# Constraining the slow-diffusion zone size and electron injection spectral
index for the Geminga pulsar halo

Kun Fang<EMAIL_ADDRESS>Key Laboratory of Particle Astrophysics,
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China

###### Abstract

Measuring the electron diffusion coefficient is the most straightforward task
in the study of gamma-ray pulsar halos. The updated measurements of the
spatial morphology and spectrum of the Geminga halo by the HAWC experiment
enable us to constrain parameters beyond the diffusion coefficient, including
the size of the slow-diffusion zone and the electron injection spectrum from
the pulsar wind nebulae (PWN). Based on the two-zone diffusion model, we find
that the slow-diffusion zone size ($r_{*}$) around Geminga is within the range
of $30-70$ pc. The lower boundary of this range is determined by the goodness
of fit of the model to the one-dimensional morphology of the Geminga halo. The
upper limit is derived from fitting the gamma-ray spectrum of the Geminga
halo, along with the expectations for the power-law index of the injection
spectrum based on simulations and PWN observations, i.e., $p\gtrsim 1$. With
$r_{*}$ set at its lower limit of $30$ pc, we obtain the maximum $p$ permitted
by the HAWC spectrum measurement, with an upper limit of $2.17$ at a $3\sigma$
significance. Moreover, we find that when $r_{*}=30$ pc and $p=2.17$, the
predicted positron spectrum generated by Geminga at Earth coincides with the
AMS-02 measurement in the $50-500$ GeV range.

##  I Introduction

According to the standard evolution model, pulsar wind nebulae (PWNe)
associated with middle-aged pulsars ($t\sim 100$ kyr) are typically in the
bow-shock stage Gaensler and Slane (2006), during which they traverse the
interstellar medium (ISM) while maintaining a compact size of $\lesssim 1$ pc.
High-energy electrons and positrons111Electrons will denote both electrons and
positrons hereafter if not specified. that escape from the PWNe and diffuse
into the ISM generate gamma-ray halos around their respective pulsars via
inverse Compton (IC) scattering of background photons. These phenomena are
referred to as pulsar halos Sudoh  _et al._ (2019); Fang (2022); Liu (2022);
López-Coto  _et al._ (2022). Since the morphology of pulsar halos traces the
spatial distribution of their parent electrons, they serve as ideal indicators
for investigating cosmic-ray propagation in localized regions of the Galaxy
Evoli  _et al._ (2018); Fang  _et al._ (2019a); Liu  _et al._ (2019); Wang
_et al._ (2021); Fang  _et al._ (2021a). Furthermore, the gamma-ray spectrum
of pulsar halos is determined by the energy spectrum of the escaped electrons
rather than those remaining confined within the PWNe, thus providing unique
insight into the process of electron escape from the PWNe Fang and Bi (2022);
Fang  _et al._ (2022).

The diffusion coefficient inferred from the morphology of the identified
pulsar halos is about two orders of magnitude smaller than the Galactic
average Abeysekara  _et al._ (2017); Aharonian  _et al._ (2021); Fang  _et
al._ (2022); Albert  _et al._ (2023a), which also ensures that these halos are
bright enough to be detected by current experiments. The slow-diffusion
phenomenon around the pulsars has a crucial impact on interpreting issues such
as the cosmic-ray positron excess Hooper  _et al._ (2017); Fang  _et al._
(2018); Tang and Piran (2019); Xi  _et al._ (2019); Fang  _et al._ (2019b);
Manconi  _et al._ (2020) and the diffuse TeV gamma-ray excess Linden and
Buckman (2018); Dekker  _et al._ (2023); Yan  _et al._ (2023). To accurately
address these issues, comprehension of the slow-diffusion mechanism is
essential. This slow diffusion could be attributed to a turbulent environment
generated by the host supernova remnants of the pulsars Fang  _et al._ (2019a)
or by the escaped electrons themselves Evoli  _et al._ (2018); Mukhopadhyay
and Linden (2022). Alternatively, it may be explained by the projection
effects under anisotropic diffusion without needing an additional turbulent
magnetic field Liu  _et al._ (2019); De La Torre Luque  _et al._ (2022); Fang
_et al._ (2023).

