1. Pulsar timing ASTRONOMY AND SPACE SCIENCE George Hobbs October 2015, Urumqi

2. Overview of pulsar timing 1.Pulsars are rapidly rotating neutron stars 2.They emit a beam of radiation that produces the observed radio pulses 3.They are incredibly stable rotators 4.The pulses can be used like the tick of a clock Presentation title | Presenter name | Page 2

3. Pulsar timing Counselman & Shapiro (1968) “pulsars … can be used to test general relativity, to study the solar corona, and to determine the earth's orbit and ephemeris time … and the average interstellar electron density.” Presentation title | Presenter name | Page 3 脉冲星测时有多种应用，

4. Pulsar timing Now also … - searching for dark matter clumps - searching for gravitational waves - studying extra-solar planets - probing the interior of neutron stars … Presentation title | Presenter name | Page 4 The pulsar timing method!

5. What is the pulsar timing method? Need to know the rotational rate of the pulsar to “fold” the pulses Presentation title | Presenter name | Page 5 Parkes digital filterbank system 多个单个脉冲的 叠加能形成高信 噪比的积分轮廓。

6. Processing the data: from raw data to site-arrival-time CSIRO. Gravitational wave detection Raw observation Create a template Polarisation and flux calibration Cross correlated template with data Obtain pulse time of arrival (or site- arrival-time) All standard pulsar data calibration can be done using the PSRCHIVE software package (http://psrchive.sourceforge.net/) Use “pav” to view files Use “pac” and “pcm” to calibrate files Use “paas” to make a template

7. TOA (measured using the observatory clock) Residual What is the pulsar timing method? CSIRO. Measuring the mass of Jupiter using pulsars Fold Model Slide from D. Champion

8. Presentation title | Presenter name | Page 8  测量脉冲到达时间。  用简单的脉冲星旋转模型，能预测脉冲到达时间。  预测会非常准确。

9. Assume that we have just discovered PSR J1539-5521 1.Know Position = 15:39:08, -55:21:11 2.Know period = 1.00495845 sec 3.Dispersion measure = 380 cm -3 pc 4.Assume that we have an observation with the Parkes telescope of this pulsar (at 1400 MHz) and the pulse arrived at: t 0 = MJD 54000.28069921358 5.We should be able to predict when all other pulses should arrive from this pulsar. t 1 = 54000.28069921358 + 1*1.00495845*sec2day t 2 = 54000.28069921358 + 2*1.00495845*sec2day t 3 = 54000.28069921358 + 3*1.00495845*sec2day … t n = 54000.28069921358 + N*1.00495845*sec2day Presentation title | Presenter name | Page 9

10. Now we measure another pulse arrival time (one day later) t 1day = 54001.26700433189 Presentation title | Presenter name | Page 10 能用我们的模型预测到达时间吗？

11. Does our prediction work for the next observation? t0] 54000.28069921358 t1] 54001.26700433189 Presentation title | Presenter name | Page 11 Closest match, okay, but not perfect – prediction is out by -0.31 seconds Note: always choose the closest predicted value – maximum reported error can therefore be +/-0.5P – but we might be simply wrong! Pulse number 54000.28069921358 + 84795*1.00495845*sec2day = 54001.2669891646 54000.28069921358 + 84796*1.00495845*sec2day = 54001.2670007961 54000.28069921358 + 84797*1.00495845*sec2day = 54001.2670124275

12. Does our prediction work for the next few observations … no it doesn’t – we cannot predict the pulse arrival times Presentation title | Presenter name | Page 12 Why doesn’t this work? 模型预测得不好？为 什么？

13. Why doesn’t the prediction work? The telescope is on the Earth – the Earth is orbiting the Sun => Refer the observations to the solar system barycentre Presentation title | Presenter name | Page 13 R 1 is the vector between the observatory and the centre of the Earth R 2 is the vector from the centre of the Earth to the solar system barycentre Extra time delay:  t = |R 3 |/c.

