# Logits are predictive of network type

Ali Borji
Quintic AI, San Francisco, CA
<EMAIL_ADDRESS>
###### Abstract

We show that it is possible to predict which deep network has generated a
given logit vector with accuracy well above chance. We utilize a number of
networks on a dataset, initialized with random weights or pretrained weights,
as well as fine-tuned networks. A classifier is then trained on the logit
vectors of the trained set of this dataset to map them to their corresponding
network index. The classifier is then evaluated on the test set of the same
dataset. Results are better with randomly initialized networks, but also
generalize to pretrained networks as well as fine-tuned ones. Classification
accuracy is higher using unnormalized logits than normalized ones. We find
that there is little transfer when applying a classifier to the same networks
but with different sets of weights. In addition to help better understand deep
networks and the way they encode uncertainty, we anticipate our finding to be
useful in some applications (_e.g._ tailoring an adversarial attack to a
certain type of network). Code is available at link.

##  1 Motivation

Deep neural networks are prevalent in many domains and achieve astonishing
performance. Our understanding of their behavior, however, is still limited.
Here, we present a new surprising phenomenon which is a bit hard to explain.
We show that it is possible to predict the type of deep network from the
logits that it generates. Exploring the reasons underlying this phenomenon
helps better understand high dimensional representations that are learned by
deep networks. Our finding suggests that perhaps deep networks have different
signatures of uncertainty encoding. For some applications, it may be useful to
know which network has generated the output, for example when a customized
treatment is necessary for a task (_e.g._ knowing which network has generated
an adversarial attack for efficient defense against it).

##  2 Method

###  2.1 Experimental setup

We choose a dataset and a number of deep networks. We first feed the training
set of this dataset to the networks and collect the logits (real-valued vector
$s$). Next, we form the following input-output pairs $(s_{ij},i)$ where
indices $i$ and $j$ represent network $i$ and input image $j$, respectively.
We then train a classifier to map the logits to the network index. The goal is
to predict which network has generated a given logit vector $s$. The
classifier trained on the training set of the dataset (here a SVM with linear
or RBF kernels) is evaluated on the test data of the same dataset.

Since the logits produced by different networks might have different ranges of
values, due to network hyperparameters and internal dynamics, we experiment
with both unnormalized and normalized (logits divided by their sum) logits.
The PyTorch code to extract the logits is as follows:

        Unnormalized: outputs = model(inputs).cpu()
        Normalized: outputs = outputs/outputs.sum(axis=1)[:,None]

Three scenarios are considered: 1) networks initialized with random weights,
2) networks initialized with pretrained ImageNet weights, and 3) fine-tuned
networks. On some dataset, we consider training the networks from scratch. We
also study the effect of the number of training data on network prediction
accuracy. Additionally, we explore whether a classifier trained on logits of
some networks can predict the network type from logits of the same networks
but when they are initialized or trained differently (_i.e._ generalization or
transfer).

###  2.2 Deep networks

We choose 9 state-of-the-art object recognition networks111Networks are
available in PyTorch hub: https://pytorch.org/hub/. We used a 12 GB NVIDIA
Tesla K80 GPU to conduct the experiments.. These networks have been published
over the past several years and have been immensely successful over the
ImageNet benchmarks [1]. They include ResNet101 and ResNet152 [2], AlexNet
[4], DenseNet [3], GoogleNet [7], MobileNetV2 [6], ResNext [9], ResNext_WSL
[5], and Deit [8]. The code to download the weights is as follows:

    torch.hub.load(‘pytorch/vision:v0.10.0’, ‘resnet101’,
                    pretrained=trained, force_reload=True, skip_validation=True)

The variable trained can be False or True, corresponding to Random or
Pretrained weights.

Figure 1: Classification accuracy of deep networks. Left) Binary
classification over BeesVsAnts dataset (trained for 10 epochs). Right) 10-way
classification over CIFAR-10 dataset (trained for 6 epochs; only the last
layer is fine-tuned). Networks are initialized with ImageNet weights. Results
are averaged over 3 runs.

##  3 Experiments and Results

We conduct four experiments over three datasets including BeesVsAnts,
CIFAR-10, and MNIST to validate our hypothesis. The networks over the former
two datasets are the ones mentioned above. Over the MNIST dataset, we train
simple CNNs with few layers of convolution and pooling.

We fine-tune (or train) networks using Cross Entropy loss, SGD optimizer, and
learning rate (LR) of 0.001, and momentum of 0.9. LR is decayed by a factor of
0.1 every 7 epochs. Networks are fine-tuned (or trained) for 10 epochs on
BeesVsAnts and MNIST, and 6 epochs on CIFAR-10 dataset. The datasets to train
and test the network prediction classifier are balanced, _i.e._ the same
number of logits per network type.

###  3.1 Experiment I: Bees vs. Ants classification

We trained the deep networks for binary classification of bees vs. ants using
the BeesVsAnts dataset from
Kaggle222https://www.kaggle.com/datasets/gauravduttakiit/ants-bees. This
dataset has 244 training images and 153 testing images. Performance of deep
networks over the test images are shown in the left panel of Fig. 2. Deep nets
are performing very well (above 80%) and close to each other. The SVM
classifiers, for network type prediction, are trained on 2196 training logits
(244 per deep net; _i.e._ 153 $\times$ 9), and are evaluated on 1377 testing
logits (153 per deep net; _i.e._ 153 $\times$ 9). Notice that here the
classification layer of deep nets consists of 2 neurons, and thus the logits
are 2D vectors.

Results using random and pretrained weights. We fed the images to the deep
nets with random weights or networks pretrained on ImageNet, and collected the
logits. We then trained the SVM classifier on logits.

Results are shown in Fig. 2. With random weights, both SVM classifiers (linear
and RBF) perform well above chance (1/9) using both normalized and
unnormalized logits. With pretrained weights, both SVMs perform well in the
unnormalized case. The RBF SVM performs consistently above chance in all cases
on this dataset. Increasing the number of training data improves network
prediction accuracy (except linear SVM in the unnormalized case over
pretrained weights).

Figure 2: Results over the BeesVsAnts dataset. Left (right) column shows
network prediction accuracy using logits from randomly initialized (pretrained
on ImageNet) networks. The logit vector is 2D. Second and third rows show
accuracy as a function of the number of training data, using both normalized
and unnormalized logits. The last row shows transfer results where a
classifier trained on some networks is applied to the same networks but with
different set of weights. Results are averaged over 5 runs.

Results using trained weights: Here we consider two cases: 1) logits from
networks with their last layer fine-tuned, and 2) logits from networks with
all weights fine-tuned. The networks are initialized with ImageNet weights.
Results are shown in Fig. 3. Here, both SVMs perform very well using
unnormalized logits. Using normalized logits, however, only RBF SVM does
better than chance. Increasing the number of training data improves network
prediction accuracy (except linear SVM in the normalized case). Results are
similar and consistent when training the last layer or all layers.

Figure 3: Results over the BeesVsAnts dataset (cont’d). Left (right) column
shows network prediction accuracy using logits from networks with their last
layer (all layers) fine tuned (using networks pretrained on ImageNet). The
logit vector is 2D. Second and third rows show accuracy as a function of the
number of training data, using both normalized and unnormalized logits. The
last row shows transfer results where a classifier trained on some networks is
applied to the same networks but with different set of weights. Results are
averaged over 5 runs.

Transfer Results. According to the bottom panels in Figs. 2 and 3, a
classifier trained on normalized logits does not transfer much, whereas in the
unnormalized case RBF SVM shows a noticeable degree of generalization.

###  3.2 Experiment II: CIFAR-10 classification

Performance of the deep networks over the test set of CIFAR-10 dataset is
shown in the right panel of Fig. 2. Most of the networks perform above 60%
with Deit achieving around 80%. SVM classifiers are trained over 8991 logits
(999 per network; _i.e._ 9 $\times$ 999) and tested over 8991 testing logits.
Notice that here the classification layer of deep nets consists of 10 neurons,
and thus the logits are 10D vectors.

Results using random and pretrained weights: Consistent with results over the
BeesVsAnts dataset, network prediction accuracy is very high here using logits
derived from networks with random weights (Fig. 4). RBF SVM performs well
using unnormalized logits from the pretrained networks.

Figure 4: Results over the CIFAR-10 dataset. Left (right) column shows network
prediction accuracy using logits from randomly initialized (pretrained on
ImageNet) networks. The logit vector is 10D. Second and third rows show
accuracy as a function of the number of training data, using both normalized
and unnormalized logits. The last row shows transfer results where a
classifier trained on some networks is applied to the same networks but with
different set of weights. Results are averaged over 5 runs.

Results using trained weights: Due to limited computational resources, here we
only fine-tune the weights of the last layer of deep networks (networks are
initialized with ImageNet weights). Results are shown in Fig. 5. Consistent
with corresponding results over the BeesVsAnts dataset, both SVMs perform
significantly better than chance using unnormalized logits. RBF SVM performs
significantly above chance using normalized logits over BeesVsAnt and CIFAR-10
datasets.

Figure 5: Results over the CIFAR-10 dataset (cont’d). Network prediction
accuracy using logits from networks with their last layer fine tuned (using
networks pretrained on ImageNet). The logit vector is 10D. Second and third
rows show accuracy as a function of the number of training data, using both
normalized and unnormalized logits. The last row shows transfer results where
a classifier trained on some networks is applied to the same networks but with
different set of weights. Results are averaged over 5 runs.

