Progress Report Sabrina Berger

Texas A&M Commerce

REU 2014

Mentor Dr. William Newton

Utilizing glitches in pulsars to attempt to describe the strong force

I. Introduction to Neutron Stars

A neutron star should arise from a massive star between 8 to 30 solar masses if its core collapses in a supernova when unable to continue to sustain nuclear fusion. Once the neutron degeneracy pressure balances with the gravity of matter contained in the star, the collapse will terminate. The neutron star is the remnant of this event with an estimated radius of 10-20 km and mass approximately 1.44 solar masses. A neutron star has a mass above the Chandrasekhar limit,~1.39 solar masses below which a star will become a white dwarf, and below the Tolman-Oppenheimer-Volkoff limit (Black Hole lower limit), less than 2.2 solar masses. There is much ambiguity surrounding the exact upper and lower limit on a neutron stars mass and radius due to the uncertainty in the equation of state.

A neutron star is extraordinarily dense. It is the densest object in the universe second only to a black hole. Neutron stars can themselves become black holes in an accreting binary system if enough mass from the neutron stars binary companion accumulates on the neutron star. The famous analogy is that you could squeeze every human being on planet Earth into the size of a sugar cube at densities found in neutron stars.

Neutron stars are classified into various types including magnetars, which have infrequent eruptions of radiation and unexpectedly strong magnetic fields, and pulsars.

Pulsars are the most significant type of neutron star to our research. Pulsars emit exceptionally precise and predictable pulses in the electromagnetic spectrum, but most often observed in the radio portion. Although there is an extremely gradual increase in the observed periods every year (only of a few milliseconds) due to the magnetic torque on the star, they have often been called the clocks of the universe for their precision in pulses. Most pulsar periods range from .1 to 2 seconds.

History

Walter Baade and Fritz Zwicky first theorized neutron stars in 1934, but PhD candidate Jocelyn Bell and her advisor Antony Hewish at Cambridge did not discover them until 1967. It was an unexpected discovery as they were building a radio telescope to study scintillation in the atmosphere of quasars when they stumbled upon the pulsar.

Structure

We are not able to visit neutron stars to observe their structures directly, but we can hypothesize to our best abilities what their structure may look like. Our best structural model is explained below.

The outer crust of a neutron star is about 1 km thick and made mostly of iron nuclei and other heavier elements up to the element tin that constitute a solid and rigid crystalline solid. The

density here is about 106 gcm3 or 1,000,000 times the density of water. In this region every nucleus lays approximately 1000 nucleus diameters apart. The so called neutron drip, where neutrons begin to actually liberate themselves from the nucleus, occurs right below the

outer crust. The density here is 4.3 x 1011 gcm3 .

The inner crust is still solid but now many neutrons are free. The neutrons are much more concentrated in the nuclei and now form a uniform ladder-like array with free neutrons scattered about in a neutron quasi superfluid. (The explanation for our assumption of this is explained in Glitches below.) A similar phenomenon is seen when Helium-3 is chilled to extremely low temperatures. There exists only 50 nucleus diameters between each nucleus in the inner crust this space decreasing approaching the transition to the core. This is the most important part of a neutron star in the two-component model discussed below. The inner crust

soon dissolves into the core at a density of about 2 x 1014 gcm3 . The transition from inner crust to core in the neutron star is rather complex. There are possible scenarios theorized where the nuclei actually form spaghetti shapes and then slowly transition to more of a swiss cheese form approaching the core as seen below in figure 1. The inner core is the most uncertain part of the neutron star but is probably made up of mostly neutron superfluid. The ratio of neutrons to protons is likely 9 to 1. At the very center

of the core, calculated densities are ~ 1015 gcm3 or ten times that which exists in an atomic nucleus.

Figure 1 courtesy of Dany Pierre Page at the Universidad Nacional Autnoma de Mxico (UNAM)

II. Glitches in Pulsar Rotational Periods

Soon after the discovery of pulsars, abrupt and abnormal changes in a few pulsars rotational periods were discovered. These were called glitchesa sudden increase in rotational velocity and therefore decrease in period from normal. The pulsars sped up and then would slowly recover to approximately their original slowdown rate.

Glitch Two-Component Model

The most widely studied glitches are the Vela and Crab pulsars. The Vela pulsar (see figure 2) has more frequent and large glitches than the Crab. Many postulations were then made for their origin. One model involved a sudden quake in the core of a neutron star, reducing its moment of inertia and therefore increasing its angular velocity. This works for smaller glitches like those in the Crab, but other models were needed to explain the larger ones in the Vela.