A definitive conclusion on the slow-diffusion mechanism remains elusive.
Nonetheless, we can still investigate the physical parameters associated with
pulsar halos phenomenologically. Recently, the High-Altitude Water Cherenkov
(HAWC) experiment provided updated observations of a prototypical pulsar halo,
namely the Geminga halo Albert  _et al._ (2023b). Compared to the initial work
on the Geminga halo Abeysekara  _et al._ (2017), the precision of the spatial
morphology measurement has been significantly enhanced, and the gamma-ray flux
has been accurately measured in distinct energy ranges. We will show that
these improvements enable us to provide valuable constraints on the size of
the slow-diffusion zone and the form of the PWN electron injection spectrum,
thereby broadening our scope beyond simple diffusion coefficient measurements.

In this study, we ground our calculation in the two-zone diffusion model,
where the slow-diffusion zone is confined to a specific vicinity around the
pulsar. This phenomenological model could appropriately characterize the
electron propagation associated with pulsar halos. In Sec. II, we present the
calculation of the morphology and energy spectrum of the Geminga halo. In Sec.
III, we fit the latest one-dimensional morphology measured by HAWC and give a
lower limit for the slow-diffusion zone size using the goodness-of-fit test.
In Sec. IV, we fit the gamma-ray energy spectrum of Geminga. The HAWC
measurement shows a very hard spectrum in $2-20$ TeV. We demonstrate that this
feature can be interpreted by a reasonable electron injection spectrum under
the two-zone diffusion model. We provide constraints on the index of the
power-law term of the injection spectrum and also estimate the maximum slow-
diffusion zone size. Based on these parameter constraints, Sec. V revisits the
potential for Geminga to account for the cosmic-ray positron excess. Sec. VI
is the conclusion.

##  II Two-zone Diffusion model

The magnetic field turbulence downstream of the SNR shock could be
significantly enhanced compared to the far upstream, resulting in a downstream
diffusion coefficient two orders of magnitude smaller than the ISM Fang  _et
al._ (2019a); Wang  _et al._ (2022). This suggests that the slow-diffusion
environment can be understood if the pulsar remains within the downstream
region of its corresponding SNR Fang  _et al._ (2019a). Among the observed
pulsar halos, the pulsar associated with the Monogem halo indeed resides
within the downstream region of its corresponding SNR, which has been strongly
supported by pulsar scintillation observations Yao  _et al._ (2022). Given the
scale of SNRs, the slow-diffusion zone around pulsars may span several tens of
parsecs.

Alternatively, if the slow diffusion is attributed to the escaped electrons
from PWNe amplifying the surrounding magnetic turbulence via streaming
instability, the diffusion coefficient could only be significantly reduced in
the vicinity of pulsars as the growth rate of turbulence is directly
proportional to the electron number density gradient Evoli  _et al._ (2018).
Therefore, a two-zone diffusion model with a slow-diffusion zone spanning
several tens of parsecs is a plausible approximation. The current paper does
not discuss the scenario of interpreting pulsar halos with anisotropic
diffusion.

High-energy electrons are accelerated by PWNe and subsequently released into
the ISM. The electron propagation in the ISM could be described by the
diffusion-loss equation as

$\frac{\partial n(E_{e},\textbf{\em r},t)}{\partial
t}=\nabla\cdot[D(E_{e},\textbf{\em r})\nabla n(E_{e},\textbf{\em
r},t)]+\frac{\partial[b(E_{e})n(E_{e},\textbf{\em r},t)]}{\partial
E_{e}}+Q(E_{e},\textbf{\em r},t)\,,$ (1)

where $n\equiv dN/dE_{e}$ is the differential electron number density, and
$E_{e}$ is the electron energy. The diffusion coefficient in the two-zone
diffusion model takes the form of

$D(E_{e},\textbf{\em r})=\left\\{\begin{aligned} &D_{100}(E_{e}/{\rm
100~{}TeV})^{\delta},\quad&|\textbf{\em r}-\textbf{\em r}_{p}|\leq r_{*}\\\
&D_{100,{\rm ism}}(E_{e}/{\rm 100~{}TeV})^{\delta_{\rm
ism}},\quad&|\textbf{\em r}-\textbf{\em r}_{p}|>r_{*}\\\
\end{aligned}\right.~{},$ (2)

where $\textbf{\em r}_{p}$ is the location of the pulsar, $r_{*}$ is the size
of the slow-diffusion zone, $D_{100}$ and $\delta$ are diffusion parameters
for the slow-diffusion zone, and $D_{100,{\rm ism}}$ and $\delta_{\rm ism}$
are those for the typical ISM. Considering the energy range of the HAWC
measurement, we set the reference energy for the diffusion coefficient at
$100$ TeV. This ensures that $D_{100}$ is minimally affected by the selection
of $\delta$. We adopt $D_{0,{\rm ism}}=3.4\times 10^{30}$ cm s-1 and
$\delta_{\rm ism}=0.38$ as inferred from the cosmic-ray boron-to-carbon ratio
Yuan  _et al._ (2017).