Transfer Results. Please see the last rows of Figs. 4 and 5. Again, consistent
with results over BeesVsAnts dataset, we do not observe a significant
transfer.

Fig. 6 shows sample normalized and unnormalized logit vectors, as well as
confusion matrices of network classifiers.

Figure 6: Visualization of logits and confusion matrices. Twenty sample logit
vectors (left column) along with sample confusion matrices of network
classifiers over BeesVsAnts (top) and CIFAR-10 dataset using both normalized
and unnormalized logit vectors. The row and column indices in the confusion
matrices correspond to deep networks. Logits come from different deep
networks. The logits on BeesVsAnts dataset are coming from a pretrained (on
ImageNet) network, whereas the logits over CIFAR-10 are from trained networks
(last layer).

###  3.3 Experiment III: Classification using networks built for ImageNet

In experiments above, we adjusted the classification layer of deep networks to
be compatible with the dataset they were applied to. Here, we use the original
deep nets constructed for ImageNet image classification. We feed images to
these networks and record the 1000D logit vectors, and then train the SVM
classifiers on these logits.

Results using random and pretrained weights. Results over BeesVsAnts and
CIFAR-10 datasets are shown in Fig. 7 and Fig.8, respectively. Over both
datasets, the linear SVM does pretty well using normalized and unnormalzied
logits from random and pretrained networks. The RBF SVM does above chance, but
lower than the linear SVM. This result suggests that network prediction might
be possible even when classifiers are trained from logits in one domain and
are tested on logits in another domain. This, however, needs further
verification. One way to verify this, is to train the classifier with logits
coming from images in domain A and to test it on logits coming from domain B
(_e.g._ training on BeesVsAnts and testing on CIFAR-10 using 1000D logit
vectors).

Figure 7: Network prediction using ImageNet networks and feeding BeesVsAnts
images. Left (right) column shows network prediction accuracy using logits
from randomly initialized (pretrained on ImageNet) networks. The logit vector
is 1000D. Second and third rows show accuracy as a function of the number of
training data, using both normalized and unnormalized logits. Results are
averaged over 5 runs.

Figure 8: Network prediction using ImageNet models and feeding CIFAR10 images.
Left (right) column shows network prediction accuracy using logits from
randomly initialized (pretrained on ImageNet) networks. The logit vector is
1000D. Second and third rows show accuracy as a function of the number of
training data, using both normalized and unnormalized logits. Results are
averaged over 5 runs.

###  3.4 Experiment IV: MNIST classification

So far we have shown that it is possible to predict the network type over
heterogeneous networks, although some were very similar (_e.g._ ResNet101 and
ResNet152). Here, we consider prediction of homogeneous networks. We ask
whether it is possible to predict the network type from modifications of the
same network. The SVM classifiers here are trained over 8997 training logits
and tested on 8997 logits. The logit vector is 10D. We perform the following
two experiments:

MNIST I. Three CNNs with different numbers of layers and kernel sizes are
constructed (left panel in Fig. 9). They have the same activation functions.
The average performance of these CNNs for MNIST digit classification in order
are 97.1%, 86.5%, and 96.7%.

We train 3-way SVMs to predict the network type from normalized or
unnormalized logits. Results are shown in the left panel of Fig. 10. Although
the network prediction accuracy now is lower than the above two datasets, it
is still much better than the chance level (1/3).

MNIST II. Three CNNs with the same number of layers (four layers) and same
kernel sizes are constructed (right panel in Fig. 9). They have different
activation functions (ReLU, linear, and Tanh). The average performance of
these CNNs in order are 97.1%, 97.1%, and 95.4%. Network prediction accuracy
is shown in the right panel of Fig. 10.

Results from both MNIST experiments show that it is still possible to separate
the networks even when they are very similar to each other. This means that
internal dynamics and small modifications inside a network cause a shift in
distribution of logits.

Figure 9: Left) CNN architectures used in the first MNIST experiment, Right)
CNN architectures used in the second MNIST experiment.

Figure 10: Network prediction accuracy over MNIST dataset. Left) network
prediction accuracy over three CNNs with different number of layers and kernel
sizes but with the same activation functions. Right) network prediction
accuracy over three CNNs with the same number of layers and same kernel sizes
but with different activation functions (ReLU, linear, and Tanh). See Fig. 9
for definition of these CNNs. The logit vector is 10D. Second and third rows
show accuracy as a function of the number of training data, using both
normalized and unnormalized logits. Results are averaged over 5 runs.

##  4 Discussion and Conclusion

We showed that logits can be regarded as signatures of deep networks. In other
words, the way deep neural networks encode uncertainty in their prediction is
different and can be used to predict their type.

Our main findings are summarized below:

  * •

It is possible to predict the network type from logits, even when networks are
very similar. Results are consistent across datasets and experiments.

  * •

Unnormalized logits are better suited for network prediction, mainly because
ranges of values are different across the networks,

  * •

On average, the SVM with RBF kernel works better than the SVM with linear
kernel in predicting the network type,

  * •

Increasing the number of training logits improves the accuracy.

  * •

Network prediction accuracy is higher using logits produced by networks with
random weights than logits from pretrained or fine-tuned networks. This means
that training the networks on the same dataset makes the logits more similar
to each other, and hence less separable,

  * •

We did not find a significant transfer of the learned classifier for network
prediction when the same networks are initialized or trained differently.
This, however, is not a big constraint for practical purposes,

  * •

Training only the last layer of the networks or all layers does not make a big
difference in network prediction accuracy,

We foresee the following directions for future research in this area:

  * •

We encourage researchers to explore the extent and scope of this phenomenon.
Understanding the underlying reason why this phenomenon happens, in particular
results using normalized logits, can provide new insights pertaining to how
deep networks learn high dimensional representations.

  * •

It might be possible to obtain better network prediction accuracy using
additional tricks. Including more features from networks such as weights from
different layers may increase the network prediction accuracy. However, it
does limit the applicability of the approach since such weights are not
usually available at the inference time.

  * •

We suggest applying the proposed method to a larger number of network types,
and to inspect for which images the network prediction is not possible and
why.

## References

  * [1] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei.  Imagenet: A large-scale hierarchical image database.  In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009.
  * [2] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 770–778, 2016.
  * [3] Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q Weinberger.  Densely connected convolutional networks.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4700–4708, 2017.
  * [4] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton.  Imagenet classification with deep convolutional neural networks.  In Advances in neural information processing systems, pages 1097–1105, 2012.
  * [5] Dhruv Mahajan, Ross Girshick, Vignesh Ramanathan, Kaiming He, Manohar Paluri, Yixuan Li, Ashwin Bharambe, and Laurens Van Der Maaten.  Exploring the limits of weakly supervised pretraining.  In Proceedings of the European conference on computer vision (ECCV), pages 181–196, 2018.
  * [6] Mark Sandler, Andrew Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh Chen.  Mobilenetv2: Inverted residuals and linear bottlenecks.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4510–4520, 2018.
  * [7] Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich.  Going deeper with convolutions.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1–9, 2015.
  * [8] Hugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre Sablayrolles, and Hervé Jégou.  Training data-efficient image transformers & distillation through attention.  In International Conference on Machine Learning, pages 10347–10357. PMLR, 2021.
  * [9] Saining Xie, Ross Girshick, Piotr Dollár, Zhuowen Tu, and Kaiming He.  Aggregated residual transformations for deep neural networks.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1492–1500, 2017.


# Systematic study for the surface properties of neutron star

Ankit Kumar1,2<EMAIL_ADDRESS>H. C. Das1,2<EMAIL_ADDRESS>Jeet
Amrit Pattnaik3<EMAIL_ADDRESS>S. K. Patra1,2<EMAIL_ADDRESS>1Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India 2Homi
Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai
400094, India 3Department of Physics, Siksha ’O’ Anusandhan, Deemed to be
University, Bhubaneswar-751030, India

###### Abstract

Background

In our earlier work [Phys. Rev. C 104, 055804 (2021)], we studied the surface
properties of a neutron star, assuming it as a huge finite nucleus containing
protons, neutrons, electrons, and muons. For the first time, we reported these
results of a neutron star for a few representative masses. In the present
paper, we give a detailed study of these quantities to draw definite
conclusions.

Method

To carry forward our earlier idea, the energy density functional of the
momentum space of neutron star matter is converted to the coordinate space in
a local density approximation. This functional is again used to derive the
neutron star surface properties within the coherent density fluctuation model
using the weight function obtained from the density profile of the neutron
star using the recently developed G3 and widely used NL3 and IU-FSU parameter
sets in the context of relativistic mean-field formalism.

Results

The systematic surface properties of the neutron star, such as
incompressibility, symmetry energy, slope parameter, and curvature
coefficient, is calculated. The volume and surface components of the total
symmetry energy are decomposed with the help of the $\kappa$ factor obtained
from the volume to surface ratio of the symmetry energies in the liquid drop
limit of Danielewicz. The magnitude of the computed surface quantities
increases with the neutron star’s mass.

Conclusion

The incompressibility $K^{\rm star}$, symmetry energy $S^{\rm star}$, slope
parameter $L_{\rm sym}^{\rm star}$ and curvature coefficient $K_{\rm sym}^{\rm
star}$ of the neutron stars with different mass are analyzed and found to be
model dependent. NL3 is the stiffest equation of state endue us with the
higher magnitude of surface quantities as compared to the G3 and IU-FSU
forces.

