The dynamics of Galactic centre pulsars: constraining pulsar distances and intrinsic spin-down

TL;DR

This study uses high-precision radio timing to measure pulsar accelerations near the Galactic Centre, constraining their distances and electron densities, and improving understanding of the Galactic gravitational potential.

Contribution

It provides the first accurate distance measurements to pulsars on the far side of the Galaxy using acceleration data, and compares electron density models with new observational constraints.

Findings

Pulsars are located on the far side of the Galaxy.

The YMW16 electron density model aligns better with observations.

Pulsar accelerations are consistent with Galactic gravitational models.

Abstract

Through high-precision radio timing observations, we show that five recycled pulsars in the direction of the Galactic Centre (GC) have anomalous spin period time derivative ($\dot P$) measurements -- PSRs J1748$-$3009, J1753$-$2819, J1757$-$2745, and J1804$-$2858 show negative values of $\dot P$ and PSR J1801$-$3210 is found to have an exceptionally small value of $\dot P$. We attribute these observed $\dot P$ measurements to acceleration of these pulsars along their lines-of-sight (LOSs) due to the Galactic gravitational field. Using models of the Galactic mass distribution and pulsar velocities, we constrain the distances to these pulsars, placing them on the far-side of the Galaxy, providing the first accurate distance measurements to pulsars located in this region and allowing us to consider the electron density along these LOSs. We find the new electron density model YMW16 to be more consistent with these observations than the previous model NE2001. The LOS dynamics further constrain the model-dependent intrinsic $\dot P$ values for these pulsars and they are consistent with measurements for other known pulsars. In the future, the independent distance measurements to these and other pulsars near the GC would allow us to constrain the Galactic gravitational potential more accurately.

Tables

Table 1. Table 1: The previously published parameters of the pulsars in this study. The distance is estimated using the electron density models given in Yao et al. ( 2017 ) (YMW16) and Cordes & Lazio ( 2002 ) (NE2001). Since there are no multi-frequency observations in this study, we use these published DM measurements and keep them fixed in our timing models.

PSR	$l$	$b$	$P$	$\dot{P}$	DM	Ref.	$D_{\mathrm{YMW16}}^{††}$	$D_{\mathrm{NE2001}}^{††}$
	(∘)	(∘)	(ms)	(s/s)	(cm-3 pc)		(kpc)	(kpc)
J1746$-$2758†	0.97	0.49	487.53	–	422.0	1	4.2	5.2
J1748$-$3009	359.27	$-$1.15	9.68	–	420.2	2	5.1	5.1
J1753$-$2819	1.46	–1.25	18.62	–	298.0	3	4.1	4.5
J1757$-$2745†	2.40	$-$1.72	17.69	$2.1\left(2\right)\times {10}^{-19}$	334.0	1	5.2	5.2
J1801$-$3210	358.92	$-$4.58	7.45	$-4\left(4\right)\times {10}^{-23}$	177.7	4, 5	6.1	4.0
J1804$-$2858	1.99	$-$3.50	1.49	–	232.4	6, 7	8.2	4.9

Table 2. Table 2: The observation details of pulsars in this analysis. From left to right, we list the pulsar name, telescope used, centre observing frequency and bandwidth, number of bins in the integrated pulse profile and number of observations.

PSR	Telescope	Centre Freq	Bandwidth	Phase bins	Data span	No. of
	Backend	(MHz)	(MHz)			observations
J1746$-$2758	LT/ROACH	1532	400	1024	$12/2013-4/2018$	82
J1748$-$3009	LT/ROACH	1532	400	1024	$5/2012-11/2018$	146
	PKS/AFB	1374	288	32	$12/1998-8/2001$	23
J1753$-$2819	LT/ROACH	1532	400	1024	$9/2012-10/2018$	94
J1757$-$27	LT/ROACH	1532	400	1024	$3/2014-10/2018$	80
J1801$-$3210	LT/ROACH	1532	400	1024	$4/2011-10/2018$	74
	LT/DFB	1520	384	1024	$7/2009-4/2011$	60
	PKS/CASPSR	1369	400	256	$7/2010-4/2018$	53
J1804$-$2858	LT/ROACH	1532	400	256	$12/2013-3/2018$	46
	PKS/CASPSR	1369	400	256	$12/2013-4/2018$	55

Table 3. Table 3: The timing model parameters of the newly discovered PSR J1753 − - 2819 and the updated timing model parameters of PSRs J1746 − - 2758, and J1748 − - 3009. Due to the lack of multi-frequency observations, we present the DM values (and their uncertainties, if available) reported in their discovery analyses and kept them fixed. The 1 σ 𝜎 \sigma uncertainties of measurements are given in parentheses. The characteristic age ( τ = P / 2 ​ P ˙ 𝜏 𝑃 2 ˙ 𝑃 \tau=P/2\dot{P} ) and the surface magnetic field ( B S = 3.2 × 10 19 ​ P ​ P ˙ subscript 𝐵 𝑆 3.2 superscript 10 19 𝑃 ˙ 𝑃 B_{S}=3.2\times 10^{19}\sqrt{P\dot{P}}  G) are computed based on the intrinsic P ˙ ˙ 𝑃 \dot{P} limits estimated using the dynamics of each pulsar as described in §  5.3 .

Timing parameter	J1746$-$2758	J1748$-$3009	J1753$-$2819
Data span (MJD)	56639 – 58423	51148 – 58428	56174 – 58423
Number of TOAs	82	169	94
Weighted rms timing residual ($\mu$s)	545	86	82
Reduced ${\chi }^2$ value	1.00	0.99	0.99
Units	TCB	TCB	TCB
Measured parameters:
Right ascension RA (J2000)	17:45:52.248(3)	17:48:23.7582(3)	17:53:56.1837(4)
Declination Dec. (J2000)	$-$27:58:38.0(5)	$-$30:09:11.28(4)	$-$28:19:30.03(7)
Spin frequency $f$ ($s^{-1}$)	2.051164249099(9)	103.26352668931(4)	53.69521954620(1)
Spin frequency $1^{\mathrm{st}}$ derivative $\dot{f}$ ($s^{-2}$)	$-1.86\left(2\right)\times {10}^{-17}$	$3.817\left(2\right)\times {10}^{-16}$	$1.423\left(4\right)\times {10}^{-16}$
Spin frequency $2^{\mathrm{nd}}$ derivative $\ddot{f}$ ($s^{-3}$)	$-4.9\left(19\right)\times {10}^{-26}$	$-2.2\left(3\right)\times {10}^{-26}$	–
Reference epoch (MJD)	57531	54788	57299
Dispersion measure DM (cm-3 pc)	422(9)	420.2	298
Orbital period $P_b$ (d)	–	2.93382034(2)	0.387677373(3)
Epoch of periastron $T_o$ (MJD)	–	56999.183308(4)	56999.821829(4)
Projected semi-major axis $x$ (lt-s)	–	1.320085(9)	0.20967(1)
Orbital period derivative $\dot{P_b}$ (s/s)	–	$-2.6\left(10\right)\times {10}^{-11}$	–
Derived parameters:
Gal. longitude $l$ (deg)	0.8473(1)	359.272301(8)	1.46179(2)
Gal. latitude $b$ (deg)	0.45243(8)	$-1.147126\left(6\right)$	$-1.25174\left(1\right)$
Spin period $P$ (s)	0.487527998033(2)	0.009683961337179(4)	0.018623631832617(5)
Spin period derivative $\dot{P}$ (s/s)	$4.42\left(5\right)\times {10}^{-18}$	$-3.580\left(2\right)\times {10}^{-20}$	$-4.94\left(2\right)\times {10}^{-20}$
Intrinsic $\dot{P}$ (s/s)	$1.48\times {10}^{-17}$	$1.38\times {10}^{-19}$	$2.08\times {10}^{-19}$
Characteristic age $\tau$ (Gyr)	$>$0.5	$>$1.1	$>$1.4
Surface magnetic field $B_s$ ($\times {10}^9$ G)	85	1.1	2.0
Companion mass (M⊙)	–	$0.08-0.20$	$0.05-0.12$

Table 4. Table 4: Same as Table  3 , but for PSRs J1757 − - 2745, J1801 − - 3210, and J1804 − - 2858.