14. Centre of the Earth to the Solar System Barycentre 1.Use a solar system planetary ephemeris. Currently JPL DEXXX series, Russian: EPM series or European INPOP series (also a Chinese Purple Mountain Observatory ephemeris – not used yet in pulsar observations) 2.Let’s use JPL DE405. 3.Various tools are available to extract the position of the Earth with respect to the solar system barycentre from an ephemeris. 4.t0] R earth2ssb = (502.258986937, -5.9788413, -2.65276363) lt-sec t1] R earth2ssb = (502.18883981, 1.765524654, 0.70504989) lt-sec Note: be careful about units. The ephemeris software often gives positions in units of Astronomical Units. Presentation title | Presenter name | Page 14 转换到太阳系质心

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16. Calculate delay 1.We need to know the position of the pulsar: RAJ 15:39:08, DECJ -55:21:11 2.We actually want a unit vector pointing at the pulsar, k. 3.K = [cos(RAJ)*cos(DECJ), sin(RAJ)*cos(DECJ), sin(DECJ)] = [-0.327847, -0.464466, -0.822671] 4.Extra time delay for a pulse to reach the solar system barycentre:  t earth2ssb = K.R earth2ssb 5.Measure barycentric arrival times BAT0 = t0 +  t earth2ssb = 54000.2788507786 BAT1 = t1 +  t earth2ssb = 54001.2650825592, error in prediction = -0.0011sec (cf. -0.31s before) Presentation title | Presenter name | Page 16

17. Now can we predict the pulse arrival times using the barycentric arrival times? … sort of! Presentation title | Presenter name | Page 17

18. Zooming in Presentation title | Presenter name | Page 18 Why can’t we predict the arrival times perfectly? 好的。预 测得不错， 但并不完 美。

19. What if our original guess for the pulse period was slightly wrong? Presentation title | Presenter name | Page 19 Slope is 0.0005901914836 s/day S = 6.8309199500e-9 s/s Initial period = P0 = 1.00495845s Let’s change it to P0 + S.P0 = 1.0049584568647907 s 对于周期的猜测并 不准确。我们能改 正。

20. Presentation title | Presenter name | Page 20 Getting better!

21. Let’s talk in terms of phase and frequency Pulse frequency = 1/P Presentation title | Presenter name | Page 21 Time evolution of the pulse phase Nearest integer to the phase

22. Can measure a spin-down rate Presentation title | Presenter name | Page 22 Can fit for: a + b*(t-54200) + c*(t-54200)^2 Update period using “b” Use “c” to measure the slow- down rate 脉冲星在自转 减慢。必须考 虑此因素。

23. Now, how well can we predict the arrival times? Presentation title | Presenter name | Page 23 Slight wobbles

24. What about our measurements of the arrival times? 1.The observatory clock was on the Earth. We’re referring our observation times to the solar system barycentre. 2.Must account for relativistic time-dilation issues 3.Irwin & Fukushima (1999) give a time ephemeris of the Earth which related the times of clocks measured at the centre of the Earth and at the barycentre. This time ephemeris can be downloaded and the time correction determined for any given observation. 4.Time corrections,  t IF, are around 14.5s, but change through the year. BAT0 = OBS0 +  t earth2ssb +  t IF = 53999.9684497604 BAT1 = OBS1 +  t earth2ssb +  t IF = 54007.0443042452 Presentation title | Presenter name | Page 24 必须考虑相对论时间延迟效应

25. How good is our prediction? Presentation title | Presenter name | Page 25

26. Using a new pulse frequency Presentation title | Presenter name | Page 26 Look carefully and you’ll notice a few wobbles still

27. For a simple ~1 second pulsar it is moderately straight- forward to get a model that predicts the arrival times Need: - pulse frequency - pulse frequency time derivative (slow down rate) - epoch of pulse determination - pulsar position Must account for the extra delays required to convert to Barycentric arrival times. - geometric time delay - time-dilation issues Presentation title | Presenter name | Page 27 此简单模型能预测该 脉冲星的到达时间

28. But … 1.Interstellar medium dispersion delays can add ~1s (more or less depending on observing frequency) 2.Observatory position to Earth’s centre (~10ms) 3.Shapiro delay caused by the Sun (~100 us) 4.Earth wobbling (1 us) 5.Inaccuracies in the observatory clock (~1 us) 6.… Must also account for pulsar motions (proper motion and orbital affects) Presentation title | Presenter name | Page 28 Note that gravitational waves are at the <100ns level!