##  I Introduction

Among all the known objects in the universe, the neutron star (NS) is
considered one of the densest. It is known that the cores are $10^{3}$ times
or more denser than the density at the “neutron drip” line Engvik  _et al._
(1994). To understand the properties of NS, a thorough knowledge of both
nuclear physics and astrophysics is demanded. In a broad concept, the NS can
be treated as a giant asymmetric nucleus comprised mostly degenerated neutrons
gas with a small fraction of protons and electrons to maintain a charge-
neutral body along with some exotic particles, such as hyperons Sharma  _et
al._ (2007). The NS is bounded by the attractive gravitational force balanced
through the short-range strong nuclear interaction generated by the baryons
and mesons. In addition to these two opposite forces, the electromagnetic
interactions are also vital for NS’s stability. In contrast, the normal
nucleus is governed by strong nuclear and relatively weaker electromagnetic
interactions. As a consequence, inside a nucleus, the density is flat, and
inside the NS, the density increases. Because of this difference in the
distribution of corresponding densities, it is not straightforward that the
behavior of similarly defined properties is the same. One can not generalize
the specific nuclear properties to the NS and needs a separate analysis to
understand the NS properties.

In Ref. Kumar  _et al._ (2021a), the conventional Brückner energy density
functional (B-EDF) Brueckner  _et al._ (1968a, 1969) is replaced by the
effective field theory motivated relativistic mean-field (E-RMF) density
functional in a local density approximation (LDA) to calculate the nuclear
surface properties in the framework of the coherent density fluctuation model
(CDFM). They demonstrated that the microscopic E-RMF energy functional is able
to incorporate the structural effects of the nucleus and reproduce the peak in
the symmetry energy of the Pb isotopic chain at neutron number $N=126$, which
is generally failed by the B-EDF functional Quddus  _et al._ (2020a); Gaidarov
_et al._ (2011). Recently, this formalism has been extended successfully to
study the surface properties of NS Kumar  _et al._ (2021b) and estimated for
the first time the incompressibility $K^{\rm star}$, symmetry energy $S^{\rm
star}$, slope parameter $L_{\rm sym}^{\rm star}$ and curvature coefficient
$K_{\rm sym}^{\rm star}$ for some specific masses of the star. These
properties of NS are pretty informative for experimental observations and
theoretical modeling. For example, assuming the NS as a giant nucleus, with
the mass number $A\sim 10^{57}$, it must possess most of the properties of a
standard finite nucleus Kumar  _et al._ (2021b). It should have the multipole
moments and all possible collective oscillations like $f$-mode and $g$-mode,
etc. Das  _et al._ (2021a).

The parameters obtained by expanding the symmetry energy near the saturation
density (slope and curvature parameter) control the cooling rate of NS, and
the core-crust transition density and transition pressure Alam  _et al._
(2016); Schneider  _et al._ (2019). These parameters also play an important
role to constraining the nuclear equation of state (EoS), which is a key
ingredient for the study of NS properties as well as the properties of
supernovae explosion, binary NS merger, and the physics of gravitational wave
Gil  _et al._ (2021). A prominent bridge between the finite nuclei and the
nuclear/neutron matter (interstellar bodies) is the comprehensive knowledge of
the nuclear EoS. The EoS is the key component for the determination of the
properties of NS, and also it controls the dynamics of core-collapse
supernovae remnants, and the cooling of NS Schneider  _et al._ (2019); Bombaci
and Logoteta (2018). With the help of observational gravitational wave
(GW170817), Abbott  _et al._ (2017), Einstein Observatory (HEAO-2) Boguta
(1981) and X-ray radio telescopes Observatory ; Greif  _et al._ (2020), a
large number of constraints had been implemented to get a proper EoS at high
density regime.

From last few decades, the non-relativistic (Skyrme Skyrme and Schonland
(1961); Chabanat  _et al._ (1997, 1998); Dutra  _et al._ (2008, 2012), Gogny
forces Dechargé and Gogny (1980)) and relativistic Walecka (1974); Reinhard
_et al._ (1986); Pannert  _et al._ (1987); Gambhir  _et al._ (1990); Müller
and Serot (1996); Del Estal  _et al._ (2001); Patra  _et al._ (2002); Arumugam
_et al._ (2004)) theoretical approaches have been used as consistent formalism
to construct the EoS and calculate the properties of strongly-interacting
dense matter systems. The relativistic class of models is the alternative
approach for low-energy Quantum Chromodynamics with all the built-in non-
perturbative properties Adam  _et al._ (2020, 2015). In the present paper, we
use the latest form of E-RMF Lagrangian to evaluate the EoS with the recently
developed G3 Kumar  _et al._ (2017a) parameter set, and the results are
compared with the familiar NL3 Lalazissis  _et al._ (1997), and IU-FSU Carbone
and Schwenk (2019) forces.

The paper is organized as follows: In Sub-Sec. II.1, the relativistic mean-
field formalism is briefly described. The coherent density fluctuation model
is detailed in Sub-Sec. II.2 and the NS properties are calculated in Sub-Sec
II.3. The results and discussions are given in Sec. III. In this section, the
mass, radius, moment of inertia, density profile, and weight function of NS
obtained from the E-RMF equation of states are exemplified and explicitly
discussed. The parameters $K^{\rm star}$, $S^{\rm star}$, $L_{\rm sym}^{\rm
star}$ and $K_{\rm sym}^{\rm star}$ of NS are illustrated in Sub-Sec. III.3.
The summary and concluding remarks are drawn in Section IV.

Table 1: The nuclear matter properties at saturation for the EoS of NL3
Lalazissis  _et al._ (1997), G3 Kumar  _et al._ (2017a) and IU-FSU Carbone and
Schwenk (2019) parameter sets. The NM parameters are in MeV, except $\rho_{0}$
which is in fm-3. The references are $[a]$,$[b]$, $[c]$ $\&$ $[d]$ Zyla  _et
al._ (2020), $[e]$&$[f]$ Bethe (1971), $[g]$ Garg and Colò (2018), $[h]$&$[i]$
Danielewicz and Lee (2014), and $[j]$ Zimmerman  _et al._ (2020).

Parameter | NL3 | G3 | IU-FSU | Empirical/Expt. Value
---|---|---|---|---
$\rho_{0}$ | 0.148 | 0.148 | 0.154 | 0.148 – 0.185 $[e]$
$E/A$ | -16.29 | -16.02 | -16.39 | -15.00 – 17.00 $[f]$
$K$ | 271.38 | 243.96 | 231.31 | 220 – 260 $[g]$
$J_{0}$ | 37.43 | 31.84 | 32.71 | 30.20 – 33.70 $[h]$
$L_{\rm sym}$ | 120.65 | 49.31 | 49.26 | 35.00 – 70.00 $[i]$
$K_{\rm sym}$ | 101.34 | -106.07 | 23.28 | -174 – -31 $[j]$
$Q_{\rm sym}$ | 177.90 | 915.47 | 536.46 | ———–

Table 2: The fitted coefficients $a_{i}$, $b_{i}$ and $b_{e}$ of the Eq. (1) for NL3, G3 and IU-FSU forces. The values are scaled by $10^{-8}$ factor i.e. each should be multiplied by a factor of $10^{8}$ to get the exact magnitude of the coefficient. | NL3 | G3 | IU-FSU
---|---|---|---
be | 0.00017 | 0.00011 | 0.00011
b3 | -0.00054 | -0.000085 | -0.000088
b4 | 0.00898 | 0.00048 | 0.00043
b5 | -0.08078 | -0.00158 | -0.00091
b6 | 0.04609 | 0.00346 | 0.00036
b7 | -1.774 | -0.00547 | 0.00241
b8 | 4.742 | 0.00648 | -0.00592
b9 | -8.896 | -0.00579 | 0.00698
b10 | 11.65 | 0.00379 | -0.00498
b11 | -10.43 | -0.00174 | 0.00227
b12 | 6.0713 | 0.00523 | -0.00064
b13 | -2.069 | -0.000092 | 0.000105
b14 | 0.3132 | 0.000007 | -0.000007
a3 | -0.00088 | -0.000198 | -0.00019
a4 | 0.02289 | 0.002919 | 0.002475
a5 | -0.02539 | -0.01797 | -0.013555
a6 | 1.639 | 0.06395 | 0.04266
a7 | -6.864 | -0.1472 | -0.08664
a8 | 19.60 | 0.2303 | 0.1198
a9 | -39.03 | -0.2502 | -0.1156
a10 | 54.27 | 0.1891 | 0.07793
a11 | -51.73 | -0.09.762 | -0.03605
a12 | 32.25 | 0.03281 | 0.01092
a13 | -11.84 | -0.00647347 | -0.00194919
a14 | 1.945 | 0.000569 | 0.000155

##  II Theory

###  II.1 Effective field theory relativistic mean field model

As mentioned earlier, we used NL3 Lalazissis  _et al._ (1997), G3 Kumar  _et
al._ (2017a) and IU-FSU Carbone and Schwenk (2019) parameter sets of the E-RMF
Lagrangian. The NL3 set is the stiffest, and the newly reported G3 parameter
set provides the softest EoS. The numerical values of nuclear matter (NM)
properties at saturation are listed in Table I. The empirical/experimental
data are also given for comparison. The nuclear matter incompressibility $K$,
which controls the stiffness/softness of the EoS, are 271.38, 243.96, and
231.31 MeV for NL3, G3, and IU-FSU, respectively. The NM symmetry energies are
37.43, 31.84, and 32.71 MeV for the corresponding parameter sets. These values
are within the range set by various experimental observations and theoretical
predictions (see Table I).

