B = 10 G, t = 10 years . Forthese values of parameters the equality takes place if B = 1 . × G , i.e., .this growth is rather slow. Indeedfor the exponential growth: B = B exp (cid:18) tτ B (cid:19) , (19)3e have τ B ≈ In this model braking of the neutron star connect with currents on the surface and their interaction with itsmagnetic field. This process leads to the evolution equation (Beskin et al., 1983): I Ω d Ω dt = − bi B R ∗ Ω cos βc , (20)where b is the numerical coefficient equaled to 0.33 - 0.48 when the angle β changes from 0 ◦ q to 90 ◦ , i isdimensionless longitudinal current depending on β also. We have in this case instead of (15): B cos β dA dβ dβdt + 2 A cos β dBdt − A B sin β dβdt − A B cos 2 βP , (21)where A ( β ) = 4 π R ∗ Ic bi (22)Neglecting as earlier the dependence of the angle β on time we conclude that the positive second derivativeis possible for dBdt > dBdt ≥ A B cos β P (23)Carrying out calculations as for the case of the magneto-dipole braking we obtain the magnetic induction B = 1 . × G after 1 billion years. Hence in the current model it is necessary the slow secular growth ofmagnetic field to achieve d Pdt > Michel and Dessler (1981) have discussed the possibility of the explanation of pulsar peculiarities suggestingthe existence of a relic disk near the neutron star. Matter of this disk determines the structure of the pulsarmagnetosphere and its braking. The corresponding equation of such a braking can be written in the followingform: I Ω d Ω dt = − πB R ∗ Ω GM , (24)where G is the gravitational constant, M the mass of the neutron star. It follows from this equation: dPdt = 2 π R ∗ IGM B = A B , (25) d Pdt = 2 A B dBdt (26)Thus in this model also the second derivative can be positive for dBdt > Current losses in the magnetosphere
Electric fields and currents in the magnetosphere can lead to losses of energy (de Jager and Net, 1988). Theselosses can be described by the following equation: I Ω d Ω dt = − kB R ∗ Ω c , (27)where k is a constant coefficient (less than 1). From (27) we have dPdt = kR ∗ Ic P B = A P B (28)and d Pdt = A (cid:18) dPdt B + 2 P B dBdt (cid:19) (29)This case differs from the previous models by the possibility of the positive second derivative not only forthe growing magnetic field but for the falling with time as well. In the last case the following inequality mustbe fulfilled: (cid:12)(cid:12)(cid:12)(cid:12) dBdt (cid:12)(cid:12)(cid:12)(cid:12) < A B kB R ∗ Ic , (30)For the used values of parameters this means that (cid:12)(cid:12) dBdt (cid:12)(cid:12) < . G/sec . The circular motion of neutrons in the neutron star can lead to emission of neutrino-antineutrino pairs and tothe dipole radiation (Huang et al., 1982). In this case energy of neutrons is passes to super-fluid vortexes andthe neutron star is braking by the law: I Ω d Ω dt = − γ m n R p Ω6 h c h n ∗ i ∆ B n ∗ , (31)where ∆ is the energy gap connected with the Cuper’s pairs , n ∗ a circulation quantum number of vortex, R p radius of the super-fluid region, γ the neutron gyromagnetic ratio, m n its mass, h the Plank’s constant, thebar denotes the average for all the vortex lines. The equality (31) leads to the equation:: dPdt = 11 γ m n R p P πh c h n ∗ i I ∆ B n ∗ = A B P (32)It is suggested that the mean magnetic field inside the star is equal to the field at the surface. Taking as inthe work of Huang et al. (1982) ∆ = 2 . M eV, n ∗ = 10 , R p = 0 . R ∗ , we have (Deng et al., 1987) A = 5 A (33)It follows from (32) that d Pdt = 2 BP A (cid:18) dBdt + A P B (cid:19) (34)and to obtain the positive second derivative we must suggest either the growth of magnetic field or its fallingwith the rate (cid:12)(cid:12)(cid:12)(cid:12) dBdt (cid:12)(cid:12)(cid:12)(cid:12) < A B P = 40 π R ∗ P B Ic (35)For the used parameters this gives (cid:12)(cid:12) dBdt (cid:12)(cid:12) < . × − G/sec.5
Pulsar wind
Particles escaping from the magnetosphere carry away an angular momentum. As a result there is the brakingof the neutron star with the rate (Harding et al.,1999) I Ω d Ω dt = − L / p BR ∗ Ω (6 c ) / , (36)where L p is the power of the pulsar wind. The equality (36) gives dPdt = A BP (37)Here A = L / p R ∗ I (6 c ) / (38)and the value of the second derivative is determined by the following equality: d Pdt = A P (cid:18) dBdt + A B (cid:19) (39)This quantity is positive if dBdt >
0. For dBdt <
0, it is necessary to fulfill the following condition: (cid:12)(cid:12)(cid:12)(cid:12) dBdt (cid:12)(cid:12)(cid:12)(cid:12) < L / p B R ∗ I (6 c ) / (40)Taking L p = 10 erg/sec , we obtain (cid:12)(cid:12) dBdt (cid:12)(cid:12) < . × − G/sec . This corresponds to the decay time of orderof 10 billions years. Hence in the model of the pulsar wind both signs of the second derivative are possible.