The second term on the right-hand side of Eq. (1) is the energy-loss term,
where $b(E_{e})\equiv|dE_{e}/dt|$ is the electron energy-loss rate. We take
the magnetic field strength of $B=3$ $\mu$G for the synchrotron loss rate. For
the IC scattering, we adopt the seed photon field given in Ref. Abeysekara
_et al._ (2017) and the parametrization method given in Ref. Fang  _et al._
(2021b) to calculate the energy-loss rate.

The third term on the right-hand side of Eq. (1) is the source term. Assuming
the time profile of the source function follows the pulsar spin-down
luminosity, it takes the form of

$Q(E_{e},\textbf{\em r},t)=\left\\{\begin{aligned} &q(E_{e})\,[(t_{p}+t_{\rm
sd})/(t+t_{\rm sd})]^{2}\,\delta(\textbf{\em r}-\textbf{\em r}_{p})\,,&t\geq
0\\\ &0\,,&t<0\end{aligned}\right.\,,$ (3)

where $q(E_{e})$ is the current electron injection spectrum, $t_{p}$ is the
characteristic age of the pulsar, $t_{\rm sd}$ is the pulsar spin-down time
scale, and $t=0$ corresponds to the birth time of the pulsar. The
characteristic age of Geminga is $t_{p}=342$ kyr as given by the Australia
Telescope Nation Facility (ATNF) pulsar catalog Manchester  _et al._ (2005),
and $t_{\rm sd}$ is set to be $10$ kyr.

The injection energy spectrum is assumed to be a power law with a super-
exponential cutoff as

$q(E_{e})\propto E_{e}^{-p}\,{\rm
exp}\left[-\left(\frac{E_{e}}{E_{c}}\right)^{2}\right]\,,$ (4)

where the form of the cutoff term is suggested by the mechanism of the
relativistic shock acceleration Dempsey and Duffy (2007).

We numerically solve Eq. (1) using the finite volume method Fang  _et al._
(2018). A semi-analytical method is also applicable for two-zone diffusion
Osipov  _et al._ (2020). We integrate the resulting electron number density
$n$ over the line of sight from Earth to the vicinity of the pulsar to obtain
the electron surface density. This allows us to derive the one-dimensional
gamma-ray morphology of the halo, $s(\theta,E_{\gamma})$, using the standard
calculation of IC scattering Blumenthal and Gould (1970), where $\theta$ is
the angle away from the pulsar. The gamma-ray spectrum within an arbitrary
angular radius around the pulsar is then calculated by

$\frac{dF}{dE_{\gamma}}(\theta<\theta_{0},E_{\gamma})=\int_{0}^{\theta_{0}}s(\theta,E_{\gamma})2\pi\theta
d\theta~{}.$ (5)

The distance of the Geminga pulsar to the Earth takes $250$ pc as measured by
trigonometric parallax Faherty  _et al._ (2007).

##  III Lower limit of the slow-diffusion zone size

The one-dimensional gamma-ray morphology of the Geminga halo provided by HAWC,
also known as the surface brightness profile (SBP), exhibits a median energy
of $\approx 20$ TeV Albert  _et al._ (2023b). This corresponds to a parent
electron energy of $\approx 100$ TeV Abeysekara  _et al._ (2017). Therefore,
the shape of the SBP is predominantly determined by the diffusion coefficient
at $100$ TeV and is minimally influenced by parameters describing energy
dependence, such as $\delta$ or $p$. In the fitting process of this section,
we consider $D_{100}$ and the normalization as free parameters. The values of
$\delta$ and $p$ remain fixed as in the initial HAWC paper Abeysekara  _et
al._ (2017), and we find that variations in these parameters have negligible
impact on the fitting results. To compare with the HAWC data, we also convolve
a point spread function with a $68\%$ containment radius of $0.3^{\circ}$ on
the model.

The SBP centered on Geminga provided by HAWC includes contributions from both
the Geminga and Monogem halos. The component of the Monogem halo is determined
using the SBP measurement center on Monogem, where only the data within a
radius of $3^{\circ}$ is considered to avoid significant influence from the
Geminga halo at larger angles. The Monogem halo is characterized using a one-
zone diffusion model, yielding $D_{100}=4.6\times 10^{27}$ cm s-1. We fix the
Monogem halo component in the following fitting process.