Timing parameter	J1757$-$2745	J1801$-$3210	J1804$-$2858
Data span (MJD)	56728 – 58422	55032 – 58423	56638 – 58213
Number of TOAs	80	187	101
Weighted rms timing residual ($\mu$s)	12	28	8
Reduced ${\chi }^2$ value	0.98	0.99	0.97
Units	TCB	TCB	TCB
Measured parameters:
Right ascension RA (J2000)	17:57:54.7826(1)	18:01:25.8872(1)	18:04:01.52323(4)
Declination Dec. (J2000)	$-$27:45:40.16(2)	$-$32:10:53.733(9)	$-$28:58:46.608(7)
Proper motion in RA, PMRA (mas/yr)	$-11.4\left(11\right)$	$-4.0\left(6\right)$	$-6.4\left(5\right)$
Proper motion in Dec., PMDEC (mas/yr)	54(17)	$-4\left(3\right)$	$-11\left(5\right)$
Spin frequency $f$ ($s^{-1}$)	56.538013330516(8)	134.16363857905(1)	669.93358254482(5)
Spin frequency $1^{\mathrm{st}}$ derivative $\dot{f}$ ($s^{-2}$)	$4.17\left(5\right)\times {10}^{-17}$	$-1.3\left(2\right)\times {10}^{-18}$	$3.399\left(9\right)\times {10}^{-16}$
Spin frequency $2^{\mathrm{nd}}$ derivative $\ddot{f}$ ($s^{-3}$)	$-4.7\left(18\right)\times {10}^{-26}$	$-2.5\left(5\right)\times {10}^{-26}$	$-2.6\left(9\right)\times {10}^{-25}$
Spin frequency $3^{\mathrm{rd}}$ derivative $\stackrel{\dot{}\dot{}\dot{}}{f}$ ($s^{-4}$)	$-5.1\left(14\right)\times {10}^{-33}$	–	–
Reference epoch (MJD)	57575	56635	57425
Dispersion measure DM (cm-3 pc)	334	177.713(4)	232.4
Orbital period $P_b$ (d)	–	20.77169953(3)	–
Epoch of periastron $T_o$ (MJD)	–	56518.268546(1)	–
Projected semi-major axis $x$ (lt-s)	–	7.809320(3)	–
Derived parameters:
Gal. longitude $l$ (deg)	2.390743(5)	358.921963(2)	1.995292(2)
Gal. latitude $b$ (deg)	$-1.726491\left(3\right)$	$-4.577230\left(2\right)$	$-3.497440\left(1\right)$
Spin period $P$ (s)	0.017687215045108(3)	0.0074535843734652(5)	0.0014926852841164(1)
Spin period derivative $\dot{P}$ (s/s)	$-1.30\left(2\right)\times {10}^{-20}$	$7.4\left(8\right)\times {10}^{-23}$	$-7.57\left(2\right)\times {10}^{-22}$
Intrinsic $\dot{P}$ (s/s)	$1.42\times {10}^{-19}$	$3.50\times {10}^{-20}$	$7.92\times {10}^{-21}$
Characteristic age $\tau$ (Gyr)	$>$2.0	$>$3.4	$>$3.0
Surface magnetic field $B_s$ ($\times {10}^9$ G)	1.6	0.5	0.1
Companion mass (M⊙)	–	$0.14-0.35$

Table 5. Table 5: Summary of the parameters of pulsars used in our dynamic analysis. The angular separation θ 𝜃 \theta of the pulsar from the GC is calculated based on the Galactocentric coordinates ( l , b ) 𝑙 𝑏 (l,b) . The uncertainties of l 𝑙 l , b 𝑏 b , and θ 𝜃 \theta are estimated based on the uncertainties of pulsar positions measured from timing (Table  3 and 4 ). The projected separation r ⟂ subscript 𝑟 perpendicular-to r_{\perp} of the pulsar from the GC in the sky plane is calculated based on a GC distance of 8.2 ± 0.1 plus-or-minus 8.2 0.1 8.2\pm 0.1  kpc from the Sun (Bland-Hawthorn et al., 2019 ) . Note that the uncertainty in r ⟂ subscript 𝑟 perpendicular-to r_{\perp} is dominated by the uncertainty in the GC distance.

PSR	$l$	$b$	$\theta$	$r_{⟂}$	${\dot{P}}_{\mathrm{obs}}/P$
	(∘)	(∘)	(∘)	(pc)	($\times {10}^{-10}$ yr-1)
J1746$-$2758	0.8473(1)	0.45243(8)	0.9605(1)	137(2)	$2.86\left(3\right)$
J1748$-$3009	359.272301(8)	$-$1.147126(6)	1.358445(7)	194(2)	$-1.1657\left(6\right)$
J1753$-$2819	1.46176(2)	$-$1.25174(1)	1.92438(1)	275(3)	$-0.836\left(3\right)$
J1757$-$2745	2.390743(5)	$-$1.726491(3)	2.94868(5)	422(5)	$-0.233\left(3\right)$
J1801$-$3210	358.921963(2)	$-$4.577230(2)	4.702204(2)	672(7)	$0.0031\left(4\right)$
J1804$-$2858	1.995292(2)	$-$3.497440(1)	4.025958(1)	576(6)	$-0.1600\left(4\right)$

Table 6. Table 6: The parameters of the main Galactic mass distribution model SOF13 used in this analysis, i.e.  Sofue ( 2013 ) with updated values given in Sofue ( 2017 ) . This model uses two bulges (i.e. inner and main), a stellar disc, and a dark halo. For comparison, we use MCM17 in the analysis and its relevant parameters are also given (McMillan, 2017 ) . This model uses only a single bulge with two stellar (i.e. thin and thick) and two gas discs (i.e. H 1 subscript 𝐻 1 H_{1} and H 2 subscript 𝐻 2 H_{2} ), and a dark halo. The central volume ( ρ 0 , b subscript 𝜌 0 b \rho_{\rm 0,b} ) and surface mass density ( Σ 0 , d subscript Σ 0 d \Sigma_{\rm 0,d} ) parameters are given in units of M ⊙ /pc 3 and M ⊙ /pc 2 , respectively, and the radii are given in kpc.

Model	Bulge	Disc	Dark halo
	${\rho }_{0,b}$ , $r_{\mathrm{cut}}$	${\Sigma }_{0,d}$ , $R_d$	${\rho }_{0,h}$ , h
SOF13	Inner: $3.7\times {10}^4$,	$292$ , 4.9	$0.029$ , 10
	$0.0035$
	Main: $2.1\times {10}^2$,
	0.12
MCM17	$98.4$ , 2.1	Thin: 896 , 2.5	0.00854 , 19.6
		Thick: 183 , 3.02
		H1: 53.1 , 7
		H2: 2180 , 1.5

Table 7. Table 7: The derived parameters of pulsars in the study. The minimum Galactic acceleration a min / c subscript 𝑎 min 𝑐 a_{\rm min}/c is obtained from the Galactic model given in Sofue ( 2013 ) . The model-dependent upper limit on the intrinsic P ˙ ˙ 𝑃 \dot{P} is derived using Eq.  6 (see §  5.3 ). The last column represents the model-dependent distance to the pulsar determined through dynamics as described in §  5.4 . Note that for PSRs J1757 − - 2745, J1801 − - 3210, and J1804 − - 2858, we quote the derived distances obtained using their timing measured proper motion limits.

PSR	$a_{\min }/c$	${\dot{P}}_{\mathrm{int}}$	$D$
	(${10}^{-10}$ yr-1)	(${10}^{-19}$ s/s)	(kpc)
J1746$-$2758	$-6.70$	148	7.50 or $>$8.16
J1748$-$3009	$-5.64$	1.38	$8.24-8.99$
J1753$-$2819	$-4.36$	2.08	$8.24-9.08$
J1757$-$2745	$-2.77$	1.42	$8.27-9.40$
J1801$-$3210	$-1.48$	0.35	$8.20-15.50$
J1804$-$2858	$-1.83$	0.08	$8.32-9.40$

Equations

$$\frac{{\dot{P}}_{\mathrm{obs}}}{P}=\frac{{\dot{P}}_{\mathrm{int}}}{P}+\frac{a}{c}+\frac{a_{\mu }}{c},$$

$${\rho }_{\mathrm{b}}(R,z)=\frac{{\rho }_{0,\mathrm{b}}}{(1+r/r_0{)}^{\alpha }}\exp [-(r/r_{\mathrm{cut}}{)}^{\zeta }],$$

$${\rho }_{\mathrm{d}}(R,z)=\frac{{\Sigma }_{0,\mathrm{d}}}{(2z_{\mathrm{d}}{)}^{\eta }}\exp \left(-\frac{|z|}{z_{\mathrm{d}}}-\frac{R}{R_{\mathrm{d}}}\right).$$

$${\rho }_{\mathrm{g}}(R,z)=\frac{{\Sigma }_{0,\mathrm{d}}}{4z_{\mathrm{d}}}\exp \left(-\frac{R_{\mathrm{m}}}{R}-\frac{R}{R_{\mathrm{d}}}\right){\mathrm{sech}}^2(z/2z_{\mathrm{d}}),$$

$${\rho }_{\mathrm{h}}(R,z)=\frac{{\rho }_{0,\mathrm{h}}}{X(1+X{)}^2},$$

$${\left(\frac{{\dot{P}}_{\mathrm{int}}}{P}\right)}_{\mathrm{upper}}=\frac{{\dot{P}}_{\mathrm{obs}}}{P}-\frac{a_{\min }}{c}-\frac{a_{\mu }}{c}$$