Motivated by the work of Brückner et al. Brueckner  _et al._ (1968b, a), using
the LDA, the momentum space energy functional is converted to the coordinate
space through a generator coordinate ‘$x$’. The detailed procedure can be
found in Refs. Kumar  _et al._ (2021b, a). It is worth mentioning that the
Brückner energy density functional Brueckner  _et al._ (1968b, a) fails to
solve the Coester band problem Coester  _et al._ (1972); Pattnaik  _et al._
(2022a). Consequently, the peak that appears in the symmetry energy at the
magic number for heavier nuclei (like Pb isotopes) does not match the
appropriate neutron number Pattnaik  _et al._ (2021, 2022a). The coordinate
space E-RMF energy density functional within the LDA for NS matter is defined
as Kumar  _et al._ (2021b):

$\displaystyle{\cal E}$ $\displaystyle=$ $\displaystyle
C_{k}n^{2/3}+C_{e}n^{4/9}+\sum_{i=3}^{14}(b_{i}+a_{i}\alpha^{2})n^{i/3},$ (1)

where
$C_{k}=0.3(\hbar^{2}/2M)(3\pi^{2})^{2/3}[(1+\alpha)^{5/3}+(1-\alpha)^{5/3}]$
is the coefficient of the kinetic energy for protons and neutrons and
$C_{e}=b_{e}(1-\alpha)^{5/9}$ is the kinetic energy coefficient for electrons
and muons, with $b_{e}$ as a variable obtained from the conversion of the
E-RMF energy density from momentum space to coordinate space Kumar  _et al._
(2021b). The last term is the potential interaction of the nucleons and the
coefficients $b_{i}$ and $a_{i}$ obtained from the fitting for different E-RMF
models. It is shown in Ref. Kumar  _et al._ (2021a) that the accuracy of the
fitting increases with increase in the number of coefficients $a_{i}$ and
$b_{i}$ in the series of the potential term of Eq. (1). The mean deviation
$\delta=\sum_{j=1}^{N}[(E/A)_{j,\mathrm{Fitted}}-(E/A)_{j,\mathrm{RMF}}]/N$, =
$18\%$, $6\%$ and $0.5\%$ for 8, 10 and 12 terms, respectively. Here $N$ is
the total number of points. The obtained coefficients of the energy functional
in Eq. (1) i.e. $b_{e}$, $b_{i}$ and $a_{i}$ are tabulated in Table 1.

###  II.2 Coherent density fluctuation model

The CDFM is a well-established formalism to calculate the properties of finite
nuclei Antonov  _et al._ (1980, 1994); Gaidarov  _et al._ (2020) by
superimposing the structure of infinite nuclear matter. This method is
recently extended to calculate the properties of NS Kumar  _et al._ (2021b).
The CDFM utilise a generator coordinate ‘$x$’ to evaluate the one-body density
matrix $n(r,r^{\prime})$ of a finite nucleus/NS as the superposition of
infinite number of one-body density matrices $n_{x}(r,r^{\prime})$, called
“Fluctons” Gaidarov  _et al._ (2011); Kaur  _et al._ (2020). The density of a
Flucton is written as Antonov  _et al._ (1980, 1994); Gaidarov  _et al._
(2020):

$n_{x}({\bf r})=n_{0}(x)\,\Theta(x-|{\bf r}|).$ (2)

The saturation density of the Flucton is $n_{0}(x)=3A/4\pi x^{3}$, $A$ is the
total number of protons and neutrons in the neutron star matter (NSM). In the
CDFM, the density of the spherical finite NSM of radius ‘$r$’ is Antonov  _et
al._ (2018, 2016),

$\displaystyle n(r)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dx\,|F(x)|^{2}\,n_{0}(x)\,\Theta(x-|{\bf r}|),$
(3)

where $|F(x)|^{2}$ is the weight function in the generator coordinate ‘$x$’
with the local density $n(r)$ written as Antonov  _et al._ (2018):

$|F(x)|^{2}=-\frac{1}{n_{0}(x)}\frac{dn(r)}{dr}\Bigg{|}_{r=x}.$ (4)

The incompressibility, symmetry energy, slope parameter, and curvature
coefficient of the NS are expressed by folding the weight function with the
respective NS matter as Kumar  _et al._ (2021b); Gaidarov  _et al._ (2011,
2012); Antonov  _et al._ (2016):

$\displaystyle K^{\rm star}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dx\,|F(x)|^{2}\ K^{\rm NSM}(n(x)),$ (5)
$\displaystyle S^{\rm star}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dx\,|F(x)|^{2}\,S^{\rm NSM}(n(x)),$ (6)
$\displaystyle L_{\rm sym}^{\rm star}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dx\,|F(x)|^{2}\,L_{\rm sym}^{\rm NSM}(n(x)),$
(7) $\displaystyle K_{\rm sym}^{\rm star}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dx\,|F(x)|^{2}\ K_{\rm sym}^{\rm NSM}(n(x)),$
(8)

where $K^{\rm NSM}$, $S^{\rm NSM}$, $L^{\rm NSM}_{\rm sym}$ and $K^{\rm
NSM}_{\rm sym}$ are the incompressibility, symmetry energy, slope parameter
and curvature of the NSM.

The converted energy density functional of the NSM from momentum space to the
coordinate space ‘$x$’ in a local density approximation is Eq. (1). The
expressions for $K^{\rm NSM}$, $S^{\rm NSM}$, $L^{\rm NSM}_{\rm sym}$ and
$K^{\rm NSM}_{\rm sym}$ are obtained from this Eq. (1) with the definitions
Fetter and Walecka (1971); Chen  _et al._ (2009); Chen and Piekarewicz (2014),
i.e., the NM parameters $K^{\rm NM}$, $S^{\rm NM}$, $L_{\rm sym}^{\rm NM}$ and
$K_{\rm sym}^{\rm NM}$ are obtained from the following standard relations
Gaidarov  _et al._ (2011); Antonov  _et al._ (2016); Gaidarov  _et al._
(2012); Kumar  _et al._ (2021b):

$\displaystyle K^{\rm NM}$ $\displaystyle=$ $\displaystyle
9\rho_{0}^{2}\frac{\partial^{2}({\cal
E}/\rho)}{\partial\rho^{2}}\Big{|}_{\rho=\rho_{0}},$ (9) $\displaystyle S^{\rm
NM}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\frac{\partial^{2}({\cal
E}/\rho)}{\partial\alpha^{2}}\Big{|}_{\alpha=0},$ (10) $\displaystyle L_{\rm
sym}^{\rm NM}$ $\displaystyle=$ $\displaystyle 3\rho_{0}\frac{\partial
S(\rho)}{\partial\rho}\Big{|}_{\rho=\rho_{0}}=\frac{3P}{\rho_{0}},$ (11)
$\displaystyle K_{\rm sym}^{\rm NM}$ $\displaystyle=$ $\displaystyle
9\rho_{0}^{2}\frac{\partial^{2}S(\rho)}{\partial\rho^{2}}\Big{|}_{\rho=\rho_{0}},$
(12)

which are given as follows using Eq. (1)

$\displaystyle K^{\rm NSM}$ $\displaystyle=$
$\displaystyle-150.12\,n_{0}^{2/3}(x)-2.22\,b_{e}\,n_{0}^{4/9}(x)$ (13)
$\displaystyle+\sum_{i=4}^{14}i\,(i-3)\,b_{i}\,n_{0}^{i/3}(x),$ $\displaystyle
S^{\rm NSM}$ $\displaystyle=$ $\displaystyle
41.7\,n_{0}^{2/3}(x)-0.12\,b_{e}\,n_{0}^{4/9}(x)$ (14)
$\displaystyle+\sum_{i=3}^{14}a_{i}\,n_{0}^{i/3}(x),$ $\displaystyle L_{\rm
sym}^{\rm NSM}$ $\displaystyle=$ $\displaystyle
83.4\,n_{0}^{2/3}(x)-0.16\,b_{e}\,n_{0}^{4/9}(x)$ (15)
$\displaystyle+\sum_{i=3}^{14}i\,a_{i}\,n_{0}^{i/3}(x),$ $\displaystyle K_{\rm
sym}^{\rm NSM}$ $\displaystyle=$
$\displaystyle-83.4\,n_{0}^{2/3}(x)+0.266\,b_{e}\,n_{0}^{4/9}(x)$ (16)
$\displaystyle+\sum_{i=4}^{14}i\,(i-3)\,a_{i}\,n_{0}^{i/3}(x).$

The symmetry energy for finite nuclei, i.e., NS with nucleon number A, can be
further expressed as the components of volume $S_{V}$ and surface $S_{S}$
contributions using Danielewicz’s liquid drop prescription, which is written
as Danielewicz (2003, 2006); DANIELEWICZ ; Antonov  _et al._ (2018, 2016);
Pattnaik  _et al._ (2021, 2022b):

$\displaystyle
S=\frac{S_{V}}{1+\frac{S_{S}}{S_{V}}A^{-1/3}}=\frac{S_{V}}{1+A^{-1/3}/\kappa},$
(17)

where the ratio $\kappa\equiv\frac{S_{V}}{S_{S}}$ is defined as Danielewicz
(2003, 2006); DANIELEWICZ ; Pattnaik  _et al._ (2021, 2022b):

$\displaystyle\kappa=\frac{3}{R\rho_{0}}\int_{0}^{\infty}dx|F(x)|^{2}x\rho_{0}(x)\left[\left(\frac{\rho_{0}}{\rho(x)}\right)^{\gamma}-1\right].$
(18)

The value of $\gamma=0.3$ is used in Eq. (18) following Ref. Antonov  _et al._
(2018). An alternative method has been reported by Gaidarov et al. to obtain
the volume, and surface symmetry energy components Gaidarov  _et al._ (2021).