Figure 1: The fitting results for the Geminga halo SBP measured by HAWC
assuming different slow-diffusion zone sizes ($r_{*}$). The left panel shows
the minimum reduced chi-square statistic with 16 DOF. The right panel depicts
the corresponding diffusion coefficient of the slow-diffusion zone at the
energy of $100$ TeV.

We assume varying sizes for the slow-diffusion zone and fit these models to
the SBP data centered on Geminga. The minimum reduced chi-square statistic
$\chi^{2}$ for each $r_{*}$ is illustrated in the left panel of Fig. 1. With
$16$ degrees of freedom (DOF), the $95\%$ confidence bound for the reduced
$\chi^{2}$ is $1.64$, as marked by the dashed line in the figure. It can be
seen that models with $r_{*}\lesssim 30$ pc are excluded at the $95\%$
confidence level by the chi-square test. The chi-square statistic is minimized
at $\approx 1.0$ when $r_{*}=35$ pc. Beyond $r_{*}\gtrsim 50$ pc, there is no
significant change in the goodness of fit. The corresponding $D_{100}$ for
each $r_{*}$ is depicted in the right panel of Fig. 1. When $r_{*}\gtrsim 40$
pc, the best-fit $D_{100}$ tends to be $\approx 4.5\times 10^{27}$ cm2 s-1,
consistent with that given by the HAWC paper. For models with $r_{*}<40$ pc, a
larger $D_{100}$ is required to widen the SBP to match the data.

Figure 2: Best-fit models for the SBP centered on Geminga under different
assumptions of the slow-diffusion zone size. This figure serves to explain the
exclusion of models with a small $r_{*}$ intuitively. The case of $r_{*}=35$
pc provides the best fit among all models, with $\chi^{2}/{\rm DOF}=0.93$, as
shown in the left panel of Fig. 1.

Fig. 2 provides an intuitive explanation for excluding models with a small
$r_{*}$. In the two-zone diffusion model, electrons disperse rapidly once they
leave the slow-diffusion zone, resulting in a significant flux reduction in
the fast-diffusion zone compared to the slow-diffusion zone. The flux of the
fast-diffusion zone is so low that it is difficult to distinguish them from
the background. As a result, the SBP of the pulsar halo under two-zone
diffusion appears to be more contracted if $r_{*}$ is smaller, which accounts
for the trend shown in the right panel of Fig. 1. If $r_{*}$ is too small, a
significant reduction in flux will occur at a location too close to the
pulsar. For instance, in the case of $r_{*}=20$ pc depicted in Fig. 2, the
model significantly underestimates the data in the angular range of $\approx
2^{\circ}$ to $6^{\circ}$ due to the transition from the slow- to fast-
diffusion zone. This results in a very poor goodness of fit that cannot be
rectified by assuming a larger $D_{100}$.

On the other hand, the current SBP measurement does not constrain the upper
limit of $r_{*}$. For Geminga, an angular distance of $10^{\circ}$ corresponds
to a transverse spatial distance of $\approx 45$ pc. This means that when
$r_{*}$ exceeds $45$ pc, the variation in SBP within $10^{\circ}$ predicted by
different models is insignificant. However, as we will show in the next
section, the gamma-ray energy spectrum could enable us to estimate the upper
limit of $r_{*}$.

Although HAWC also provides the preliminary SBP centered on Monogem, since the
Monogem halo is significantly dimmer than the Geminga halo, the SBP is heavily
influenced by the Geminga halo at large angles. The Geminga halo may display
asymmetry in its morphology Chen  _et al._ (2023), but in the absence of two-
dimensional data, we cannot estimate the contribution of the Geminga halo to
this SBP accurately. Therefore, we do not investigate the slow-diffusion zone
size corresponding to the Monogem halo in this study.

##  IV Constraints of gamma-ray spectrum on parameters

The HAWC spectrum measurement of the Geminga halo ranging from $\approx 1-100$
TeV indicates that the electron injection spectrum is consistent with the form
presented in Eq. (4), with $E_{c}\approx 100$ TeV. Nevertheless, the gamma-ray
spectrum from $2$ to $20$ TeV exhibits a hard nature, indicating the necessity
for a hard power-law term in the injection spectrum. However, under one-zone
diffusion, even though a very hard injection spectrum of $p\approx 1$ is
assumed, the model still does not fit the observed gamma-ray spectrum well
Albert  _et al._ (2023a).