The idea of superfluidity of neutrons in the core of a pulsar comes from the observed long recovery times after glitches. If the star werent superfluid at its core, the friction of the fluid with the faster rotating outer core would slow it down much more quickly. The fact that it takes so long to recover to its original period indicates the superfluidity of the neutrons in the core.

Figure 2 courtesy of Andrew G. Lyne and Francis Graham-Smith in Pulsar Astronomy

The concept that larger glitches were the result of the neutron superfluid penetrating the inner crust of the star and of there being two distinct differently spun up portions of the star was proposed by Anderson and Itoh in 1975. This was called the two-component model. In Anderson and Itohs model, the neutron superfluid forms vortices that percolate around and between the inner crustal nuclei. Many of the vortices do get pinned or stuck to the nuclei in the lattice structure of the inner crust. In the equipoise of a pulsars behavior, the superfluidic vortices slowly seep out through the lattice.

There are two components to the pulsar; each is rotating at a different frequency.

The moment of inertia of the first is I which is defined as the neutron superfluidic vortices that transfer their angular momentum to I*, which is the moment of inertia of the charged part of the star where we observe the rotational velocity. I is the faster moving parts moment of inertia while I* is the slower moving components moment of inertia that includes:

o outer crust o inner crust lattice of iron nuclei o core protons o electrons in the star o the fraction of the neutron superfluid that is spun up during a glitch (I coupled during

the glitch)

The Magnus force (the sidewise force a spinning object encounters moving through a fluid), comes into play here on the quantized vortices. The Magnus force is proportional to the difference in frequency as seen below:

f = fI fI* So as this f increases, the Magnus force acting on the pinned vortices increases proportionally. There is a limit when this Magnus force overwhelms the pinning force keeping the vortices in place, and the vortices are then able to migrate away from their pinning site and share their angular momentum with the rest of the neutron star. This has been informally described as a sort of tug of war between the two components. We observe the glitch.

Whats certain?

From the data collected of the Vela pulsar, there is one very important piece that we can use to prove and disprove our ideas about the two-component model for glitches. If the coupling/decoupling of I to I* is correct, then we can say that the ratio between I and I* in the Vela is at least 1.6% (Link et al. 1999).

II * = GGvela 1.6%

III. What is the neutron star equation of state?

An equation of state (EoS) is the relationship between the density of matter, pressure, and temperature (an elementary example is PV=nRT). It defines how much a certain substance can be compressed. If a substance has a very stiff EoS, it will be very difficult to change its density by applying pressure and vice versa for a soft EoS. We wish to be able to understand the properties of dense matter mathematically using neutron stars. This is possible with an equation of state. The temperature in neutron stars is relatively low on the cosmic scale of temperatures so we may exclude it from the EoS for now.

Figure 3 - Various EoSs (named APR, GM1, L, SLY) that generate masses and radii for neutron stars. One of

the Skyrme interaction generated EoSs is listed as APR.

There are many possible theoretical models for how neutrons and protons may interact (strong interaction) from which we derive equations of state. The British physicist Tony Skyrme theorized the Skyrme phenomenological (describing the potential energy between protons and neutrons in medium in nuclear matter) potential in the mid 20th century. This model describes the strong force or the residual force between quarks that holds neutrons and protons together. It is still applied today in research of subatomic structure and will also be employed in our research into the EoS of neutron stars.

From the dissertation, The Phase Transition to Uniform Nuclear Matter in Supernovae and Neutron Stars by William Newton

The equation above however is full of unknown constants, e.g. t0, t1, t3. To determine these constants without estimating, we need to know the nuclear symmetry energy at nuclear

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saturation density (0 = 2.5 1014gcm3 or n0 0.16/fm

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function of density and proton fraction. S could theoretically be calculated by determining the amount of energy input needed to produce nuclear matter with 0% protons from 50% protons. We are more specifically looking at the slope of symmetry energy at 0 , which we call L. Laboratory experiments have been conducted to measure L in various other research projects in particle physics, including those attempting to measure the neutron skin. In the most recent experiments, the best measurements have fallen in the range 25 L(MeV ) 115 .

Figure 4 Varying L changes the overall energy calculated in the Skyrme model and therefore the EoSs

generate different properties for the neutron stars.

From the Skyrme interaction, we may derive the equation of state for neutron stars. We know that the Skyrme model describes E as a function of density and proton fraction.

We may use the first thermodynamic identity and disregard temperature.

(where S = entropy, and E is the internal energy)

We can then arrive at an EoS which describes pressure as function of density and proton fraction-x.

Equation of state will be P(, x). See figure 5 for various EoSs and their outputs of pressure and energy density.