$$\frac{{\dot{P}}_{\mathrm{b}}}{P_{\mathrm{b}}}={\left(\frac{{\dot{P}}_{\mathrm{b}}}{P_{\mathrm{b}}}\right)}_{\mathrm{GR}}+\frac{a}{c}+\frac{a_{\mu }}{c},$$

$${\vec{v}}_{\odot }=[U,V+v_R(R_{\odot }),W],$$

$${\vec{v}}_{\mathrm{psr}}=[U_p+v_R(R_p)\sin {\theta }^{\prime },V_p+v_R(R_p)\cos {\theta }^{\prime },W_p],$$

$$\tan ({\theta }^{\prime }+l)=\frac{R_{\odot }\sin l}{R_{\odot }\cos l-d\cos b}.$$

$$\begin{array}{rl}{\vec{v}}_{\mathrm{psr}}-{\vec{v}}_{\odot }=& [U_p+v_R(R_p)\sin {\theta }^{\prime }-U,\\ & V_p+v_R(R_p)\cos {\theta }^{\prime }-V-v_R(R_{\odot }),W_p-W].\end{array}$$

$$\begin{array}{rl}\hat{l}& =[-\sin l,\cos l,0]\\ \hat{b}& =[-\cos l\sin b,-\sin l\sin b,\cos b].\end{array}$$

$$\begin{array}{rl}{v}_l=& \hat{l}\cdot ({\vec{v}}_{\mathrm{psr}}-{\vec{v}}_{\odot })\\ =& -(U_p+v_R(R_p)\sin {\theta }^{\prime }-U)\sin l\\ & +(V_p+v_R(R_p)\cos {\theta }^{\prime }-V-v_R(R_{\odot }))\cos l\end{array}$$

$$\begin{array}{rl}{v}_b=& \hat{b}\cdot ({\vec{v}}_{\mathrm{psr}}-{\vec{v}}_{\odot })\\ =& -(U_p+v_R(R_p)\sin {\theta }^{\prime }-U)\cos l\sin b\\ & -(V_p+v_R(R_p)\cos {\theta }^{\prime }-V-v_R(R_{\odot }))\sin l\sin b\\ & +(W_p-W)\cos b,\end{array}$$

Full text

1 Introduction

Pulsars are among the most reliable macroscopic periodic sources in the universe and their long-term stability is comparable to that of atomic clocks (e.g. Petit & Tavella, 1996; Hobbs et al., 2012). This allows the parameters of pulsars to be measured to exquisite precision through measuring and modelling the arrival times of their pulses (e.g. Kaspi et al., 1994). These measurements are sensitive to any dynamical changes caused by external forces on the pulsar such as the gravitational forces exerted by other masses (e.g. Freire et al., 2017; Perera et al., 2017).

Pulsars are thought to be powered by the loss of rotational kinetic energy and thus, gradually spin down over time, leading to positive values of $\dot{P}$. The current pulsar population strongly supports this hypothesis with measured positive $\dot{P}$ values for the vast majority of pulsars111http://www.atnf.csiro.au/people/pulsar/psrcat (see Manchester et al., 2005). However, some pulsars, located in globular clusters, show significant negative $\dot{P}$ measurements (e.g Wolszczan et al., 1989; Freire et al., 2017; Jacoby et al., 2006; Corongiu et al., 2006; Lynch et al., 2012). These $\dot{P}$ values are not entirely intrinsic, but induced by the dynamics due to the LOS component of the acceleration of the pulsar, mainly in the gravitational potential of the cluster itself, locating them on the far-side of the cluster (see, e.g., Freire et al., 2003; Freire et al., 2017).

Similar to the globular clusters, the Galaxy has a strong gravitational potential within the central regions due to the high stellar and gas densities. We therefore expect negative $\dot{P}$ measurements from pulsars that are located near the Galactic centre (GC) on the far-side of the Galaxy (see § 5.1 for details). However, prior to this work, there was no conclusive evidence for this. As we show below using our timing observations, PSRs J1748$-$3009 (Knispel et al., 2013), J1753$-$2819 (originally announced by Mickaliger, 2013), J1757$-$2745 (Ng et al., 2015), and J1804$-$2858 (Morello et al., 2019, Barr et al., in prep) all show significant negative $\dot{P}$ measurements, and PSR J1801$-$3210 (Ng et al., 2014) shows a remarkably small but positive $\dot{P}$ measurement when compared to the known pulsar population.

In addition to the pulsars listed above, we note that our timing of PSR J1746$-$2758 (Ng et al., 2015) does not show an unusual $\dot{P}$ measurement, however, its location in the Galaxy and some measured properties have similarities to those of above mentioned pulsars, making it useful to consider in this work. The previously reported rotational, and some other measured and derived parameters, for these pulsars are given in Table 1, noting that four of them had no previous $\dot{P}$ measurements. All six pulsars considered here are located along LOSs which pass near the GC with an angular separation of $<5^{\circ}$. With their comparably large dispersion measure (DM – which accounts for the frequency-dependent time delay of the radio pulses due to electrons in the inter-stellar medium along the LOS) values, it is likely that these pulsars are located within the inner regions of the Galaxy around the GC (Cordes & Lazio, 2002; Yao et al., 2017). Given the special locations of these pulsars, it is likely that their unusual observed $\dot{P}$ measurements are not entirely intrinsic, but rather induced by the pulsar acceleration in the Galactic gravitational potential. We further note that these pulsars are useful in studying the DM environment of the inner region of the Galaxy.

The paper is organized as follows. In § 2, we describe the discovery observations of PSR J1753$-$2819. In § 3, we present our observations and data processing of all six pulsars. We then discuss the timing analysis of these pulsars in § 4 and report their negative/small observed $\dot{P}$ measurements. We analyse their dynamics in § 5, including the LOS accelerations combining the Galactic mass distribution and pulsar velocity models (§ 5.1 and 5.2), leading to limits on their intrinsic $\dot{P}$ values (§  5.3) and their possible distances (§ 5.4). We compare our dynamically-derived distances with DM-derived distances obtained using electron density models in § 5.5. Finally in § 6, we discuss our results and present conclusions.

2 Discovery of PSR J1753$-$2819

The Parkes Multibeam Pulsar Survey (PMPS – Manchester et al., 2001) has significantly expanded the known pulsar population, resulting in over 800 new pulsar discoveries to-date (Morris et al., 2002; Kramer et al., 2003; Hobbs et al., 2004a; Faulkner et al., 2004; Lorimer et al., 2006; Eatough et al., 2009; Eatough et al., 2010, 2013; Keith et al., 2009; Mickaliger et al., 2012; Knispel et al., 2013), including 21 millisecond pulsars and 30 rotating radio transients. The PMPS was carried out as a blind pulsar survey along the Galactic plane with $|b|<5^{\circ}$ using the 13-beam multibeam receiver (with a centre frequency of 1374 MHz and a bandwidth of 288 MHz) on the Parkes radio telescope (PKS) in Australia. Due to advanced and improved pulsar search pipelines being applied to the archival PMPS data, and independent visual inspection of the candidate output plots, five new millisecond pulsars were discovered and reported by Mickaliger et al. (2012).

These methods also resulted in the discovery of PSR J1753$-$2819 (originally announced by Mickaliger, 2013). As described by Mickaliger et al. (2012), the PMPS data were initially dedispersed using dedisperse_all222https://github.com/swinlegion/sigproc, a multi-threaded dedisperser compatible with the sigproc software package (Lorimer, 2011). The resulting time series were then searched using seek333http://sigproc.sourceforge.net. For statistically significant candidates, the raw data were folded at the candidate period using prepfold from the presto software package444https://www.cv.nrao.edu/~sransom/presto (Ransom, 2011). The number of candidates was reduced by selecting those within a parameter space from which pulsars were likely to have been previously missed, i.e. spin period $P<50$ ms, DM $>10$ pc cm*-3*. The resulting folded candidates were viewed by eye and confirmation observations were performed and were reported by Mickaliger et al. (2012). This search was undertaken as part of a larger search for fast radio bursts, which classified single pulses from known pulsars detected in the PMPS (Mickaliger et al., 2018). Like the five pulsars reported in Mickaliger et al. (2012), PSR J1753$-$2819 was probably missed in earlier searches of the PMPS due to its high DM, short spin period, and binary nature, as well as the large number of candidates present in those searches. Further details of the search are reported in Mickaliger et al. (2012).

3 Observations and data processing

We observed PSRs J1748$-$3009, J1746$-$2758, J1757$-$2745, J1801$-$3210, and J1804$-$2858 using the Lovell Telescope (LT) at the Jodrell Bank Observatory in the UK approximately monthly or semi-monthly since their discoveries. After confirming the discovery of PSR J1753$-$2819, it was observed using the LT approximately every 17 days. All these observations were carried out at ‘L band’ at a centre frequency of 1532 MHz and a bandwidth of 400 MHz. The data were recorded using the ‘ROACH’ pulsar backend (Bassa et al., 2016a). The observation details are given in Table 2. In addition to the ROACH data sets, PSR J1801$-$3210 was observed using the LT with the ‘DFB’ backend between February 2010 and April 2011. The ‘DFB’ observations were carried out at a slightly different centre frequency of 1520 MHz and a bandwidth of 384 MHz. We used both data sets of this pulsar in the analysis to extend its time baseline.

