Particle Emission-dependent Timing Noise of Pulsars?
aa r X i v : . [ a s t r o - ph . H E ] N ov Particle Emission-Dependent Timing Noise of Pulsars?
LIU Xiong-Wei, NA Xue-Sen, XU Ren-Xin, QIAO Guo-Jun
School of Physics and State Key Laboratory of Nuclear Physics and Technology,Peking University, Beijing 100871, China;[email protected]
ABSTRACT
Though pulsars spin regularly, the differences between the observed and predictedToA (time of arrival), known as “timing noise”, can still reach a few milliseconds ormore. We try to understand the noise in this study. As proposed by Xu and Qiaoin 2001, both dipole radiation and particle emission would result in pulsar braking.Accordingly, possible fluctuation of particle current flow is suggested here to contributesignificant ToA variation of pulsars. We find that the particle emission fluctuation couldlead to timing noise which cannot be eliminated in timing process and that a longerperiod fluctuation would arouse a stronger noise. The simulated timing noise profileand amplitude are in agreement with the observed timing behaviors on the timescale ofyears.PACS: 97.60.Gb, 05.40.CaWhy do pulsars spin down? This is a question still not fully answered even more than 40years later since the discovery of the first pulsar. It is generally suggested that pulsars spin downvia magneto-dipole radiation, by which the ages and the surface magnetic fields are estimatedaccordingly. However, it was proposed by Xu and Qiao (2001) that both dipole radiation andrelativistic particle emission powered by a unipolar generator can result in the loss of pulsar rotationenergy, and the observed braking indices ( <
3) could be understood then. [1 , This opinion isconsistent with later simulation [3] and observation. [4]
In this Letter, we focus on further implicationof the braking mechanism to timing behavior in Xu and Qiao’s model.Timing noise is the residual of pulsar time of arrival (ToA) after fitted by the timing model.It reflects the effects of unknown elements to ToA. A lot of models were proposed to explaintiming noise, such as the random walk in pulse frequency, [5] the free-precession of neutron star, [6] the unmodelled companions, [7 , and the effect of gravitational waves. [9] However, the noise stillcannot be eliminated completely, especially in long timescale of years. [10 − On the other hand,the pulsar flux density monitoring of the Green Bank [14] indicates that the pulsar emission maynot be absolutely stable. It is then reasonable in Xu and Qiao’s model that there exits fluctuationin the relativistic particle emission, which would consequently contribute to the timing noise. Wewill take the fluctuation in pulsar emission into account in timing process, in this work, and try tofind the relationship between the fluctuation and timing noise. 2 –The rotational energy loss rate is − I Ω ˙Ω = ˙ E = ˙ E d + ˙ E u , (1)where I is the moment of inertia of a pulsar, Ω and ˙Ω are its angular velocity and the firstderivative, ˙ E is the loss rate of rotational energy, and ˙ E d and ˙ E u are the powers of dipole radiationand relativistic particle flow, respectively. [1] When there is a fluctuation in ˙ E u , it becomes˙ E u = ¯˙ E u (1 + δ ) , (2)where ¯˙ E u is the stable value of ˙ E u , and δ is the fluctuation. For different pulsars the relativequantities of ˙ E d and ˙ E u are different because the magnetic inclinations are distinct and maybe theradiation mechanisms are not the same. However, for an individual pulsar these two componentsare sufficiently decided in a period of time and generally in a same order of magnitude. For theabove reasons, and considering the dipole radiation is stable, we take˙ E d = n × ¯˙ E u , (3)where n is a constant and decided by the magnetic inclination and radiation mechanism.From Eqs. (1), (2) and (3) we obtain − I Ω ˙Ω = ¯˙ E u ( n + 1 + δ ) . (4)Performing integration to both sides it becomes12 I (cid:2) Ω − Ω( T ) (cid:3) = ¯˙ E u (cid:20) ( n + 1) T + Z T δ ( t ) dt (cid:21) , (5)where Ω is the value of Ω at the beginning time, and we suppose that the moment of inertia I isconstant because it changes sufficiently small. When there is no fluctuation in ˙ E u , Eq. (5) becomes12 I (cid:2) Ω − Ω ′ ( T ) (cid:3) = ¯˙ E u ( n + 1) T, (6)where Ω ′ ( T ) is the expected value when fluctuation is zero. Equation (5) minus Eq. (6) is12 I (cid:2) Ω ′ ( T ) − Ω( T ) (cid:3) = ¯˙ E u Z T δ ( t ) dt. (7)Considering the spin of pulsar changes very slowly, we obtainΩ ′ ( T ) − Ω( T ) = ¯˙ E u I Ω Z T δ ( t ) dt. (8)From Eqs. (4) and (8) we haveΩ ′ ( T ) − Ω( T ) = − ˙Ω n + 1 + δ Z T δ ( t ) dt. (9) 3 –Performing integration to both sides we obtain − ˙Ω n + 1 + δ Z τ Z T δ ( t ) dtdT = Z τ (cid:2) Ω ′ ( T ) − Ω( T ) (cid:3) dT = Φ ′ ( τ ) − Φ( τ ) = − ∆Φ( τ ) = − Ω R, (10)where Φ is the phase of the pulsar, and R is a provisional timing residual. Thus one has R = ˙Ω ( n + 1 + δ )Ω Z τ Z T δ ( t ) dtdT = − ˙ P ( n + 1 + δ ) P Z τ Z T δ ( t ) dtdT, (11) P and ˙ P are the period and its first derivative at beginning time. Equation (11) reflects therelationship between the fluctuation and timing residual. We can obtain the real timing residual ℜ by performing least-squares-fitting to R .To understand more clearly about Eq. (11), we try to provide a simple example. Let δ ( t ) = a sin(2 πt/t ), one has ℜ ∼ = a ˙ P t π ( n + 1 + δ ) P sin(2 π tt ) . (12)From Eq. (12) we can see that longer timescale variation will cause stronger noise because ℜ ∝ t .For a normal pulsar with P = 0 . P = 1 × − , when a = 0 . t = y × . × s and n = 1, we obtain ℜ ∼ = 0 . × y × sin(2 πt/t ) s. It is a very strong noise at the timescale of years.We further do a simulation with Eq. (11). Three sets of random data with different Hurstparameter H , which reflects the time dependence of a time series data, [15] are produced to simulatethree types of irregular fluctuations in ˙ E u . As is shown in Fig. 1, each set of data has 10000 points.The first set has more short period components, with H = 0 .