###  II.3 Neutron star properties

The neutron star EoS is calculated using the E-RMF model with the assumption
that the NS is in $\beta-$equilibrium and charge neutrality conditions
Glendenning (1997). The EoS of the NS depends on the model and also on the
types of extra particles such as hyperons Schaffner and Mishustin (1996);
Bhowmick  _et al._ (2014); Fortin  _et al._ (2017); Kumar  _et al._ (2017b);
Bhuyan  _et al._ (2017); Biswal (2019); Biswal  _et al._ (2019), kaons Pal
_et al._ (2000); Gupta and Arumugam (2012), dark matter Das  _et al._ (2019,
2020, 2021b, 2021c, 2021d) etc. are present in the systems. Here, we limit to
the nucleons and leptons only, which are in $\beta-$ equilibrium and charge
neutrality.
To calculate the NS macroscopic/structural properties such as $M$ and $R$, one
has to solve the Tolman-Oppenheimer-Volkoff (TOV) equations Tolman (1939);
Oppenheimer and Volkoff (1939)

$\displaystyle\frac{dP}{dr}$ $\displaystyle=$ $\displaystyle-\frac{[{\cal
E}+P][m+{4\pi r^{3}P}]}{r^{2}\Big{(}1-\frac{2m}{r}\Big{)}},$
$\displaystyle\frac{dm}{dr}$ $\displaystyle=$ $\displaystyle 4\pi r^{2}{{\cal
E}}.$ (19)

The coupled equations are solved by using boundary conditions as: $r=0$,
$P=P_{c}$ and $r=R$, $P=0$ at fixed central density. The maximum mass and
radius of the NS are calculated assuming the pressure vanishes at the surface
of the star.

For slowly and uniformly rotating NS, the metric is given by Stergioulas
(2003)

$\displaystyle ds^{2}=-e^{2\nu}dt^{2}+e^{2\psi}(d\phi-\omega
dt^{2})+e^{2\alpha}(r^{2}d\theta^{2}+d\phi^{2}).$ (20)

The moment of inertia ($I$) of the NS is calculated with the slow rotation
approximation and is given as Stergioulas (2003); Jha  _et al._ (2008); Sharma
and Jha (2009); Friedman and Stergioulas (2013); Paschalidis and Stergioulas
(2017); Quddus  _et al._ (2020b); Koliogiannis and Moustakidis (2020):

$I\approx\frac{8\pi}{3}\int_{0}^{R}\ dr\ ({\cal E}+P)\
e^{-\phi(r)}\Big{[}1-\frac{2m(r)}{r}\Big{]}^{-1}\frac{\bar{\omega}}{\Omega}\
r^{4},$ (21)

where $\bar{\omega}$ is the dragging angular velocity for a uniformly rotating
star. The $\bar{\omega}$ satisfies the following boundary conditions,

$\bar{\omega}(r=R)=1-\frac{2I}{R^{3}},\qquad\frac{d\bar{\omega}}{dr}\Big{|}_{r=0}=0.$
(22)

##  III Results and Discussions

In this section, we present the macroscopic properties of the NS, such as mass
($M$), radius ($R$), and moment of inertia ($I$) for NL3, G3, and IU-FSU
parameter sets. After getting a broad knowledge of the bulk properties, we
extend our calculations to the surface properties of NS with the above three
forces. For this, we estimate the symmetry energy $S^{\rm star}$,
incompressibility $K^{\rm star}$, slope parameter $L^{\rm star}_{\rm sym}$ and
curvature co-efficient $K^{\rm star}_{\rm sym}$ of NS with respect to their
masses from $0.8\ M_{\odot}$ to $M_{\rm max}$. To calculate these properties,
the NS densities are extracted by feeding the EoSs in the TOV equations.
Considering the obtained density as the local density of the star, with the
help of CDFM, we construct the weight function $|F(x)|^{2}$, which is folded
with $S^{\rm NSM}$, $K^{\rm NSM}$, $L_{\rm sym}^{\rm NSM}$ and $K_{\rm
sym}^{\rm NSM}$, respectively to evaluate the surface properties of the NS.
The detailed procedure of the evaluation scheme of the results is available in
Ref. Kumar  _et al._ (2021b). Further, the results are discussed in the
following sub-sections.

Figure 1: (color online) Internal and the maximum mass-radius profiles of a NS
for NL3 (red), G3 (green) and IU-FSU (magenta) parameter sets. The different
masses $1.0\ M_{\odot}$, $1.4\ M_{\odot}$ and $M_{\rm max}$ are shown with
solid, dashed and dotted lines. Figure 2: (color online) Moment of inertia of
NS for NL3 (red), G3 (green) and IU-FSU (magenta) parameter sets. The
different masses $1.0\ M_{\odot}$, $1.4\ M_{\odot}$ and $M_{\rm max}$ are
shown with solid, dashed and dotted lines.

###  III.1 Mass, radius and momentum of inertia of neutron star

The mass-radius profiles of the NS are calculated with three different cases
$M=1.0\ M_{\odot}$, $1.4\ M_{\odot}$ and $M_{\rm max}$ by fixing the central
densities, which are depicted in Fig. 1 using NL3, G3, and IU-FSU parameter
sets. Being the stiffest EoS, NL3 predicts the maximum mass. Since we fix the
central densities corresponding to these three masses $1.0\ M_{\odot},1.40\
M_{\odot}$ and $M_{\rm max}$, the radii are also found to be different for
each of the parameter sets. This can be seen clearly from Fig. 1. The NL3 set
predicts both larger mass and radius as compared to G3 and IU-FSU forces.
Similarly, we calculate the moment of inertia $I$ of the NS for these three
sets, which are shown in Fig. 2. The value of $I$ increases with the mass of
the NS due to their nearly linear relationship. The NL3 predicts higher $I$ as
compared to G3 and IU-FSU. This is because of the stiffer EoS of NL3 than G3
and IU-FSU sets.

###  III.2 Neutron star density and it’s weight function

The densities and their corresponding weight functions ${|F(x)|^{2}}$ versus
radius of the NS with masses $1.0\ M_{\odot}$, $1.4\ M_{\odot}$ and $M_{\rm
max}$ are depicted in Fig. 3. The densities are in the upper panel, and their
weight functions are in the lower panel. The results are presented for NL3,
G3, and IU-FSU parameter sets. The chosen masses cover the lower, canonical,
and maximum mass of the NS. The maximum mass for NL3, G3, IU-FSU are $2.85\
M_{\odot}$, $2.004\ M_{\odot}$, $1.940\ M_{\odot}$ respectively. The $M_{\rm
max}$ for NL3 is distinctly larger than the other two sets. This behavior
reflects not only in the $I-M$ and $M-R$ profiles but is also clearly seen in
the densities and weight functions. Unlike the normal nucleus, which is bound
by strong interaction, the NS is balanced by the attractive gravitational and
the repulsive force due to the degenerated neutrons gas. It is worthy of
mentioning that the nuclear force is state-dependent, i.e., (i) singlet-
singlet, (ii) triplet-triplet, and (iii) singlet-triplet. The former two
interactions are attractive, while the latter category is repulsive
interaction Patra and Praharaj (1992); Satpathy and Patra (2004); Kaur  _et
al._ (2020); Das  _et al._ (2021b). Because of the excessive neutrons in the
NS, the repulsive part is subject to instability, which is counterbalanced by
the huge gravitational attraction. Thus, NS’s density distribution is quite
different from the normal nucleus. In addition, the density is influenced by
the presence of electrons and muons, and the density obtained by the NL3 set
has the minimum central density followed by IU-FSU and G3 models. However, the
$M_{\rm max}$ of NL3 is more as compared to the maximum mass of the star
acquired by the G3 and IU-FSU parameter sets.

The corresponding weight functions for NL3, G3, and IU-FSU sets are given
below to their densities in Fig. 3. The shape of the ${|F(x)|^{2}}$ is like an
exponential rise, and it is maximum at the surface of the NS. The values of
$K^{\rm star}$, $S^{\rm star}$, $L^{\rm star}_{\rm sym}$ and $K_{\rm sym}^{\rm
star}$ are determined by folding the weight function with the corresponding
NSM quantities are $K^{\rm NSM}$, $S^{\rm NSM}$, $L_{\rm sym}^{\rm NSM}$ and
$K_{\rm sym}^{\rm NSM}$ (see Eqs. 5, 6, 7, 8). Thus, the maximum contribution
comes from the surface of the NS and is termed a surface phenomenon.
Precisely, the values of ${|F(x)|^{2}}$ gather momentum at $\sim{6}$ km, and
it is maximum at the surface ($\sim{10-12}$ km).