In addition to the LT observations, PSRs J1801$-$3210 and J1804$-$2858 were observed using the Parkes radio telescope since their discoveries. These observations were also taken at ‘L band’ with a centre frequency of 1369 MHz and a bandwidth of 400 MHz and recorded with the CASPSR backend (see Table 2). We noticed that some of the calibration observations obtained from PKS as a part of PMPS in the period between December 1998 and August 2001 happened to overlap with the position of PSR J1748$-$3009, that is, prior to its discovery. Therefore, we used these data and folded them using the pulsar ephemeris and included the resultant detections in our timing analysis. These observations were carried out at 1374 MHz with a bandwidth of 288 MHz and recorded using the ‘analogue filter bank’ (AFB) backend. When combined with the existing data set this extended the observation baseline of PSR J1748$-$3009 by more than 13 years, we were able to significantly improve the timing measurements while fitting for new timing model parameters (see § 4).

We processed these data using the pulsar analysis software package psrchive555http://psrchive.sourceforge.net(Hotan et al., 2004; van Straten et al., 2012; van Straten et al., 2011). We folded each observation for its entire length and summed all the frequency channels together to form the integrated pulse profile. A time-of-arrival (TOA) was generated by cross-correlating each of the integrated pulse profiles with a noise-free profile template (Taylor, 1992) using pat666we used the FDM (Fourier domain with Markov chain Monte Carlo) method. in psrchive. This resulted in 83, 169, 98, 75, 187, and 101 TOAs for PSRs J1746$-$2758, J1748$-$3009, J1753$-$2819, J1757$-$2745, J1801$-$3210, and J1804$-$2858, respectively.

Fig. 1 shows the integrated pulse profile of PSR J1753$-$2819 as it was not published in the original announcement (Mickaliger, 2013). We estimate the flux density of the pulsar using the radiometer equation (see Lorimer & Kramer, 2005), with the system temperature of $T_{\rm sys}=42$ K (i.e. $T_{\rm sky}=12$ K in the direction of the pulsar according to Haslam et al. (1982) with a spectral index of $-2.6$ and $T_{\rm rec}+T_{\rm spill}=30$ K) and the telescope gain of 0.9 K/Jy, and scale the profile accordingly. These calculations estimate the mean flux density of the pulsar to be 0.14 mJy. We further measure the pulse width, at 50% and 10% of the peak to be $W_{\rm 50}=1.51$ ms and $W_{\rm 10}=3.98$ ms, respectively.

4 Timing pulsars

We fit timing models including astrometric and rotational frequency parameters for PSRs J1746$-$2758, J1748$-$3009, J1753$-$2819, J1757$-$2745, J1801$-$3210, and J1804$-$2858 to the TOAs. The timing solutions are improved by minimising the $\chi^{2}$ value of the timing residuals between the observed and model-predicted TOAs using the pulsar timing software tempo2 (Edwards et al., 2006; Hobbs et al., 2006; Hobbs & Edwards, 2012). Since we do not have multi-frequency observations, we keep the DM of the pulsar fixed at its previously published value (see Table 1). Note that PSRs J1748$-$3009, J1753$-$2819, and J1801$-$3210 are in binary orbits and thus, we include Keplerian parameters in their timing models (see Lorimer & Kramer, 2005) using the binary model ‘BT’ (Blandford & Teukolsky, 1976). When TOAs from different telescopes/backends are available (see Table 2), we combined them by fitting for a time offset (‘JUMP’) in the timing model of the pulsar to take into account any systematic delays between the data sets. During the fitting process, the topocentric TOAs are converted to the Solar-system barycentric coordinate time (TCB) using the DE436777This new solar-system ephemeris DE436 is based on Folkner et al. (2014). ephemeris (Folkner et al., 2009) and placed on the Terrestrial Time standard BIPM2015888This time standard has been obtained according to principles given in Guinot (1988) and Petit (2003).. The resultant timing parameters of our pulsars are given in Table 3 and 4.

4.1 PSR J1748$-$3009

This binary millisecond pulsar has a spin period of $9.7$ ms and is located with an angular separation of $1\aas@@fstack{\circ}36$ from the GC (see Table 5). Based on about 20 years of our timing data, by combining the early PKS observations with LT observations, the timing parameters of the pulsar improved significantly. The timing model constrains the $\dot{P}$ measurement of the pulsar for the first time to be $-3.580(2)\times 10^{-20}$ s/s. This negative value indicates that the measured $\dot{P}$ is not entirely intrinsic, rather induced by the acceleration of the pulsar due to the Galactic potential, and we constrain the limit on the intrinsic value based on dynamics in § 5.3. We further find that the timing solution of the pulsar requires a second time derivative of the rotational frequency ($\ddot{f}$), with an approximately 7$\sigma$ significance (see Table 3), to achieve white residuals. Our timing analysis also places a limit on the orbital period time derivative $\dot{P}_{b}$ of the pulsar with a significance of 2.6$\sigma$. We use this limit independently to further constrain the intrinsic period derivative of the pulsar in § 5.3.

4.2 PSR J1753$-$2819

The newly discovered PSR J1753$-$2819 (Mickaliger, 2013) is a 18.6 ms mildly-recycled pulsar in a 9.3-hr binary orbit (see Table 3) with a DM of 298 cm*-3* pc. It is located at an angular separation of $1\aas@@fstack{\circ}92$ from the GC, resulting in a projected separation of 275 pc in the sky-plane at the distance of the GC (see Table 5). Initial timing observations during the discovery showed a mild evidence for a negative $\dot{P}$, and the current value is measured to be $-4.94(2)\times 10^{-20}$ s/s based on our 6 years of LT observations. This negative $\dot{P}$ value indicates that this measurement is also affected by the pulsar dynamics in the Galactic potential (see § 5.3).

4.3 PSR J1757$-$2745

PSR J1757$-$2745 is an isolated mildly-recycled pulsar with a spin period of 17.7 ms and located at an angular separation of $2\aas@@fstack{\circ}95$ from the GC (see Table 5). Our 4.5 years of observations improved its timing solution, updating the $\dot{P}$ measurement to be $-1.30(2)\times 10^{-20}$ s/s (see Table 4). This negative value indicates that this $\dot{P}$ measurement is also affected by the pulsar dynamics in the Galactic potential, and we study this in detail (see § 5.3). The timing analysis also places a limit on the proper motion, particularly in Right Ascension (see Table 4).

The previous timing analysis of this pulsar reported a $\dot{P}$ measurement of $2.1(2)\times 10^{-19}$ s/s (Ng et al., 2015) based on a short time span of 206 days. However, we now confirm that this value is not correct due to the short data span and resultant large covariance between the rotational and astrometric parameters.

4.4 PSR J1801$-$3210

PSR J1801$-$3210 has a spin period of 7.45 ms and is located at an angular separation of $4\aas@@fstack{\circ}7$ from the GC (see Table 5). As shown in Table 4 based on our 9 yr data set, we find improved timing parameters of the pulsar, indicating that $\dot{P}$ measurement is constrained to be a positive value of $7.4(8)\times 10^{-23}$ s/s. This measurement is still smaller than the lowest positive $\dot{P}$ measured to-date for any known pulsar999http://www.atnf.csiro.au/people/pulsar/psrcat.

The previous timing analysis of this pulsar showed a low-significance $\dot{P}$ measurement of $-4(4)\times 10^{-23}$ s/s based on a 4 yr long data set (Ng et al., 2014). They argued that the pulsar is possibly located behind the sky-plane which passes through the GC along the LOS. In that case the negative Galactic acceleration affects the positive intrinsic spin-down of the pulsar and results in an extremely low measured $\dot{P}$ value (see § 5.1 for details). Although the DM-derived distance from the NE2001 electron density model places the pulsar in front of the sky-plane (see Table 1), they argued that the electron density models can contain large uncertainties. Therefore, our exceptionally small $\dot{P}$ measurement indicates the scenario of the pulsar being located behind the sky-plane through the GC proposed by Ng et al. (2014) is still valid (which we will show independently in § 5.4).

4.5 PSR J1804$-$2858

PSR J1804$-$2858 is a solitary millisecond pulsar and has the third lowest spin period known to-date, a spin period of $1.49$ ms (Morello et al., 2019, Barr et al., in prep). It is located at an angular radius of 4∘ from the GC (Table 5) and the DM is 232 cm*-3* pc, which places the pulsar at a distance of 8.2 kpc from the Earth based on the YMW16 electron density model (see Table 1). The discovery analysis of the pulsar did not have a $\dot{P}$ measurement, however our 4.3 years of timing data measures a significant $\dot{P}$ value of $-7.57(2)\times 10^{-22}$ s/s (see Table 4). As mentioned above for other pulsars, this measured $\dot{P}$ value is not entirely intrinsic, rather induced by the acceleration of the pulsar along its LOS due to the Galactic potential. We study its dynamics in detail combining with the Galactic mass distribution and constrain the intrinsic $\dot{P}$ value in § 5.3.