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Then we may use the Tolman-Oppenheimer-Volkoff (TOV) equation (see figure 3 and 4) to determine properties of neutron stars from possible EoSs. The TOV equation sets contraints on various aspects of the structure of a spherical object in hydrostatic equilibrium. This is the entire TOV equation with relativistic considerations:

Figure 5 Pressure vs. energy density for four possible equations of state.

IV. Research Procedure

This research will be focused on constraining the equation of state in neutron stars. From this equation of state, it will be possible to understand the strong forcethe residual or color force from the interaction of quarks. The most compelling argument we can make is that Gvela 1.6% . For the first few weeks of this research, I have been using this to look at an EoS from the Skyrme model to generate moment of inertia ratios greater than this by varying certain parameter

GNSprop

For this project, I will be using the program of Dr. William Newton used to generate properties of neutron stars from various equations of states, called GNSprop. I have been using different values of L between 25 and 115 to see how it affects the properties of neutron stars that it generates, especially G. We will be varying L accordingly to fit both observations

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of neutron stars and laboratory experiments. We are trying to pinpoint the equation of state of dense matter through this.

For the second half of the research, GNSprop was modified to account for a more constrained version of Yg using mutual friction.

Other variables of the two-component model

There are also other modifiable aspects to the two-component model in pulsars relating to the more accurate definition of I.

Strong crustal vs. crustal neutrons

In the outputs of GNSprop, all the neutron superfluid in the inner crust may be considered in the pinning and unpinning of the vortices or just the strong crustal neutrons as seen in shaded grey in the figure below. The strong crustal neutrons are those completely embedded in the inner crust in the strong pinning regions. I have been examining G when considering the strong crustal neutrons and crustal neutrons to hopefully eventually be able to further analyze and constrain the data. See figure 6 and 7 to see the differences in the data generated for differing L.

Figure 6 shows the strong pinning region shaded as grey (Figure from Hooker, Newton, Li 2014)

Figure 7 and 8 Gcn takes all the crustal neutrons into account while Gscn just takes those in the strong pinning region as explained above.

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Entrainment

Recently it was found that much of the neutron superfluid could actually be pulled onto the inner crust lattice and not be able to contribute to the transfer of angular momentum during the glitch. The inner crust could entrain the neutron superfluid. Some articles even claimed that this disproved the two-component model completely [Andersson et. al 2012]. However the calculations for the effects of entrainment were recently completed (Chamel 2013) and were found to occur in only 80% of the nuclear superfluid, leaving the rest to contribute to the rise of a glitch. Changing the percentage of neutrons entrained affects the outputs for G and can be used to set further limits on the EoS.

Yg

We also have to consider the quantity of neutron superfluid in the core that is actually spun up at the time of the glitch. Yg is used to indicate this and is the ratio between the total amount of neutron superfluid spun up in the core during a glitch (the coupled portion) and the total star. It is a parameter alterable to fit Gvela.

Yg = IcoupledItotal

Mutual Friction and Yg In this model before the glitch, the neutron superfluid in the strong pinning region is uncoupled and moving faster that the rest of the star and the portion we observe. Then during the glitch, when the faster moving neutron superfluid releases its angular momentum to the outer crust, not all of the inner core neutron superfluid is spun up simultaneously. Depending on the intensity of mutual friction on a certain part of the neutron superfluid in the core, the neutron superfluid in the core will take an uneven amount of time to recouple to the star. Mutual friction (B) is the force that causes the change of speed of the rest of the neutron star (other than the outer crust). When the crust changes its spin, the electrons and protons in the core spin up to match its speed very quickly; however, the neutron superfluid having practically no viscosity takes much longer to spin up. The electrons hit the neutron and proton superfluidic vortices and cause them to increase their speed (or decrease). Mutual friction is an observationally quantifiable amount thats intensity during a glitch can be calculated from the estimated time a glitch takes to lose 50% of the increase in rotational velocity it obtained during the glitch (- and is angular frequency):

B = 12

(Haskell 2014) From the 2000 and 2004 glitches in the Vela pulsar, this value is about one minute, leading to a mutual friction range:

3105 < B

From the mutual friction, we can find the cylindrical radius over which we can integrate to find the portion of neutron superfluid in the core that ever becomes coupled to the star (top of Yg) for the calculated values of mutual friction.

Figure 9-10 Amount of mutual friction at a certain distance from the inner core for different masses for L=50 and L=110. The radial coordinate starts at the innermost portion of the star and moves outwards towards the

crust.

The parts of the star above the lower dashed line could be a part of the coupled portion of the star during a glitch. A portion of the part below the top dashed line is the uncoupled portion during a glitch. The dashed lines just set the maximum values for this.