Figure 3: (color online) The NS densities ($\rho$) for NL3 (red), G3 (green)
and IU-FSU (magenta) parameter sets as a function of radius. The mass number
($A$) of the maximum mass of the NS for NL3, G3 and IU-FSU are $3.35\times
10^{54}$, $2.32\times 10^{54}$ and $2.23\times 10^{54}$ respectively. The
different masses $1.0\ M_{\odot}$, $1.4\ M_{\odot}$ and $M_{\rm max}$ are
shown with solid, dashed and dotted lines.  Figure 4: (color online) The NS
incompressibility $K^{\rm star}$ as a function of NS mass for NL3 (red), G3
(green) and IU-FSU (magenta) parameter sets.  Figure 5: (color online) The
symmetry energy of the NS $S^{\rm star}$ as a function of NS mass for NL3
(red), G3 (green) and IU-FSU (magenta) parameter sets. Figure 6: (color
online) The $L^{\rm star}_{\rm sym}$ coefficient of the NS $S^{\rm star}$
obtained from Eq. (7) as a function of NS mass for NL3 (red), G3 (green) and
IU-FSU (magenta) parameter sets. Figure 7: (color online) The curvature
coefficient $K^{\rm star}_{\rm sym}$ of the NS obtained from Eq. (8) as a
function of NS mass for NL3 (red), G3 (green) and IU-FSU (magenta) parameter
sets. Figure 8: (color online) The ratio of the volume to surface components
of the symmetry energy $\kappa$ as a function of NS mass for NL3 (red), G3
(green) and IU-FSU (magenta) parameter sets. The $\kappa$ values are evaluated
from Eq. (18).

###  III.3 Surface properties of neutron star

In this sub-section, we analyse the surface properties of our results obtained
from the CDFM calculations for NS as a function of mass. Here, the
incompressibility $K^{\rm star}$, symmetry energy $S^{\rm star}$, slope
parameter $L^{\rm star}_{\rm sym}$ and curvature co-efficient $K^{\rm
star}_{\rm sym}$ of the NS are discussed. The results are depicted in Figs.
(4,5,6,7) in the following sub-sections.

####  III.3.1 Neutron star incompressibility

The incompressibility $K$ of an object is a prominent characteristic to know
its nature. It defines how much the object can be compressed or expanded. So,
it has a direct connection with the collective motion of the system. As a
matter of fact, $K$ is a significant quantity for the NS. The EoS, governed by
incompressibility, plays a crucial role in determining the mass and radius of
the NS. It is shown in Ref. Kumar  _et al._ (2021b) that the incompressibility
of NS $K^{\rm star}$is much less than the nuclear matter incompressibility at
saturation $K_{\infty}$. For example, $K_{\infty}=271.38$ MeV for NL3 set as
compared to the $K^{\rm star}=44.956$ MeV for the maximum mass of the NS Kumar
_et al._ (2021b). This can also be related to the asymmetric nature of the
medium $\alpha$. The $K_{\infty}$ is obtained at the asymmetric limit
$\alpha=0$, and $K^{\rm star}$ is evaluated at $\alpha\sim 1$, i.e., the
incompressibility of a system decreases with asymmetric of the system Kumar
_et al._ (2021b). One can see the trend of $K^{\rm star}$ as a function of
mass $M$ ranging from $M=0.8\ M_{\odot}-M_{max}$ of the NS in Fig. 4 for the
three considered sets. The three forces predict almost similar
incompressibilities up to mass $M=1.8\ M_{\odot}$. More explicitly, the G3 and
IU-FSU give almost similar results, while NL3 predicts a comparatively larger
value of $K^{\rm star}$ as can be seen from the inset of the figure. Beyond
mass $M=1.8\ M_{\odot}$, the incompressibility increases suddenly. The
similarity of $K^{\rm star}$ between G3 and IU-FSU forces could be related to
the $K_{\infty}$, which are 243.96 and 231.31 MeV for G3 and IU-FSU,
respectively. On the other hand, the incompressibility of NL3 at saturation is
quite high as compared to G3 and IU-FSU. After realizing that the EoS can be
made softer by reducing the $K$, which in turn reduces considerably the
maximum mass of the NS and vice-versa. This shows that the value of $K^{\rm
star}$ increases marginally up to $M=1.8\ M_{\odot}$ and suddenly increases
beyond, indicating the mass dependence of the incompressibility.

####  III.3.2 Symmetry energy and its higher-order derivatives

The symmetry energy of the NS in its maximum mass is predicted to be higher in
comparison to the value of symmetric nuclear matter at saturation. This
observation is noticed in all the three considered E-RMF models. The symmetry
energy at saturation $J_{0}$ for NM with NL3, G3, and IU-FSU sets are $37.43$,
$31.84$, and $32.71$ MeV, respectively. These results are $146.002$, $66.813$,
and $60.758$ MeV for the NS at the limit of maximum mass. These $S^{\rm star}$
are small for smaller masses as compared to the maximum mass of the star. The
results of $S^{\rm star}$ are depicted in Fig. 5 as a function of mass for all
the three-parameter sets of NS. The symmetry energy for G3 and IU-FSU are
found to be almost similar, while the values with the NL3 set are a bit
higher. This can be seen clearly from the inset of the figure, which depicts
the structural dependence of the symmetry energy. The symmetry energy is
obtained from the derivative of the energy density with respect to
asymmetricity $\alpha$, which shows a significant variation in the $S^{\rm
star}$ as compared to $K^{\rm star}$. The higher derivatives of the symmetry
energy, i.e., the slope $L^{\rm star}_{\rm sym}$ and curvature parameter
$K^{\rm star}_{\rm sym}$ are quite useful quantities. These are shown in Figs.
6 and 7 as a function of NS mass. The magnitude of $L^{\rm star}_{\rm sym}$
and $K^{\rm star}_{\rm sym}$ increases with the mass of the star. The $L^{\rm
star}_{\rm sym}$ values for all the models are found to be positive, contrary
to the negative nature of $K^{\rm star}_{\rm sym}$ as shown in the figures.

The negative sign of $K^{\rm star}_{\rm sym}$ is correlated by the 1-$\sigma$
constraint, and $90\%$ confident limits on its saturation value of normal
nuclear matter as reported by Zimmerman et al. from the experimental data of
PSR J0030+0451 and GW170817 event Zimmerman  _et al._ (2020); Riley  _et al._
(2019); Abbott  _et al._ (2017). It is noted that these bounds are not well
matched to explain the $K^{\rm star}_{\rm sym}$ of NS; however, it indicates
the possibility of a negative value of the curvature parameter and predicts
the range with $\sim{90}\%$ confidence limit of the observational data.
Sometimes it is beneficial to analyze the terrestrial data related to the
exotic nuclei and heavy-ion collisions Chen  _et al._ (2007); Centelles  _et
al._ (2009) by separating the contribution of iso-vector incompressibility or
curvature parameter.

Finally, the total symmetry energy of the NS can be divided into its volume
$S_{V}$, and surface $S_{S}$ components, which are derived from Eq. (17)
through $\kappa$ (the ratio of the volume to surface symmetry energies) and
$\kappa$ is obtained from Eq. (18). The value of $\kappa$ as a function of the
mass of the NS is shown in Fig. 8. In our calculation, $\gamma=0.3$ is used
following Ref. Antonov  _et al._ (2018). The $\kappa$ values are consistently
larger for the NL3 set followed by IU-FSU and least for the G3 force. These
trends are in accordance with the forces used in the calculations. It goes on
increasing with the mass of the NS to some value, as shown in the figure.
Beyond that, the $\kappa$ decreases considerably. When the mass number of the
system approaches a large value, i.e., in the limit of
$A{\rightarrow{\infty}}$, the system deals mostly with the volume. For
example, for 1.4 $M_{\odot}$ of the G3 set, the total symmetry energy is 8.904
MeV, while the volume part contributes as a whole and the surface component
contributes nearly 9% only of total $S^{\rm star}$. In such a case, the major
contribution of symmetry energy comes from the volume part of the NS as
referring Eq. (17). To take care of such a system properly, the alternative
method of Ref. Gaidarov  _et al._ (2021) may be useful. The $\kappa$ is
defined as $\kappa=S_{V}/S_{S}$, and even though the total symmetry energy is
contributed by the volume component, i.e., $S\to S_{V}$, it has a surface
component due to the finite value of $\kappa$.

##  IV Summary and conclusions

The structural properties of NS, along with the mass, radius, and moment of
inertia, are studied within the well-known E-RMF formalism. Three established
forces (NL3, G3, and IU-FSU) are used in the calculations. Although NL3 is one
of the oldest sets, it gives an understanding of the properties of finite
nuclei. Also, this set is used extensively so that it may provide a piece of
known information, and a comparison of other sets with the results of NL3 may
be more familiar. The primary motivation of the present work is the surface
properties of NS in terms of incompressibility, symmetry energy, and its
higher coefficients, such as the slope and curvature coefficients.