4.6 PSR J1746$-$2758

This isolated $0.49$-s spin period pulsar has an angular separation of just $0\aas@@fstack{\circ}96$ from the GC (see Table 5). Our updated timing solution based on more than four years of observations indicates a $\dot{P}$ measurement for the first time to be $4.42(5)\times 10^{-18}$ s/s (see Table 3). Comparing this measurement with $\dot{P}$ values of known pulsars that have a similar pulse period, it is evident that this value is slightly smaller than the majority, but within the distribution. On the other hand, one can imagine, given the pulsar’s location, that it is closer to the GC than the DM-derived distance (see Table 1) and thus, the measured $\dot{P}$ is not entirely intrinsic, but rather induced by the Galactic potential. However, such a claim is not likely to be straightforward. Further note that the maximum Galactic acceleration in the direction of the pulsar is greater than the measured $\dot{P}/P$ and thus, we study its dynamics due to the Galactic potential in § 5.3 in detail.

5 Dynamics of pulsars

The timing models for PSRs J1748$-$3009, J1753$-$2819, J1757$-$2745, and J1804$-$2858 show negative observed $\dot{P}$ measurements (see § 4), indicating that these values are not entirely intrinsic and have a dynamic contribution. We also note that the observed $\dot{P}$ of PSR J1801$-$3210 is extremely small, so that it is also likely to be affected by dynamics (see § 4.4). In this section, we model the observed unusual $\dot{P}$ measurements as being due to acceleration of these pulsars along their LOSs, and then constrain their intrinsic $\dot{P}$ values and possible distances.

5.1 Pulsar line-of-sight acceleration

As noted in previous studies (e.g. Damour & Taylor, 1991; Nice & Taylor, 1995; Matthews et al., 2016; Desvignes et al., 2016; Perera et al., 2017), the observed $\dot{P}_{\rm obs}/P$ of the pulsar is a combination of individual terms and can be expressed as

[TABLE]

where, $\dot{P}_{\rm obs}$ and $\dot{P}_{\rm int}$ are the observed and intrinsic spin period derivatives, respectively, $a$ is the LOS acceleration of the pulsar with respect to the Sun due to the 3-dimensional Galactic potential, $a_{\rm\mu}$ is the acceleration due to transverse motion of the pulsar (i.e. the Shklovskii effect – Shklovskii, 1970), and $c$ is the speed of light. In previous studies, $a/c$ is computed as two separate components, namely the parallel component in the Galactic plane and the vertical component towards the Galactic plane (see Damour & Taylor, 1991; Nice & Taylor, 1995; Bassa et al., 2016b; Desvignes et al., 2016; Lazaridis et al., 2009). These components were estimated separately based on Galactic rotation curve models and Galactic vertical acceleration models for a given $(l,b)$ of a pulsar using the Galactocentric distance of the solar system ${\rm R}_{\odot}$, the circular rotational speed of the Galaxy at ${\rm R}_{\odot}$, and the pulsar distance. Since the location of the Sun is only $25\pm 5$ pc above the Galactic plane (Bland-Hawthorn & Gerhard, 2016; Bland-Hawthorn et al., 2019), its vertical acceleration towards the plane is very small and neglected in general (e.g. Damour & Taylor, 1991). In contrast, we directly use 3-dimensional Galactic mass distribution models (see § 5.2) and thus, estimate the 3-dimensional acceleration of the pulsar and the Sun separately at their locations in the Galaxy. We then use these acceleration terms to compute the relative acceleration $a/c$ of the pulsar along its LOS with respect to the Sun. We note that the acceleration of the Sun is insignificant compared to that of the pulsar when the pulsar is located near the GC. We describe the Galactic mass distribution models used in this study and the estimation of $a/c$ in § 5.2.

The Shklovskii effect can be written as $a_{\rm\mu}/c=v_{T}^{2}/cd$ $(\equiv\mu^{2}d/c)$, where $v_{T}$ and $\mu$ are the transverse velocity and the proper motions of the pulsar with respect to the Sun, respectively, and $d$ is the pulsar distance (Shklovskii, 1970). Our timing solutions do not measure significant proper motions of the pulsars considered here, rather place limits (see Table 3 and 4). Thus, we assume typical pulsar properties to estimate this term. We note that the velocity of a pulsar is an intrinsic property of the source and the proper motion is an observed measurement that depends on the pulsar distance and the relative motion of the pulsar and the Sun. Therefore, we use typical pulsar 3-dimensional peculiar velocities and then estimate the expected $a_{\rm\mu}/c$. In general, young pulsars have higher 3-dimensional velocities compared to millisecond pulsars (Lyne et al., 1998). Therefore, we assume a typical value of 100 km/s, which is consistent with the millisecond pulsar population (e.g. Gonzalez et al., 2011; Matthews et al., 2016), and 320 km/s, which is consistent with the ordinary pulsar population (e.g. Hobbs et al., 2005; Verbunt et al., 2017), in this study. We describe our calculation of $a_{\rm\mu}/c$ in Appendix A in detail. We note that if the actual velocity of the pulsar is small, then the $a_{\rm\mu}/c$ contribution to the observed $\dot{P}_{\rm obs}/P$ becomes insignificant, and vice-versa (see Eq. 1).

The Galactic term $a/c$ has both positive and negative signs depending on the source location along the LOS in the Galaxy. When the source is located behind (or in front of) the sky-plane which passes through the GC, then $a/c$ becomes negative (or positive). We note that the Shklovskii term $a_{\rm\mu}/c$ is always positive. Given that the intrinsic $\dot{P}$ is positive, the pulsar must be located behind the sky-plane in order to explain any observed negative $\dot{P}$, and the observed $\dot{P}/P$ should be dominated by the Galactic contribution (see Eq. 1). A schematic diagram of the geometry used in this study and the definitions are given in Fig. 2.

5.2 Galactic mass distribution models and pulsar line-of-sight acceleration

We use the Galactic mass distribution model proposed in Sofue (2013, 2017) in our analysis to estimate $a/c$. They analysed high-resolution longitude-velocity diagrams of the Galaxy and constructed the rotation curve including the central black hole. Then they deconvolved the rotation curve and fit several components to the observed velocities across the Galaxy, resulting in two exponential-spherical bulges (i.e. inner and main bulges), an exponential flat disk, and a Navarro-Frenk-White (Navarro et al., 1996) dark halo. Using this information, Sofue (2013) modelled the volume mass density of each component and then the mass distribution of the Galaxy (see Table 6 in Sofue, 2017, for updated values). This study used observations covering the inner region (from $\sim$1 pc) of the Galaxy and thus, required several bulge components to appropriately model the data.

For comparison, we also use the Galactic mass distribution model proposed by McMillan (2017). This model uses a main bulge, two stellar discs, two gas discs, and a dark halo. The McMillan (2017) model is an improved version of their previous model (McMillan, 2011) which now includes gas discs and uses new observations of maser sources in the Galaxy. We note that these models mainly focus on the region beyond the bulge (where $r_{\perp}>1$ kpc) in the Galaxy. Since our pulsars are likely located around the central region of the Galaxy (within a $r_{\perp}<1$ kpc, see Table 5), we use the Sofue (2013) model with updated parameters given in Sofue (2017) as our main Galactic mass distribution model in the analysis.

Throughout the analysis, we assume the Galactocentric distance of the solar system to be ${\rm R}_{\odot}=8.2\pm 0.1$ kpc and the central black hole mass to be $(4.2\pm 0.2)\times 10^{6}$ M⊙ (Bland-Hawthorn et al., 2019; Vasiliev, 2019; Bland-Hawthorn & Gerhard, 2016; Sofue, 2013, 2017; McMillan, 2017). These values are consistent with those derived in previous studies within their uncertainties (e.g. Genzel et al., 2000; Ghez et al., 2005, 2008; Gillessen et al., 2009; Reid et al., 2009; Honma et al., 2012; Schönrich, 2012; Brunthaler et al., 2011; Reid et al., 2014). We further note that this ${\rm R}_{\odot}$ is broadly consistent with the recently measured GC distance, which has an uncertainty of only $0.3$% (Abuter et al., 2019).