Figures 11-12 G or the ratio of the coupled portions of the star before and after the glitch (I strong pinning region -coupled before glitch/I coupled after glitch) calculated for L=50 and L=110 at varying masses. The

dotted line at the top of the graphs represents the observational activity parameter of the Vela pulsar G = .016. Here we are only considering the strong pinning region for the neutron superfluid.

As seen above in the graphs the Gvela value is never met, but perhaps by increasing the neutron superfluid that can contribute to the glitch from just the strong pinning region to the entire inner crust or going slightly into the core, we could reach that value. We also have many other parameters to modify that could help us reach that value.

Figures 13-14 Yg calculated (with GNSprop) for L=50 and L=110 and at varying masses.

V. Conclusion

The extraordinarily dense matter found in neutron stars cannot be fabricated in a laboratory. The core (~1015 g/cm3) of neutron stars may in fact be denser than an atomic nucleus (2.31014 g/cm3). Although the closest observed possible neutron star is still 250-1000 light years away, we can employ empirical data in comparison with our theoretical conjectures to make reasonable deductions about the equation of state of all nuclear matter at this density. The most worthwhile data from neutron stars comes in the form of observation of a glitcha sudden increase in rotational velocity of the observed component of the neutron star. Utilizing the two-component model for glitches, most notably in the Vela pulsar, presents a way for us to understand the inner workings of neutron stars and further restrain the strong force model that dictates the potential energies between nucleons of all matter of arbitrary density. We want to quantify the strong force between protons and neutrons in an atomic nucleus. The neutron star is a supreme laboratory for astrophysicists and particle physicists alike, to discover the properties of matter at such high densities.

It may be possible to constrain the model and the EoS of dense matter even further by employing mutual friction in an attempt to obtain a more accurate range for Yg. We can use an observed and approximated timescale from the Vela pulsar glitches to hopefully pinpoint mutual friction.

As we observe more Vela glitches in the future with more sophisticated and accurate instruments, we will be able to reduce the error margin for this timescale, mutual friction and Yg.

Before the research this summer, Yg has never been able to be approximated from observation but only just changed arbitrarily. Yg may prove to be an extremely valuable piece of evidence supporting our model for glitches.

VI. References

Andersson, N., K. Glampedakis, W. C. G. Ho, and C. M. Espinoza. "Pulsar Glitches: The Crust Is Not Enough." Physical Review Letters 109.24 (2012): n. pag. Web.

"Bell and Hewish Discover Pulsars." PBS. PBS, n.d. Web. 27 June 2014.

Chamel, N. "Crustal Entrainment and Pulsar Glitches." Physical Review Letters 110.1 (2013): n. pag. Web.

Dodson, R. G., P. M. Mcculloch, and D. R. Lewis. "High Time Resolution Observations of the January 2000 Glitch in the Vela Pulsar." The Astrophysical Journal 564.2 (2002): L85-88. Web. Espinoza, Antonopoulou, Stappers, and Lyne. "Neutron Star Glitches Have a Substantial Minimum Size." Royal Astronomical Society (2014): n. pag. Web. Harding, Alice K. "The Neutron Star Zoo." Frontiers of Physics 8.6 (2013): 679-92. Web.

Haskell. Private conversation. 2014.

Hirasawa, Masaki, and Noriaki Shibazaki. "Vortex Configurations, Oscillations, and Pinning in Neutron Star Crusts." The Astrophysical Journal 563.1 (2001): 267-75. Web. Hooker, Josh, William G. Newton, and Bao-An Li. "Efficacy of Crustal Superfluid Neutrons in Pulsar Glitch Models." Royal Astronomical Society (2014): n. pag. Web. Kalogera, Vassiliki, and Gordon Baym. "The Maximum Mass of a Neutron Star." The Astrophysical Journal 470.1 (1996): L61-64. Web. Kaspi, V. M. "Chandra's First Decade of Discovery Special Feature: Grand Unification of Neutron Stars." Proceedings of the National Academy of Sciences 107.16 (2010): 7147-152. Web.

Lattimer, J.M. Prakash, M. The Physics of Neutron Stars.

Link, Bennett, Richard Epstein, and James Lattimer. "Pulsar Constraints on Neutron Star Structure and Equation of State." Physical Review Letters83.17 (1999): 3362-365. Web. Lyne, Andrew G., and Francis Graham-Smith. Pulsar Astronomy. Cambridge: Cambridge UP, 1990. Print.

Newton, William. The Phase Transition to Uniform Nuclear Matter in Supernovae and Neutron Stars. Dissertation at Oxford University. (2007).