Sometimes an accretion from a debris disk on a neutron star can play a certain role in a braking of pulsars. Inthis case the so called propeller regime can be realized. In such a case we have (Illarionov and Sunyaev, 1975): I Ω d Ω dt = − GM ∗ dM/dtr eq , (41)where M ∗ is the mass of the neutron star, dMdt the rate of accretion, r eq = (cid:18) GM ∗ Ω (cid:19) / − (42)the distance where the rotation velocity is equal to the Kepler’s velocity. The equation (41) can be trans-formed to the following form: dPdt = A P / (43)Here A = dMdt ( GM ∗ / π ) / I (44)It follows from (43) that d Pdt = 73 A P / , (45)i.e. the pulsar rotation is braking during all time of its evolution with the increasing rate. It is worth notingthat the braking index is negative ( n = − /
3) in this regime.6
Discussion and conclusions
Table 1 contains estimates of the braking index for 9 pulsars (Ho, 2015) calculated using the formula (2).We have given the corresponding estimate for the Crab pulsar in the beginning of our paper. For the restobjects we have used the formula (12) taking τ = 1 .
42 billion years and obtained values of corrections ∆ n , givenin the last column of the table. We can see that these corrections are small for all 9 pulsars. However we mustpoint out once more that values of ∆ n can be noticeable for pulsars with long periods and small derivatives ofthe period. It is follows also from the table that the second derivative must be positive for the pulsars B0833-45,J1734-3333 and J1833-1034. For B0531+21, B0540-69, J1119-6127, B1509-58 and J1846-0258 this derivative isnegative. In the case of the pulsar J0537-6910 we must expect the influence of a debris disk. It is very importantto search for such a disk around J0537-6910. We can not use the formula (7) for all pulsars from the table .The value of the second derivative depends strongly on pulsar parameters. We will give one estimate onlyin the frame of the pulsar wind model. Omitting the term dBdt in (39) we obtain: d Pdt = A P B = L p R ∗ B P I c (46)Taking P = 1 sec, L p = 10 erg/sec, R ∗ = 10 cm, B = 10 G, I = 10 g cm we obtain d Pdt = 6 × − s − . Such derivatives we can expect in the precise timing measurements.The choice of the braking mechanism remains the extremely important problem for the understanding ofmany processes running in pulsars and the determining of the ways of their evolution. As follows from ouranalysis new more precise estimates of the second derivatives are necessary. They will give the possibility toadvance in the choice of the braking mechanism and conclude on the changes with time some pulsar parameters,in particular, magnetic fields and the angles between the rotation and magnetic axes.Hobbs et al. (2004) carried out the giant work on compilation of timing data for more than 300 pulsars.They gave values of the second derivatives . However these values did not characterize the basic mechanismsof braking but were caused by noises of different nature. Indeed there are no mechanisms giving values of nof order of tenths or even thousands and both signs. Unfortunately their data are not useful for the choice ofbraking mechanisms for individual pulsars. This moment only values from the table 1 can be used for this aim.There are works where sone kinds of oscillations are postulated to explain large values of n (see, for example,Birykov et. al., 2012, Xie and Zhang, 2014). They used a number of suggestions and worked out the so calledtoy models with many parameters. In any case they did not help to choise the main braking mechanism.This moment we can conclude that for pulsars with the measured second derivative corrections to the brakingindex are small and we can use formulas (2) and (3). Acknowledgements
This work has been carried out with the financial support of Basic Research Program of the Presidium of theRussian Academy of Sciences ”Transitional and Explosive Processes in Astrophysics (P-41)”. The author thanksL.B.Potapova for the help with the preparation of the manuscript.
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