To be consistent with the above Galactic models (Sofue, 2013; McMillan, 2017), we define the volume mass density of the bulge component in cylindrical Galactocentric coordinates

[TABLE]

where, $\rho_{\rm 0,b}$ is the central mass density and $r=\sqrt{R^{2}+(z/q)^{2}}$ with the axis ratio $q$. Sofue (2013) includes two bulges with $\zeta=1$, $\alpha=0$, $r_{0}=1$ kpc, and $q=1$ (i.e. spherically symmetric assumption), while McMillan (2017) includes a single bulge with $\zeta=2$, $\alpha=1.8$, $r_{0}=0.075$ kpc, and $q=0.5$ (i.e. an axially symmetric assumption). All best-fit parameter values in their models are given in Table 6. The volume mass density of the stellar disc component,

[TABLE]

Here, $\Sigma_{\rm 0,d}$ is the central surface density, $z_{d}$ is the scale height, and $R_{d}$ is the scale length. Sofue (2013) uses a flat stellar disc approximation (i.e. $z_{d}=0$ an $\eta=0$), while McMillan (2017) uses thin and thick stellar discs with $z_{d}=0.3$ and $0.9$ kpc, respectively, and $\eta=1$. The surface density and the scale radius of the two models are given in Table 6. In addition to stellar discs, McMillan (2017) uses two gas discs representing $H_{1}$ and $H_{2}$ gas in the Galaxy. These gas discs are defined with mass densities

[TABLE]

with holes in their centres. The $H_{1}$ and $H_{2}$ discs have scale height $z_{\rm d}$ of $0.085$ kpc and $0.045$ kpc, and central hole scale radius $R_{\rm m}$ of 4 kpc and 12 kpc, respectively. In both Sofue (2013) and McMillan (2017), the dark matter halo is defined as having mass density

[TABLE]

where $X=\sqrt{R^{2}+z^{2}}/h$ and $\rho_{\rm 0,h}$ and $h$ are the scale density and core radius, respectively. The fitting procedure and the best-fit values for the above parameters are given in Sofue (2017) and McMillan (2017), and also in Table 6 of this study.

Using the above components and the central BH, we estimate the total volume mass density of the Galaxy as a function of radius $R$ (see the top panel in Fig. 3). Based on mass density, we then estimate the mass distribution (i.e. the total mass interior to a given radius $R$) of the Galaxy (see the bottom panel in Fig. 3). For further comparison, we also use previously published Galactic mass distribution models (Piffl et al., 2014; Binney & Tremaine, 2008; Dehnen & Binney, 1998) with their best fit parameters and calculate the volume mass density and the mass distribution as a function of radius (see Fig. 3). It can be seen that all these additional models were focused on the regions beyond the bulge and thus, derive similar variations in their density and mass curves to those derived from McMillan (2017) model. We note that the Galactic models in Dehnen & Binney (1998) are derived for the outer Galaxy and thus, its density and mass estimates are not meaningful at radii $R\lesssim 8$ kpc. As shown in Fig. 3, the model derived in Sofue (2013) provides more features and better constraints within the inner region of the Galaxy, particularly in the bulges where our pulsars are likely to be located.

We further note that some Galactic mass models suggest that there is a need for a “barred bulge” component in the central region of the Galaxy (e.g., McWilliam & Zoccali, 2010; Nataf et al., 2010; Gardner & Flynn, 2010; Ness et al., 2012). These models are non-axisymmetric and are typically computationally expensive to implement. By comparing the Sofue (2013) model and the bar/bulge model given in Portail et al. (2017), we noticed that the mass distribution at a given radius of the two models are consistent with each other. Compared to an axially symmetric model, a bar/bulge model can cause a slight difference in the Galactic gravitational potential along the LOS of the pulsar depending on the orientation of the bar component, leading to a slight change in our results. However, this change is within the uncertainties of our estimated $\dot{P}_{\rm int}$ values and pulsar distances. Therefore, we simply use the above mentioned axially symmetric Galactic mass models in this study.

We first calculate the gravitational potential $\Phi$ at the pulsar location $(R,z)$ due to the Galactic mass distribution, and then estimate the Galactic acceleration in $R$ and $z$ directions, namely $-d\Phi/dR$ and $-d\Phi/dz$. To do this, we use the computer software GalPot101010https://github.com/PaulMcMillan-Astro/GalPot (McMillan, 2017, 2016), which computes the Galactic potential and the acceleration for any given Galactic mass distribution model. This software was based on Dehnen & Binney (1998) and the details of the computation are given in McMillan (2017). We note that since all these Galactic mass models are axially symmetric (see Eq. 2 – 5), the acceleration does not depend on the azimuthal angle. Once the acceleration components of the pulsar and the Sun at their given locations in the Galaxy in $R$ and $z$ directions are computed, we can estimate the relative LOS acceleration $a/c$ of the pulsar with respect to the Sun (see § 5.1). The acceleration $a/c$ varies with the distance to the pulsar, providing a negative (or positive) value when the pulsar is located behind (or in front of) the sky-plane.

We estimate the maximum possible LOS acceleration curves $a_{\rm max}/c$ due to the Galactic potential as a function of Galactic longitude for a given Galactic latitude (see Fig. 4). We plot the absolute values of $|\dot{P}_{\rm obs}/P|$ of PSRs J1748$-$3009, J1753$-$2819, J1757$-$2745, and J1804$-$2858 in the figure. Due to the axially symmetric mass distribution and the acceleration, we map the pulsars that are located in the fourth quadrant of the Galaxy in the first quadrant and include their $\dot{P}_{\rm obs}/P$ in the figure. It is evident that, as shown in Fig. 4, all our six PSRs J1746$-$2758, J1748$-$3009, J1753$-$2819, J1757$-$2745, J1801$-3210$, and J1804$-$2858 have much lower $|\dot{P}_{\rm obs}/P|$ than the maximum Galactic accelerations along their LOSs obtained from our main Galactic model (Sofue, 2013). Therefore, we cannot simply neglect the Galactic contribution in their observed $\dot{P}$. We investigate the LOS accelerations of these pulsars in detail in § 5.3, leading to limits on their intrinsic $\dot{P}$ values. Fig. 4 also shows that most of the known pulsars that are located within the same field of view have much larger $\dot{P}_{\rm obs}/P$ compared to their Galactic maximum accelerations. If their Shklovskii terms are small, we can simply assume that the observed $\dot{P}$ measurements of these pulsars are approximately equal to their intrinsic values. In addition, PSRs J1723$-$2837, J1727$-$2946, J1751$-$2857, and J1804$-2717$ have observed $\dot{P}_{\rm obs}/P$ that are comparable to the maximum Galactic accelerations along their LOSs (see black dots in Fig. 4). We note that the DM measurements of these pulsars are relatively small (between $19-60$ cm*-3* pc; Hobbs et al., 2004b; Crawford et al., 2013; Lorimer et al., 2015; Desvignes et al., 2016) compared to those of our pulsars in the analysis (see Table 1). The DM-derived distances (using YMW16 model) to these pulsars are in the range of $0.72-1.9$ kpc (note that the NE2001 model derived distances are also in a similar range). This indicates that they are located close to the Sun and thus, the Galactic contributions in their observed $\dot{P}$ values are insignificant, which means that the observed $\dot{P}$ values are approximately equal to their intrinsic values.

5.3 Constraining the intrinsic period derivatives of pulsars

We know that $\dot{P}_{\rm int}$ and thus, $\dot{P}_{\rm int}/P$, of a non-accreting pulsar is positive. As described before, $a_{\mu}/c$ is a positive contribution and thus, the measured negative $\dot{P}_{\rm obs}$ measurements of PSRs J1748$-$3009, J1753$-$2819, J1757$-$2745, and J1804$-$2858 can only be explained with a negative $a/c$ Galactic contribution (see Eq. 1). Using a Galactic model as shown in § 5.2, we can compute the minimum possible Galactic acceleration $(a_{\rm min}/c)$ along a given LOS. Therefore, we can write the upper limit of the intrinsic period derivative for these four pulsars to be

[TABLE]

using Eq. 1. We first estimate the Galactic acceleration curves of these pulsars along their LOSs and plot in Fig. 5. We then estimate $a_{\mu}/c$ assuming the pulsar moves with a space velocity of either 100 km/s (black dashed line) or 320 km/s (black dotted line) and over-plot in the same figure. For clarity, the x-axis in Fig. 5 represents the distance $L$ measured from the sky-plane (see Fig. 2 for definitions). Appendix A describes the calculation of $a_{\mu}/c$ in detail. We then estimate the model-dependent upper limit on the intrinsic $\dot{P}$ of PSRs J1748$-$3009, J1753$-$2819, J1757$-$2745, and J1804$-$2858 using Eq. 6 with the Galactic accelerations shown in Fig. 5. The estimated limits are given in Table 7. PSRs J1757$-$2745 and J1804$-$2858 have measured values on their proper motions (see Table 4). For comparison, we estimate the Shklovskii contribution based on these measured proper motions in Right Ascension and over-plot in Fig. 5 (see the orange curves).

