Constraining the magnetic field geometry of millisecond pulsar PSR J0030+0451 using NICER and Fermi data
Anu Kundu*
Post-Doc @ CSR, NWU
Collaborators :
Constantinos Kalapotharakos (NASA GSFC/UMCP/CRESST II) Alice K. Harding (LANL)
Zorawar Wadiasingh (NASA GSFC/USRA) Demosthenes Kazanas (NASA GSFC) Christo Venter (CSR, NWU)
SAIP 2021 : Virtual Conference
29 July 2021, Thursday
*anukundu02@yahoo.com
Constraining the magnetic field geometry of millisecond pulsar PSR J0030+0451 using NICER and Fermi data
● Pulsars : Remnant of a star after supernova
● Millisecond pulsars (MSP) : Period < 10 ms
● Older (age > 100 million years)
● Origin theory : spun-up from accretion*
● Minority are isolated MSP : Formation not fully understood, could be ablation of companion
PSR J0030+0451
● an isolated MSP
● spin period 4.865 ms / 205.53Hz (Lommen et al. 2000)
● distance of 325 ± 9 pc (Arzoumanian et al. 2018)
● Multiwavelength emission :
○ discovered as a radio pulsar using Arecibo telescope (Lommen et al. 2000)
○ identified as an (11th) X-ray pulsar with ROSAT (Becker et al. 2000)
○ first gamma-ray MSP announced by Fermi (Abdo et al. 2009)
Image credit : NASA
(https://asd.gsfc.nasa.gov/blueshift/index.php/2017/08/03/rxt es-greatest-pulsar-hits/)
* Bisnovatyi-Kogan & Komberg 1974; Alpar et al. 1982;
Bhattacharya & Van den Heuvel 1991; Urpin et al. 1998
Constraining the magnetic field geometry of millisecond pulsar PSR J0030+0451 using NICER and Fermi data
Cases for multipolar field in neutron stars
● High braking index observations (Archibald et al. 2016)
● Impact of an off-centred dipole on neutron star binaries (Pétri 2019)
● Joint radio and X-ray modelling of PSR J1136+1551 (Pétri & Mitra 2019)
● NICER multi-wavelength light curve (LC) fitting of PSR J0030+0451 (Chen et al. 2020)
● Magnetic strength of non-dipolar components influencing trigger rate in magnetars (Dehman et al. 2020)
Cases for offset field
● Offsets increase pair production in pulsars (Harding and Muslimov 2011)
● Offset dipole invoked for modelling observed Fermi LCs (Barnard et al. 2016)
● Consequences of an offset dipole magnetic field on broadband pulsar emission (Kundu & Pétri 2017)
● Offset dipole in neutron stars to explain the polarization features (Pétri 2017)
Constraining the magnetic field geometry of millisecond pulsar PSR J0030+0451 using NICER and Fermi data
● NICER soft X-ray pulse waveform of J0030 - two peaks
● Possible origin : hotspots on surface
● Modelling of hotspots via LC fitting (Bogdanov et al. 2019)
Bogdanov et al. 2019
Constraining the magnetic field geometry of millisecond pulsar PSR J0030+0451 using NICER and Fermi data
Miller et al. 2019 Riley et al. 2019
The consistency of the two independent results - most favoured result being two same-temperature hotspots in same hemisphere - adds confidence about their accuracy
Suggested strong evidence for an offset and multipolar magnetic field of J0030 - inspiration for follow-up study
Constraining the magnetic field geometry of millisecond pulsar PSR J0030+0451 using NICER and Fermi data
Kalapotharakos et al. 2021
● Started with static vacuum field - analytic expressions exist, calculations faster
● Used M and R from Riley et al. 2019 and Miller et al. 2019
● Fitted X-ray LCs
● Central dipole : Two hotspots/polar caps, almost circular, antipodal (different hemispheres)
● Offset dipole + offset quadrupole : similar shape and pattern as Riley et al. 2019 and Miller et al. 2019
● 11 parameters (position, direction, relative strength of components)
● Force-free (FF) configurations (Kalapotharakos et al. 2012, 2014)
Constraining the magnetic field geometry of millisecond pulsar PSR J0030+0451 using NICER and Fermi data
Kalapotharakos et al. 2021
● Started with static vacuum field - analytic expressions exist, calculations faster
● Used M and R from Riley et al. 2019 and Miller et al. 2019
● Fitted X-ray LCs
● Central dipole : Two hotspots/polar caps, almost circular, antipodal (different hemispheres)
● Offset dipole + offset quadrupole : similar shape and pattern as Riley et al. 2019 and Miller et al. 2019
● 11 parameters (position, direction, relative strength of components)
● Force-free (FF) configurations (Kalapotharakos et al. 2012, 2014)
● Explored same configuration - fitted Fermi gamma-ray LCs (Kalapotharakos et al. 2014)
● Degeneracies… or not!