To our knowledge, there are no prior theoretical results or
empirical/experimental data are available to support our calculations of NS
symmetry energy, incompressibility, slope, and curvature parameters. In the
recent work, we suggested the formalism for the computation of these
quantities Kumar  _et al._ (2021b) for NS. In the present paper, we extended
the model to study the properties systematically. Thus, the results reported
here are the first of such kind. We know that the NS is a highly iso-spin
asymmetric system. We expected that the NS surface properties could be very
different from the standard NM. The more significant $L_{\rm sym}^{\rm star}$
for NL3 force is in agreement with the stiff EoS Bednarek  _et al._ (2020) and
moderate slope parameter predicted for the softer G3 and IU-FSU sets. A proper
understanding of the range of $S^{\rm star}$, $L^{\rm star}_{\rm sym}$ and
$K^{\rm star}$ generally helps to fix the radius of the NS. A precise
correlation is established via Danielewicz’s liquid drop prescription with the
factor $\kappa$, i.e., the volume ratio to the surface component of the
symmetry energy. This correlation can be extended to $L_{\rm sym}^{\rm star}$
in the liquid drop mass formula with the help of surface $S^{\rm star}_{S}$,
and the volume $S^{\rm star}_{V}$ symmetry energy Lattimer (2015). From the
analysis of the surface properties, we noticed that almost all these
parameters, along with $\kappa$ for all the three sets, coincide with each
other in the vicinity of mass range $M\sim 1.8M_{\odot}$ indicating a possible
correlation with the mass of the NS. It is shown in Lattimer (2015); Lattimer
and Lim (2013) that the static dipole and quadrupole polarizability and the
neutron-skin thickness are strongly related between symmetry energy and slope
parameter. Thus, we expect the NS radius to be synchronized with its surface
properties. The magnitude of $K^{\rm star}$, $S^{\rm star}$, $L^{\rm
star}_{\rm sym}$ and $K_{\rm sym}^{\rm star}$ increases with mass of the NS.
For smaller NS, we find smaller values of all the surface properties. The
second derivative of symmetry energy is the $K_{\rm sym}^{\rm star}$, which is
obviously more sensitive to the mass and also most ambiguous.

The present systematic calculations may provide a better theoretical bound on
$K^{\rm star}_{\rm sym}$ and a better pathway to constraint the experimental
setup for the isoscalar giant resonances for the properties of astrophysical
objects. Despite the absence of direct experimental data for these surface
properties of NS (incompressibility, symmetry energy, slope, and curvature
parameters), the calculated results using the E-RMF densities and CDFM
approach seem reasonable. The present theoretical calculations can be
validated using different relativistic and non-relativistic energy density
functionals and suitable force parameters. The current method of accessibility
to NS, considering the NS as a finite nucleus system, favors a bridge between
the two unequal size objects. The theoretical approach adopted here presents a
new way for the nuclear and astrophysicist to reveal the wealth of information
on exotic nuclei and dense astronomical objects.

Acknowledgement: Mr. J. A. Pattnaik thanks the Institute of Physics,
Bhubaneswar, for its facilities. SERB partly reinforces this work, Department
of Science and Technology, Govt. of India, Project No. CRG/2019/002691.