In addition to the observed $\dot{P}$, the timing solution of PSR J1748$-$3009 places a limit on its orbital period derivative $\dot{P}_{b}$ (see Table 3). The timing parameters indicate that the pulsar is in a compact binary system with a period of 2.9-d and possibly in orbit with a white draft (the mass of the companion is $0.08-0.20$ M⊙ with a median of 0.1 M⊙). Therefore, we can assume that the observed $\dot{P}_{b}$ includes contributions from both dynamics and the emission of gravitational waves (see Taylor & Weisberg, 1989; Stairs et al., 2002; Kramer et al., 2006). Thus, the observed $\dot{P}_{b}$ provides important information to estimate the intrinsic period derivative of the pulsar independently. This approach has been used previously in several studies (e.g., Matthews et al., 2016; Freire et al., 2017; Prager et al., 2017). Similar to Eq. 1, we can express the pulsar LOS acceleration in terms of observed $\dot{P}_{b}$ as follows:

[TABLE]

where $(\dot{P}_{\rm b}/P_{\rm b})_{\rm GR}$ is the expected contribution due to gravitational wave emission. Assuming General Relativity, the expected $(\dot{P}_{b})_{\rm GR}$ is $-4.3\times 10^{-16}$ s/s (using Eq. 8.52 in Lorimer & Kramer, 2005, assuming a pulsar mass of 1.4 M⊙ and $\cos(i)=0.5$, resulting in a companion mass of 0.1 M⊙), which is significantly smaller than the observed $\dot{P}_{b}$ limit of $-2.6(10)\times 10^{-11}$ s/s. This indicates that the origin of the observed $\dot{P}_{b}$ is most likely induced by the dynamical acceleration component of the pulsar along the LOS, similar to the observed $\dot{P}$ measurement discussed earlier. Therefore, we can simply assume that $(a/c+a_{\mu}/c)\approx\dot{P}_{b}/P_{b}$ from Eq. 7 and then substitute that in to Eq. 1 to deduce $\dot{P}_{\rm int}$ independently, and it is calculated to be $(9.7\pm 8.0)\times 10^{-19}$ s/s, where the uncertainty is 2$\sigma$. Note that this measurement is greater than the upper limit constrained above using the Galactic mass model (see Table 7). The reason for this discrepancy could be that we only have a limit on $\dot{P}_{b}$ through timing, so that the above estimated $\dot{P}_{\rm int}$ from Eq. 7 is not well constrained.

As given in Table 4, the timing model of PSR J1801$-$3210 shows a positive, but extremely small $\dot{P}$. Given that PSR J1801$-$3210 is located towards the GC (with an angular separation of $4\aas@@fstack{\circ}7$ – see Table 5) and has a large DM, it is probable that the measured $\dot{P}$ is dominated by the acceleration of the pulsar due to the Galactic potential. If $\dot{P}_{\rm obs}<\dot{P}_{\rm int}$, the contribution $a/c$ must be negative (see Eq. 1). We plot the acceleration components given in Eq. 1 for this pulsar along its LOS in Fig. 6. Following the same method as above, we can then estimate the model-dependent upper limit on its intrinsic $\dot{P}$ using Eq. 6, leading to a value of $3.5\times 10^{-20}$ s/s (see Table 5). This limit is broadly consistent with the measured $\dot{P}$ values of unaccelerated millisecond pulsars with a similar period and we suspect that it is closer to the intrinsic $\dot{P}$ value of the pulsar.

PSR J1746$-$2758 has a positive $\dot{P}$ measurement of $4.42(5)\times 10^{-18}$ s/s, but this value is slightly smaller than that of the majority of other known pulsars that have a similar period (§ 4.6). Given that it is located towards the GC, with an angular separation of $0\aas@@fstack{\circ}96$, and also has a large DM, it is possible that the observed $\dot{P}$ of PSR J1746$-$2752 is also influenced by its LOS acceleration. Therefore, we estimate the acceleration curves of this pulsar following the same procedure as described above (see Fig. 6). Using Eq. 6, we then estimate the upper limit on its intrinsic $\dot{P}$ to be $1.48\times 10^{-17}$ s/s, which is approximately a factor of 3 larger than the measured value.

We finally plot the dynamically-derived model-dependent intrinsic $\dot{P}$ limits of all our pulsars with their measured $P$ values together with those of other known pulsars in Fig. 12. It is clear that the dynamically-derived $\dot{P}$ limits of our pulsars are broadyly consistent with those of other pulsars. Using these upper limits on $\dot{P}$ values, we estimate and present the lower limits on characteristic ages and upper limits on surface magnetic fields in Table 3 and 4 (using Eq. 3.12 and 3.15 given in Lorimer & Kramer, 2005).

5.4 Dynamical constraints on pulsar distances

Using the LOS acceleration values shown in Figures 5 and 6, we can simply estimate the intrinsic period derivatives $\dot{P}_{\rm int}$ of PSRs J1746$-$2758, J1748$-$3009, J1753$-$2819, J1757$-$2745, J1801$-$3210, and J1804$-$2858 as a function of distance using Eq. 1 (see Fig. 8 and 9). By definition we know that $\dot{P}_{\rm int}$ is positive and thus, these curves can be used to constrain the model-dependent distances to these pulsars. Using the known pulsar population, we can impose possible lower limits on their $\dot{P}_{\rm int}$ values. To do that, we select known pulsars within a 50 per cent window around the period of a given pulsar in our sample, and then obtain the lowest observed $\dot{P}$ in that population. This lowest $\dot{P}$ is used as the minimum possible $\dot{P}_{\rm int}$ for that given pulsar and it is plotted in Fig. 8 and 9 with horizontal dashed lines. Assuming that $\dot{P}_{\rm int}$ should be greater than this minimum limit, we constrain the distances to be $(8.24-8.99)$, $(8.24-9.08)$, $(8.22-9.65)$, $(8.25-9.64)$, and $(8.40-8.90)$ kpc for PSRs J1748$-$3009, J1753$-$2819, J1757$-$2745, J1801$-$3210, and J1804$-2858$, respectively (see Table 5). We notice that the lower limits of these distances are beyond the distances to the plane which passes through the GC perpendicular to the line joining the Sun and the GC along their LOSs, placing all these five pulsars on the far-side of the Galaxy. The distance to PSR J1746$-$2758 is not well constrained (see Fig. 9), although, it is likely to be located at a distance of either $<$7.5 kpc or $>$8.16 kpc. In all the above distance estimates, we assume Sofue (2013) as our main Galactic model and the peculiar velocity of the pulsar to be 100 km/s when calculating the Shklovskii contribution. If the velocity is larger than this assumed value, then $\dot{P}_{\rm int}$ decreases (i.e. the intrinsic curve shifts downward along y-axis) and thus, constrains the distance to a smaller range. Moreover, using the timing-measured proper motion limits of PSRs J1757$-$2745, J1801$-$3210, and J1804$-$2825, we estimate the Shklovskii contribution and then derive $\dot{P}_{\rm int}$ as before (see orange curves in Fig. 8 and 9), leading to more conservative pulsar distances of $(8.27-9.40)$, $(8.20-15.50)$, and $(8.32-9.40)$ kpc, respectively.

Note that, as we mentioned in § 5.2, the McMillan (2017) model mainly focuses beyond the bulge regions and thus, it cannot be used accurately to constrain the distances to these pulsars. This can be seen in Fig. 8 and 9 as the model-derived $\dot{P}_{\rm int}$ values (grey solid curves) are potentially smaller than the minimum possible $\dot{P}_{\rm int}$ values (dashed black horizontal lines).

5.5 Comparison of dynamic-derived distances with DM-derived distances

The thermal free electron density model NE2001131313https://www.nrl.navy.mil/rsd/RORF/ne2001 (Cordes & Lazio, 2002) is widely used to estimate the distances to pulsars based on their measured DM obtained through timing. More recently, Yao et al. (2017) (YMW16141414http://119.78.162.254/dmodel/index.php) re-evaluated the electron density model by using more recent independent pulsar distance observations. We compare our dynamically-estimated distances for PSRs J1748$-$3009, J1753$-$2819, J1757$-$2745, J1801$-$3210, and J1804$-$2825 given in § 5.4 with their DM-derived distances obtained from these electron density models. Note that we do not use PSR J1746$-$2758 in this comparison since its distance is not well constrained. We first use the YMW16 and NE2001 models separately to estimate the DM along the LOS of these pulsars (see Fig. 10). Note that we take account of a typical 20 per cent error on these DM values and indicate it with the shaded region in the figure. Based on the measured DM values, we find that the DM-derived distances obtained from the YMW16 model for PSRs J1748$-$3009, J1757$-$2745, J1801$-$3210, and J1804$-$2825 are consistent with our dynamics-derived distances within their uncertainties. For PSR J1753$-$2819, the DM-derived distance from YMW16 is small compared to its dynamics-derived distance. This indicates that the electron density predicted from YMW16 along the LOS of PSR J1753$-$2819 is overestimated, resulting in underestimated distance. As shown in the figure, the DM-derived distances obtained from NE2001 model for all of these pulsars are underestimated compared to their dynamics-estimated distances. By comparing the two models, it is seen that in general the distances estimated by the NE2001 model are underestimated (as reported before, see, e.g., Kramer et al., 2003) compared to those estimated by the YMW16 model at a given DM along the LOSs of these pulsars.