Using analytical multipolar magnetic field
Kalapotharakos et al. 2021
Pétri 2015
Extending work done by Kalapotharakos et al. 2021 Static field (simple)
Vacuum Offset dipole Quadrupole
Retarded field (realistic, closer to FF) Vacuum
No offset
Multipolar expansion (complete basis, can describe any configuration)
Magnetic field outside
neutron star
l = 0 : Dipole l = 1 : Quadrupole l = 2 : Hexapole l = 3 : Octopole ...
m = - l to l : different orientations of corresponding multipolar component
Spherical Hankel Functions Functions in terms of r eg. l = 1
Vector Spherical Harmonics Functions in terms of 𝜗 and 𝜑 eg. l = 1, m = 1
Constants
e.g. l=1; m = 0 and 1
Using analytical multipolar magnetic field
Methods
Step 0
● Replace magnetic field equations
● Spherical functions, higher field components
● Computationally efficient way
Step 1
● Find hotspots/polar caps that accurately reproduce the NICER X-ray LCs
● Explore the corresponding parameter space of field configurations
● Using Markov Chain Monte Carlo (MCMC) parallel code (Kalapotharakos et al. 2021)
● Compare modelled X-ray LC observed profile from NICER, find maximum likelihood
● Determine how many terms are needed in the expansion
Methods
Step 2
● Turn the field configuration from vacuum to FF
● Implement the MCMC code to modify the field parameters
● X-ray LCs for the FF configurations should remain consistent with observation
Step 3● For each such choice compute the corresponding gamma-ray LCs (Kalapotharakos et al. 2014)
● Compare with Fermi gamma-ray LCs
Optimum set of magnetic field parameters consistent
with both the X-ray and the gamma-ray LCs
R - Riley et al. 2019 M - Miller et al. 2019 V : Vacuum Red : high offset
Green : quadrupole strength (always higher, except one case - blue)
Result highlights from Kalapotharakos et al. 2021
Result highlights from Kalapotharakos et al. 2021
R - Riley et al. 2019 F: Force-free Red : high offset
Green : quadrupole strength (always higher)
Yellow : hotspots LCs :
Black : NICER/Fermi data Red : Model
X-ray LCs : field degeneracies
gamma-ray LCs : lifts the degeneracies
Result highlights from Kalapotharakos et al. 2021
Questions to be answered presently :
● How many components provide an adequate fitting? Does it improve the LC fit?
● How would the strength of higher components vary from previous results, and compared to each other?
● Will there be degeneracies, and if so, how do they compare with previous results?
● How would the parameter constraints change with multiwavelength fitting?
● In broad terms, how do we explain the configuration of the magnetic field physically?
Prospects
Past
● The NICER thermal X-ray LC led to attempt the exploration of multipolar field structures in MSPs, which led to the discovery of degenerate solutions
● Comparing the model gamma-ray LCs to the Fermi one lifts the degeneracies
Summary
Present
● Static field -> retarded field
● Dipolar + Quadrupolar -> Higher multipolar
● Using analytical solutions
● Degeneracies?
● Strength of higher components, how much contribution?
Future
● General relativistic field expressions
● Offset multipolar field expansion
● Self-consistent M and R determination
● Apply to other NICER MSPs