29. Errors in timing model parameters Presentation title | Presenter name | Page 29 Credit: Dick Manchester Reardon et al. (2015)

30. Let’s look at some real data Presentation title | Presenter name | Page 30

31. A history of pulsar timing: the discovery of B1919+21 CSIRO. Gravitational wave detection 1ms Rms timing residual = 500us Jodrell Bank data

32. The pulsar population before 1982 CSIRO. Gravitational wave detection

33. The pulsar population before 1983 CSIRO. Gravitational wave detection PSR B1937+21 discovered in 1982

34. PSR B1937+21 CSIRO. Gravitational wave detection Rms timing residual = 200ns 400ns Arecibo data

35. B1937 compared with B1919+21 CSIRO. Gravitational wave detection

36. Problem with B1937+21 CSIRO. Gravitational wave detection Where is this signal coming from? Arecibo + Parkes data

37. The pulsar population now (ATNF pulsar catalogue) CSIRO. Gravitational wave detection Lots of millisecond pulsars

38. Amazing timing residuals Shannon et al. (2015) Presentation title | Presenter name | Page 38 But, what is this noise?

39. What is the noise in this (simulated) data set?

40. Shrink the error bars

41. Longer data sets …

42. The answer Four very-simple (just change in F0) glitch events were simulated No red noise. 周期 “ 跃变 ”

43. Let’s try another one

44. More observing bands

45. The answer Kolmogorov turbulence in the interstellar medium 星际介质的扰动

46. … two to go …

47. The answer … This is the solar wind not being correctly modelled! 太阳风模型

48. … and one more …

49. … and one more … with better sampling

50. A few comments With a single observing band it is “very difficult” to distinguish timing noise, outliers etc. from dispersion measure variations. White noise (radiometer noise) can hide low frequency noise Fitting can hide low frequency noise The noise may “look different” as data spans increase.

51. CSIRO. Gravitational wave detection A few more simulations With one pulsar you cannot (normally) tell what unmodelled physical effect is causing the residuals GW backgroundSpin-down irregularities Clock noise Simulated data

52. CSIRO. Gravitational wave detection Spin-down irregularities No angular signature

53. CSIRO. Gravitational wave detection Terrestrial time standard irregularities Monopolar signature

54. CSIRO. Gravitational wave detection Errors in the planetary ephemerides - e.g. error in the mass of Jupiter Dipolar signature

55. CSIRO. Gravitational wave detection What if gravitational waves exist? Quadrapolar signature

56. Need more than 1 pulsar We can start to understand the noise in our data if we have: - Long time spans - Multiple observing frequencies - Multiple pulsars CSIRO. Gravitational wave detection

57. Examples of pulsar timing research Presentation title | Presenter name | Page 57

58. Millisecond pulsar timing models 1.Reardon et al. (2015) obtained pulsar timing parameters for the PPTA pulsars 2.Most exciting result for PSR J0437-4715: possible sub-light year distance Verbiest et al. (2008) – distance = 512(8) ly Deller et al (2008) – VLBI parallax = 510(5) ly New preliminary result – 509(1) ly 3.One of the most precise stellar distances measured! Presentation title | Presenter name | Page 58

59. Analysis of pulsar-based time scales Hobbs et al. (2013) 1.The pulsar model based on the stable pulsar rotation will not include errors caused by time standards 2.Therefore, any such error will induce timing residuals 3.All pulsar arrival times referred to the same realisation of Terrestrial Time. Therefore error in realisation => all pulsars will exhibit the same residuals 4.Must search for the correlated signal in pulsar timing residuals 5.Any correlated signal is likely to be caused by such an error Presentation title | Presenter name | Page 59 TT(TAI)-TT(BIPM11) versus date Issue: cannot recover linear or quadratic functional form Steering of TAI

60. An example Consider two pulsar data sets.

61. An example Have different data spans

62. An example Fit for the pulse frequency and its derivative

63. An example Add in realistic amounts of noise

64. An example Add in some unexplained timing irregularities

65. Result using Parkes Pulsar Timing Array data Presentation title | Presenter name | Page 65  Can detect deliberate steering of TAI  Pulsar-corrected realisation of terrestrial time agrees with TT(BIPM11)