4; the second set is approximatewhite noise, with H = 0 .
6; the third one has more long period components, with H = 0 .
8. In thissimulation we take ( n + 1 + δ ) P = 0 . P = 1 × − . The corresponding timing noisesare shown in Fig. 2. The figures indicate that if the particle emission has a random variation withextent of about 1% in daily timescale, the flux density from the most distant pulsars varies lessthan 5%, [14] it will lead to a timing noise with range of dozens of millisecond in 2000 days (shownon the left of Fig. 2), and several hundreds of millisecond in 10000 days (shown on the right of Fig.2). These curves also show the fluctuation with more long period components to cause strongernoise, which accords with Eq. (12) very well.Compared Fig. 2 with the observations, Fig. 1 in Ref. [11] and Figs. 1 and 2 in Ref. [12], wefind that they have some common features. (1) The majority time curves have about one period-likemain structure no matter how long the time spans are (see Refs. [10,13] for more examples), so thatone cannot distinguish which one has the long or short time span, just depend on their profiles,even for the same pulsar. (2) The range from the minimum to maximum residual with longer timespan is larger than the one with shorter time span for each pulsar, which is in agreement withEq. (12). (3) The time curve of the shorter time span is extremely similar to the correspondingtime span part of the longer one, which is natural because of the integral relation in Eq. (11). 4 –Recently, Lyne et al . [16] proposed another idea of producing timing noise to pulsar, namelyvariations of the pulsar spin-down states variations lead to timing noise. This phenomenologicallyexplains the origin of some quasi-periodic structures, which lie on lower-frequency structures ofsome timing noise. However, it cannot give rise to the ubiquitous lower-frequency structures inlong time scales, which are what we try to do in this study.The statistics results from most pulsar timing noises are in agreement with our model. Soonafter we put our work on arXiv, a statistics from Ryan et al. gives σ T N, ∝ ν . ± . | ˙ ν | . ± . , [17] which is consistent with Eq. (11) very well. From observations, Cordes and Downs, [18] D’Alessandro et al. [10] and Ryan et al. [17] all suggested that a mixture of random walks in ν and ˙ ν is compatiblewith the timing noise, whereas we propose here a natural physical origin as shown in Eq. (4). We canhave the timing noises of millisecond pulsar and AXP to be orders of 10 ns and 10 s, respectively,from Eq. (11), which are consistent with the observations.In summary, our model shows that the fluctuation of particle emission will cause significanttiming noise. We emphasize that there could be other kinds of the fluctuation (e.g., δ ), neverthelessthe long period composition of variation contributes larger to the noise. The simulation accordswith long (years) timescale noises both in range and profile features. Simultaneously, our worksupports the opinion that the pulsar emission is not always stable, which is important to theresearch of pulsar radiation and the understanding of pulsar physics. Any other possible processesthat lead to instability to pulsar spin down energy could give timing residuals similar to our result,and may be in agreement with the observations as well as ours.We thank the members at PKU pulsar group for helpful discussions. This work is supported byNSFC (10833003, 10935001, 10973002) and the National Basic Research Program of China underGrant No 2009CB824800. REFERENCES [1] Xu R X and Qiao G J 2001
Astrophys . J . Lett . L85[2] Yue Y L, Xu R X and Zhu W W 2007
Adv . SpaceRes . Astrophys . J . Science
Astrophys . J . Suppl . Nature
Astron . Soc . PacificConf . Ser . Astr . Soc . PacificConf . Ser . Astrophys . J . Mon . Not . R . Astron . Soc . Chin . J . Astron . Astrophys . Chin . J . Astron . Astrophys . Mon . Not . R . Astron . Soc . Astrophys . J .
300 5 –[15] Na X S et al. 2009 arXiv : . [ astro − ph . IM ][16] Lyne A G et al. 2010 Science arXiv : . [ astro − ph . SR ][18] Cordes J M and Downs G S 1985 Astrophys . J . Suppl . This preprint was prepared with the AAS L A TEX macros v5.0. −0.0200.02
H=0.4 δ −0.0200.02 H=0.6 δ H=0.8 δ Fig. 1.— The data used to simulate the fluctuation of relativistic particles flux. The first set ofdata has more short period component, the second set is approximate white noise, the third sethas more long period component. Here H is the Hurst parameter. T i m i ng r e s i dua l ( m s ) Time(day) T i m i ng r e s i dua l ( m s ) Time(day)13.4 519.8251.054.3 200.783.9
Fig. 2.— Curves of timing noise produced from the fluctuation data shown in Fig. 1. The Hurstparameters in the upper, middle, and bottom panels are H = 0 . , . , .
8. The first 2000 points andthe whole 10000 points are used in the left and right panels, respectively. We take ( n + 1 + δ ) P =0 . P = 1 × −14