## References

  * Engvik  _et al._ (1994) L. Engvik, M. Hjorth-Jensen, E. Osnes, G. Bao, and E. Østgaard, Phys. Rev. Lett. 73, 2650 (1994).
  * Sharma  _et al._ (2007) B. K. Sharma, P. K. Panda, and S. K. Patra, Phys. Rev. C 75, 035808 (2007).
  * Kumar  _et al._ (2021a) A. Kumar, H. C. Das, M. Kaur, M. Bhuyan, and S. K. Patra, Phys. Rev. C 103, 024305 (2021a).
  * Brueckner  _et al._ (1968a) K. A. Brueckner, J. R. Buchler, S. Jorna, and R. J. Lombard, Phys. Rev. 171, 1188 (1968a).
  * Brueckner  _et al._ (1969) K. A. Brueckner, J. R. Buchler, R. C. Clark, and R. J. Lombard, Phys. Rev. 181, 1543 (1969).
  * Quddus  _et al._ (2020a) A. Quddus, M. Bhuyan, and S. K. Patra, Journal of Physics G: Nuclear and Particle Physics 47, 045105 (2020a).
  * Gaidarov  _et al._ (2011) M. K. Gaidarov, A. N. Antonov, P. Sarriguren, and E. Moya de Guerra, Phys. Rev. C 84, 034316 (2011).
  * Kumar  _et al._ (2021b) A. Kumar, H. C. Das, and S. K. Patra, Phys. Rev. C 104, 055804 (2021b).
  * Das  _et al._ (2021a) H. C. Das, A. Kumar, S. K. Biswal, and S. K. Patra, Phys. Rev. D 104, 123006 (2021a).
  * Alam  _et al._ (2016) N. Alam, B. K. Agrawal, M. Fortin, H. Pais, C. Providência, A. R. Raduta, and A. Sulaksono, Phys. Rev. C 94, 052801 (2016).
  * Schneider  _et al._ (2019) A. S. Schneider, L. F. Roberts, C. D. Ott, and E. O’Connor, Phys. Rev. C 100, 055802 (2019).
  * Gil  _et al._ (2021) H. Gil, Y.-M. Kim, P. Papakonstantinou, and C. H. Hyun, Phys. Rev. C 103, 034330 (2021).
  * Bombaci and Logoteta (2018) I. Bombaci and D. Logoteta, Astronomy & Astrophysics 609, A128 (2018).
  * Abbott  _et al._ (2017) B. P. Abbott _et al._ (The LIGO Scientific Collaboration and the Virgo Collaboration), Phys. Rev. Lett. 119, 161101 (2017).
  * Boguta (1981) J. Boguta, Physics Letters B 106, 255 (1981).
  * (16) C. X.-R. Observatory, .
  * Greif  _et al._ (2020) S. K. Greif, K. Hebeler, J. M. Lattimer, C. J. Pethick, and A. Schwenk, The Astrophysical Journal 901, 155 (2020).
  * Skyrme and Schonland (1961) T. H. R. Skyrme and B. F. J. Schonland, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 260, 127 (1961).
  * Chabanat  _et al._ (1997) E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer, Nuclear Physics A 627, 710 (1997).
  * Chabanat  _et al._ (1998) E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer, Nuclear Physics A 635, 231 (1998).
  * Dutra  _et al._ (2008) M. Dutra, O. Lourenço, A. Delfino, J. S. Sá Martins, C. Providência, S. S. Avancini, and D. P. Menezes, Phys. Rev. C 77, 035201 (2008).
  * Dutra  _et al._ (2012) M. Dutra, O. Lourenço, J. S. Sá Martins, A. Delfino, J. R. Stone, and P. D. Stevenson, Phys. Rev. C 85, 035201 (2012).
  * Dechargé and Gogny (1980) J. Dechargé and D. Gogny, Phys. Rev. C 21, 1568 (1980).
  * Walecka (1974) J. Walecka, Annals of Physics 83, 491 (1974).
  * Reinhard  _et al._ (1986) P. G. Reinhard, M. Rufa, J. Maruhn, W. Greiner, and J. Friedrich, Z. Phys. A 323, 13 (1986).
  * Pannert  _et al._ (1987) W. Pannert, P. Ring, and J. Boguta, Phys. Rev. Lett. 59, 2420 (1987).
  * Gambhir  _et al._ (1990) Y. Gambhir, P. Ring, and A. Thimet, Annals of Physics 198, 132 (1990).
  * Müller and Serot (1996) H. Müller and B. D. Serot, Nuclear Physics A 606, 508 (1996).
  * Del Estal  _et al._ (2001) M. Del Estal, M. Centelles, X. Viñas, and S. K. Patra, Phys. Rev. C 63, 044321 (2001).
  * Patra  _et al._ (2002) S. K. Patra, M. Centelles, X. Viñas, and M. Del Estal, Phys. Rev. C 65, 044304 (2002).
  * Arumugam  _et al._ (2004) P. Arumugam, B. Sharma, P. Sahu, S. Patra, T. Sil, M. Centelles, and X. Viñas, Physics Letters B 601, 51 (2004).
  * Adam  _et al._ (2020) C. Adam, A. G. Martín-Caro, M. Huidobro, R. Vázquez, and A. Wereszczynski, Physics Letters B 811, 135928 (2020).
  * Adam  _et al._ (2015) C. Adam, M. Haberichter, and A. Wereszczynski, Physical Review C 92 (2015), 10.1103/physrevc.92.055807.
  * Kumar  _et al._ (2017a) B. Kumar, S. Singh, B. Agrawal, and S. Patra, Nuclear Physics A 966, 197 (2017a).
  * Lalazissis  _et al._ (1997) G. A. Lalazissis, J. König, and P. Ring, Phys. Rev. C 55, 540 (1997).
  * Carbone and Schwenk (2019) A. Carbone and A. Schwenk, Phys. Rev. C 100, 025805 (2019).
  * Zyla  _et al._ (2020) P. Zyla, R. Barnett, J. Beringer, and et al., “Particle data group,”  (2020).
  * Bethe (1971) H. A. Bethe, Annual Review of Nuclear Science 21, 93 (1971), https://doi.org/10.1146/annurev.ns.21.120171.000521 .
  * Garg and Colò (2018) U. Garg and G. Colò, Progress in Particle and Nuclear Physics 101, 55 (2018).
  * Danielewicz and Lee (2014) P. Danielewicz and J. Lee, Nuclear Physics A 922, 1 (2014).
  * Zimmerman  _et al._ (2020) J. Zimmerman, Z. Carson, K. Schumacher, A. W. Steiner, and K. Yagi, (2020), arXiv:2002.03210 [astro-ph.HE] .
  * Brueckner  _et al._ (1968b) K. A. Brueckner, S. A. Coon, and J. Dabrowski, Phys. Rev. 168, 1184 (1968b).
  * Coester  _et al._ (1972) F. Coester, B. Day, and A. Goodman, Phys. Rev. C 5, 1135 (1972).
  * Pattnaik  _et al._ (2022a) J. A. Pattnaik, J. T. Majekodunmi, A. Kumar, M. Bhuyan, and S. K. Patra, Phys. Rev. C 105, 014318 (2022a).
  * Pattnaik  _et al._ (2021) J. A. Pattnaik, M. Bhuyan, R. N. Panda, and S. K. Patra, Physica Scripta 96, 125319 (2021).
  * Antonov  _et al._ (1980) A. Antonov, V. Nikolaev, and I. Petkov, Zeitschrift für Physik A Atoms and Nuclei 297 (1980), 10.1007/BF01892806.
  * Antonov  _et al._ (1994) A. N. Antonov, D. N. Kadrev, and P. E. Hodgson, Phys. Rev. C 50, 164 (1994).
  * Gaidarov  _et al._ (2020) M. Gaidarov, I. Moumene, A. N. Antonov, D. N. Kadrev, P. Sarriguren, and E. M. Guerra, arXiv: Nuclear Theory (2020).
  * Kaur  _et al._ (2020) M. Kaur, A. Quddus, A. Kumar, M. Bhuyan, and S. K. Patra, Journal of Physics G: Nuclear and Particle Physics 47, 105102 (2020).
  * Antonov  _et al._ (2018) A. N. Antonov, D. N. Kadrev, M. K. Gaidarov, P. Sarriguren, and E. Moya de Guerra, Phys. Rev. C 98, 054315 (2018).
  * Antonov  _et al._ (2016) A. N. Antonov, M. K. Gaidarov, P. Sarriguren, and E. Moya de Guerra, Phys. Rev. C 94, 014319 (2016).
  * Gaidarov  _et al._ (2012) M. K. Gaidarov, A. N. Antonov, P. Sarriguren, and E. M. de Guerra, Phys. Rev. C 85, 064319 (2012).
  * Fetter and Walecka (1971) A. L. Fetter and J. D. Walecka,  _Quantum Theory of Many-Particle Systems_ (McGraw-Hill, Boston, 1971).
  * Chen  _et al._ (2009) L.-W. Chen, B.-J. Cai, C. M. Ko, B.-A. Li, C. Shen, and J. Xu, Phys. Rev. C 80, 014322 (2009).
  * Chen and Piekarewicz (2014) W.-C. Chen and J. Piekarewicz, Phys. Rev. C 90, 044305 (2014).
  * Danielewicz (2003) P. Danielewicz, Nuclear Physics A 727, 233 (2003).
  * Danielewicz (2006) P. Danielewicz, “Nuclear symmetry energy: from nuclear extremes to neutron-star matter,” (2006), arXiv:nucl-th/0411115 [nucl-th] .
  * (58) P. DANIELEWICZ, “Symmetry energy,” in  _Opportunities with Exotic Beams_, pp. 142–151, $https://www.worldscientific.com/doi/pdf/10.1142/9789812708250_{0}015$ .
  * Pattnaik  _et al._ (2022b) J. A. Pattnaik, R. Panda, M. Bhuyan, and S. Patra, Canadian Journal of Physics 100, 102 (2022b), https://doi.org/10.1139/cjp-2021-0231 .
  * Gaidarov  _et al._ (2021) M. K. Gaidarov, E. M. de Guerra, A. N. Antonov, I. C. Danchev, P. Sarriguren, and D. N. Kadrev, Phys. Rev. C 104, 044312 (2021).
  * Glendenning (1997) N. K. Glendenning,  _Compact stars_ (Springer-Verlag New York, 1997).
  * Schaffner and Mishustin (1996) J. Schaffner and I. N. Mishustin, Phys. Rev. C 53, 1416 (1996).
  * Bhowmick  _et al._ (2014) B. Bhowmick, M. Bhattacharya, A. Bhattacharyya, and G. Gangopadhyay, Phys. Rev. C 89, 065806 (2014).
  * Fortin  _et al._ (2017) M. Fortin, S. S. Avancini, C. Providência, and I. Vidaña, Phys. Rev. C 95, 065803 (2017).
  * Kumar  _et al._ (2017b) B. Kumar, S. K. Biswal, and S. K. Patra, Phys. Rev. C 95, 015801 (2017b).
  * Bhuyan  _et al._ (2017) M. Bhuyan, B. V. Carlson, S. K. Patra, and S.-G. Zhou, IJMP E 26, 1750052 (2017).
  * Biswal (2019) S. K. Biswal, AIP Conf. Proceedings 2127, 020031 (2019).
  * Biswal  _et al._ (2019) S. K. Biswal, S. K. Patra, and S.-G. Zhou, APJ 885, 25 (2019).
  * Pal  _et al._ (2000) S. Pal, D. Bandyopadhyay, and W. Greiner, Nucl. Phys. A 674, 553–577 (2000).
  * Gupta and Arumugam (2012) N. Gupta and P. Arumugam, Phys. Rev. C 85, 015804 (2012).
  * Das  _et al._ (2019) A. Das, T. Malik, and A. C. Nayak, Phys. Rev. D 99, 043016 (2019).
  * Das  _et al._ (2020) H. C. Das, A. Kumar, B. Kumar, S. K. Biswal, T. Nakatsukasa, A. Li, and S. K. Patra, Monthly Notices of the Royal Astronomical Society 495, 4893 (2020), https://academic.oup.com/mnras/article-pdf/495/4/4893/33383648/staa1435.pdf .
  * Das  _et al._ (2021b) H. Das, A. Kumar, B. Kumar, S. Biswal, and S. Patra, Journal of Cosmology and Astroparticle Physics 2021, 007 (2021b).
  * Das  _et al._ (2021c) H. C. Das, A. Kumar, and S. K. Patra, MNRAS 507, 4053 (2021c).
  * Das  _et al._ (2021d) H. C. Das, A. Kumar, and S. K. Patra, Phys. Rev. D 104, 063028 (2021d).
  * Tolman (1939) R. C. Tolman, Phys. Rev. 55, 364 (1939).
  * Oppenheimer and Volkoff (1939) J. R. Oppenheimer and G. M. Volkoff, Phys. Rev. 55, 374 (1939).
  * Stergioulas (2003) N. Stergioulas, Living Rev. Relativ. 6, 3 (2003).
  * Jha  _et al._ (2008) T. K. Jha, H. Mishra, and V. Sreekanth, Phys. Rev. C 77, 045801 (2008).
  * Sharma and Jha (2009) B. K. Sharma and T. K. Jha, (2009), arXiv:0905.1549 [nucl-th] .
  * Friedman and Stergioulas (2013) J. L. Friedman and N. Stergioulas,  _Rotating Relativistic Stars_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2013).
  * Paschalidis and Stergioulas (2017) V. Paschalidis and N. Stergioulas, Living Rev. Relativ. 20, 7 (2017).
  * Quddus  _et al._ (2020b) A. Quddus, G. Panotopoulos, B. Kumar, S. Ahmad, and S. K. Patra, Journal of Physics G: Nuclear and Particle Physics 47, 095202 (2020b).
  * Koliogiannis and Moustakidis (2020) P. S. Koliogiannis and C. C. Moustakidis, Phys. Rev. C 101, 015805 (2020).
  * Patra and Praharaj (1992) S. K. Patra and C. R. Praharaj, Europhysics Letters (EPL) 20, 87 (1992).
  * Satpathy and Patra (2004) L. Satpathy and S. K. Patra, Journal of Physics G: Nuclear and Particle Physics 30, 771 (2004).
  * Riley  _et al._ (2019) T. E. Riley, A. L. Watts, S. Bogdanov, P. S. Ray, R. M. Ludlam, S. Guillot, Z. Arzoumanian, C. L. Baker, A. V. Bilous, D. Chakrabarty, K. C. Gendreau, A. K. Harding, W. C. G. Ho, J. M. Lattimer, S. M. Morsink, and T. E. Strohmayer, The Astrophysical Journal 887, L21 (2019).
  * Chen  _et al._ (2007) L.-W. Chen, C. M. Ko, and B.-A. Li, Phys. Rev. C 76, 054316 (2007).
  * Centelles  _et al._ (2009) M. Centelles, X. Roca-Maza, X. Viñas, and M. Warda, Phys. Rev. Lett. 102, 122502 (2009).
  * Bednarek  _et al._ (2020) I. Bednarek, J. Sładkowski, and J. Syska, Symmetry 12 (2020), 10.3390/sym12060898.
  * Lattimer (2015) J. M. Lattimer, in  _Exotic Nuclei and Nuclear/Particle Astrophysics (V) From Nuclei to Stars: Carpathian Summer School of Physics 2014_, American Institute of Physics Conference Series, Vol. 1645 (2015) pp. 61–78.
  * Lattimer and Lim (2013) J. M. Lattimer and Y. Lim, The Astrophysical Journal 771, 51 (2013).