66. Remember the timing method … Need to know the position of the SSB CSIRO. Gravitational wave detection

67. Measuring planetary masses Use International Pulsar Timing Array data from Parkes, Effelsberg, Nancay and Arecibo. A planetary mass error will lead to incorrect determination of the Solar System barycentre => correlated pulsar timing residuals Can fit to multiple pulsars simultaneously to search for such a signal CSIRO. Gravitational wave detection

68. Measuring planetary mass Champion, Hobbs, Manchester et al. (2010), ApJ, 720, 201 Use data from Parkes, Arecibo, Effelsberg and Nancay M Sun Best Published (Mo)This work (Mo) Mercury1.66013(7)x10 -7 1.660(2)x10 -7 Venus2.4478686(4)x10 -6 2.44782(10)x10 -6 Mars3.227151(9)x10 -7 3.2277(8)x10 -7 Jupiter*9.547919(8)x10 -4 9.547916(4)x10 -4 Saturn2.85885670(8)x10 -4 2.858858(14)x10 -4 9.54791915(11)x10 -4

69. Remember the timing method … Need to know the position of the observatory CSIRO. Gravitational wave detection

70. Using data sets for pulsar navigation: where is Parkes? Unpublished work by G. Hobbs and X. You 1.Assume that we’re on the Earth’s surface 2.Use Parkes timing observations and fit for position of Parkes Presentation title | Presenter name | Page 70

71. Proof of concept – where is Parkes? Unpublished work by G. Hobbs and X. You 1.Assume that we’re on the Earth’s surface 2.Use Parkes timing observations and fit for position of Parkes Presentation title | Presenter name | Page 71

72. Proof of concept – where is Parkes? Unpublished work by G. Hobbs and X. You 1.Assume that we’re on the Earth’s surface 2.Use Parkes timing observations and fit for position of Parkes Presentation title | Presenter name | Page 72 Use millisecond pulsar (PSR J0437-4715) Correct position to within a few kilometers

73. Can navigate spacecraft: Earth to Mars trajectory (Deng et al. 2013) 1.Can we use millisecond pulsar observations to determine the position and velocity of a spacecraft travelling from Earth to Mars? 2.Use STK software to simulate trajectory – accounts for gravitational field, Solar pressure etc. 3.Large ground-based radio telescope to get pulsar timing model before launch (assumed PPTA data) 4.XTE-type X-ray telescope on-board the spacecraft Presentation title | Presenter name | Page 73 http://www.master-flight- training.org/images/AGI_STK.jp g

74. Two algorithms implemented into tempo2 1.Let tempo2 deal with all the pulsar-timing aspects of the problem (e.g., Shapiro delays, ISM delays etc.) 2.Algorithm 1: assumes no prior knowledge of the space-craft trajectory 3.Algorithm 2: makes use of a dynamics model for the space-craft motion 4.Use global fitting routines to fit to the timing residuals of multiple pulsars simultaneously. 5.Result: position estimation better than 10km 6.Result: velocity estimation better than 1m/s Presentation title | Presenter name | Page 74

75. Current research in pulsar timing 1.Bayesian methodology, frequency-dependent fitting, glitch detection, scattering … 2.Robust fitting algorithms 3.Timing and searching with new receivers (ultra-wide-band receivers, phased-array-feeds) 4.What are the fundamental limits to the possible pulsar timing precision (can we go <100ns over many years)? 5.Can we mitigate timing noise (“red noise”) or jitter noise (pulse shape variability)? 6.What timing precision will be reached by SKA, QTT and FAST? 7.What is the optimal observing frequency for pulsar timing? 8.… Presentation title | Presenter name | Page 75

76. Conclusions 1.Pulsar timing is a very exciting technique 2.Can measure pulsar parameters with incredible precision 3.Can use the pulsar timing method to search for errors in terrestrial time standards, navigate spacecraft, study the solar wind, study the interstellar medium and … (for tomorrow) … search for gravitational waves and test theories of gravity! Presentation title | Presenter name | Page 76 脉冲星测时是非常有趣的课题。可以有很多应用。