We consider all known pulsars that are located within an angular radius of $5^{\circ}$ from the GC (i.e. $\theta<5^{\circ}$) and plot their projected distances in the Galactic plane (see Fig. 11). Note that, we use DM-derived distances for these pulsars using the YMW16 model and assume a typical 20% uncertainty. We also plot our dynamically-derived distances for PSRs J1748$-$3009, J1753$-$2819, J1757$-$2745, J1801$-$3210, and J1804$-$2858 in the same figure. As can be seen, our distance estimates have much smaller uncertainties compared to the DM-derived distances, and these five pulsars are located on the far-side of the Galaxy.

6 Discussion and Conclusion

We report on the timing of recently discovered PSR J1753$-$2819, a 18.6 ms pulsar in a 9.3-hr binary orbit. We included this pulsar along with PSRs J1746$-$2758, J1748$-$3009, J1757$-$2745, J1801$-$3210, and J1804$-$2858 using LT and PKS observations to understand their $\dot{P}$ measurements. The improved timing baselines indicate significant negative $\dot{P}$ measurements for PSRs J1748$-$3009, J1753$-$2819, J1757$-$2745, and J1804$-$2858 (see Table 3 and 4). This is the first non-globular cluster pulsar sample that has observed negative $\dot{P}$ measurements in their timing solutions, and they all lie within a few degrees of the GC. In addition, the timing of PSR J1801$-$3210 confirmed that it has an extremely low positive $\dot{P}$ measurement, which provides the lowest measured value for a pulsar to date.

Since pulsars are powered by the loss of rotational kinetic energy, they spin-down gradually with a positive intrinsic $\dot{P}$. Therefore, the negative $\dot{P}$ measurements of PSRs J1748$-$3009, J1753$-$2819, J1757$-$2745, J1804$-$2858 and the extremely small $\dot{P}$ measurement of PSR J1801$-$3210 are not intrinsic, but induced by dynamics. We modelled the observed $\dot{P}$ values of these pulsars as being due to dynamical acceleration in the Galactic potential along their LOSs. The analysis constrained the model-dependent upper limits on the intrinsic $\dot{P}$ of these pulsars and they are broadly consistent with those of known pulsars (see Fig. 12 and Table 7). Using these model-dependent intrinsic $\dot{P}$ values, we then constrained possible pulsar distances, indicating that PSRs J1748$-$3009, J1753$-$2819, J1757$-$2745, J1801$-$3210, and J1804$-$2858 are located on the far-side of the Galaxy (see Fig. 11 and Table 7). This is the first time that dynamics have been used to measure the distances to pulsars that are located in the far-side of the Galaxy.

The timing solutions of three pulsars in our sample measured limits on their proper motions (see Table 4). Assuming the Galactic model (Sofue, 2013) is correct and $\dot{P}_{\rm int}>0$, we can determine the maximum possible total proper motion for PSRs J1757$-$2745, J1801$-$3210, and J1804$-$2858 and they are estimated to be 19.2, 12.6, and 13 mas/yr (using Eq. 1). These maximum possible total proper motions are consistent with our timing-measured limits given in Table 4. If the high-significance timing-measured proper motions in the future for these pulsars are greater than the above mentioned values, we can validate the Galactic model and also include any required additional features in the model to explain the observed pulsar motions.

As shown in Fig. 10, our dynamics-derived distances and the DM-derived distances from the YMW16 electron density model for these pulsars are consistent with each other in general within their uncertainties. However, the NE2001 model-derived distances are inconsistent with the dynamics-derived distance. This could be due to difficulties in modelling the electron densities in the inner regions of the Galaxy with high stellar densities and large gas and dust contents towards the centre. Therefore, the independent distance measurements obtained to the pulsars in this study can be used to further improve electron density models.

PSRs J1748$-$3009, J1753$-$2819, and J1801$-$3210 are in binary systems and the timing results indicate that their companions are likely to be white dwarfs. Using the measured $\dot{P}_{b}$ limit of PSR J1748$-$3009, we derived an independent limit on its intrinsic $\dot{P}$. Following a similar method, we will be able to measure $\dot{P}_{b}$ values of these binary pulsars in the future with more observations and thus, derive their intrinsic $\dot{P}$ values independently with better accuracy (see Smits et al., 2011). These estimates then can be used to improve their distance measurements.

Most of the Galactic models assume that the mass distribution in the Galaxy is either axially or spherically symmetric for simplicity, in particular within the bulge region. Having more pulsars in the Galactic centre region, we can use their observations to derive dynamics independently to study the Galactic potential, and then update and validate Galactic models including clumpiness in the mass distribution. We can also obtain independent distances to these GC pulsars in the future through parallax measurements using interferometers such as VLBI and then use them in deriving the Galactic potential. The next generation telescopes, such as the Square Kilometre Array, are useful with their outstanding capabilities to discover more pulsars in the GC region and to allow a comprehensive study to constrain the Galactic potential in the central region using pulsar timing in the future.

Acknowledgments

Pulsar research at the Jodrell Bank Centre for Astrophysics and the observations using the Lovell Telescope are supported by a consolidated grant from the STFC in the UK. The Parkes radio telescope is part of the Australia Telescope, which is funded by the Commonwealth of Australia for operation as a National Facility managed by the Commonwealth Scientific and Industrial Research Organisation (CSIRO). D.R.L. acknowledges support from NSF RII Track I award number OIA$-$1458952 and is part of the NANOGrav Physics Frontiers Center which is supported by NSF award 1430284. R.S. acknowledges support from the Australian Research Council grant FL150100148. MB acknowledges ARC Grant CE170100004 (OzGrav).

Appendix A Transverse velocity of the pulsar and the Shklovskii contribution

We denote the peculiar velocity of the Sun as $\vec{v}_{{}_{\odot},{\rm pec}}=[U,V,W]$, where the unit vectors of the components are in the direction from the Sun towards the GC, in the direction of the Galactic rotation, and in the direction perpendicular to the Galactic plane, respectively. Then the total velocity of the Sun

[TABLE]

where $v_{R}(R_{\odot})$ is the velocity of the local standard of rest for the Sun, or the circular Galactic rotational velocity at the radius of Sun $R_{\odot}$. For our calculation, we assume $[U,V,W]=[11.1,12.24,7.25]$ km/s (Schönrich et al., 2010) and obtain the Galactic rotation $v_{R}$ from Galactic mass distribution models given in § 5.2. The main Galactic model we use in this study (Sofue, 2017) results in $v_{R}(R_{\odot})=238$ km/s, which is the value they adopted in their model from Honma et al. (2012). Then the peculiar velocity, or the 3-dimensional space velocity, of the pulsar is defined as $\vec{v}_{\rm p,pec}=[U_{p},V_{p},W_{p}]$ in the same coordinate system as before. The total velocity of the pulsar including its local standard of rest,

[TABLE]

where $v_{R}(R_{p})$ is the Galactic rotational velocity of the pulsar at its projected radial distance $R_{p}$ on to the Galactic plane and $\theta^{\prime}$ is the angle subtended at the GC between the position of the Sun and the projected position of the pulsar on the Galactic plane (see Fig. B.1 in Verbunt et al., 2017, for a schematic diagram). The angle $\theta^{\prime}$ can be expressed as

[TABLE]

Using the above coordinate system, the relative velocity of the pulsar with respect to the Sun

[TABLE]

To express the relative velocity in the Galactic coordinate system $(l,b)$, we write the unit vectors $(\hat{l},\hat{b})$ in above coordinate system as

[TABLE]

We then write the relative velocity of the pulsar for its given $(l,b)$ with respect to the Sun in Galactic coordinates as

[TABLE]

and

[TABLE]

respectively. We finally calculate the transverse velocity of the pulsar to be $v_{T}=\sqrt{(v_{l}^{2}+v_{b}^{2})}$. The derivation of the above expressions are given in Verbunt et al. (2017) in detail.

In § 5, for a given 3-dimensional pulsar space velocity of $|\vec{v}_{\rm p,pec}|$, we randomly calculate $[U_{p},V_{p},W_{p}]$ (where $|\vec{v}_{\rm p,pec}|=\sqrt{U_{p}^{2}+V_{p}^{2}+W_{p}^{2}}$), leading to a transverse velocity of $v_{T}$ using Eq. 13 and 14. We perform 1000 trials for a given $|\vec{v}_{\rm p,pec}|$ and then obtain the average $v_{T}$, and then calculate the Shklovskii effect $a_{\mu}/c=v_{T}^{2}/cd$. We note that $v_{T}$ is a function of $R_{p}$ and thus, it varies along the LOS of the pulsar for a given $\vec{v}_{\rm p,pec}$, and then $a_{\mu}/c$. For instance, this variation in $a_{\mu}/c$ can be seen in Fig. 5.