Assuming the one-zone diffusion model with $D_{100}=5\times 10^{27}$ cm2 s-1
and $\delta=1/3$, as used in the HAWC work, the gamma-ray flux within
$10^{\circ}$ around the pulsar constitutes $\approx 98\%$ of the total angle-
integrated spectrum. Consequently, the spectral measurement provided by HAWC
approximately represents the integrated gamma-ray spectrum within
$\theta=10^{\circ}$.

Figure 3: This figure illustrates the impact of the energy index of the
diffusion coefficient ($\delta$) and the size of the slow-diffusion zone
($r_{*}$) on the angle-integrated gamma-ray spectrum within $10^{\circ}$
around the Geminga pulsar. The electron injection spectrum is assumed to be a
power law with $p=2.2$. The gamma-ray spectral softening at high energies is
due to the Klein-Nishina effect.

The integrated gamma-ray spectrum within a specific angle $\theta_{0}$ is
determined not only by the electron injection spectrum but also by $\delta$
and $r_{*}$, as illustrated in Fig. 3, where $\theta_{0}=10^{\circ}$. Given
that $D_{100}$ is constrained by the SBP data, a smaller $\delta$ corresponds
to faster diffusion of low-energy electrons, making them more likely to escape
from $\theta_{0}$ and consequently yielding a harder gamma-ray spectrum.
Moreover, within the energy range of interest, the characteristic diffusion
distance of low-energy electrons is larger than that of high-energy electrons.
This implies that a smaller $r_{*}$ results in a higher proportion of low-
energy electrons escaping from $\theta_{0}$, thereby producing a harder gamma-
ray spectrum. As can be seen from Fig. 3, the impact of $r_{*}$ on the
spectral shape is significant. Therefore, under the two-zone diffusion model,
a very hard injection spectrum might not be necessary. This point has also
been demonstrated in our previous studies Fang  _et al._ (2021a, 2022).

The analysis in Sec. III indicates that $r_{*}=30$ pc may serve as the lower
limit for the slow-diffusion size, and we typically regard $\delta=0$ as the
lower limit for the energy index of the diffusion coefficient. With this set
of parameters, we explore the maximum permissible injection spectral index
derived from the observation. We fit the integrated gamma-ray spectrum
obtained from the model within $\theta=10^{\circ}$ to the spectral
measurements of the Geminga halo by HAWC (including the flux upper limits with
$95\%$ confidence), with $p$, $E_{c}$, and the normalization as free
parameters. We impose constraints on $p$ using the likelihood ratio test. The
likelihood ratio is defined as $\lambda=-2\ln\left[L(p_{0})/L(p)\right]$,
where $L(p)$ is the global maximum likelihood, and $L(p_{0})$ is the local
maximum likelihood when the spectral index is fixed at $p_{0}$. According to
Wilks’ theorem, $\lambda$ follows the chi-square distribution with one DOF.
When $\lambda$ equals $4$ and $9$, we can respectively determine the $2\sigma$
and $3\sigma$ confidence bounds for $p$.

Figure 4: Fitting results of the Geminga halo gamma-ray spectrum by the two-
zone diffusion model. The model represents the angle-integrated spectrum
within $10^{\circ}$ around the pulsar, adopting the smallest $r_{*}$ and
$\delta$ to achieve the largest possible injection spectral index $p$. The
three models in the figure include the maximum likelihood model and models
using the $2\sigma$ and $3\sigma$ confidence upper limits of $p$. The best-fit
$E_{c}$ for these scenarios are all around $100$ TeV.

The maximum likelihood $p$ we obtained is $1.18$, and the upper limits at
$2\sigma$ and $3\sigma$ confidence levels are $1.85$ and $2.17$, respectively.
The fitting effects of these three scenarios on the data are illustrated in
Fig. 4. It can be seen that when $p$ is larger, the model tends to
overestimate the flux at low and high energy ranges while underestimating the
flux at the intermediate energy range. The best-fit $E_{c}$ in different
scenarios are all around $100$ TeV. For larger $\delta$ values, such as
$\delta=1/3$ or $\delta=1/2$, as predicted by Komolgorov’s or Kraichnan’s
theory, the resulting injection spectral index is correspondingly larger. We
have reiterated the fitting procedure and summarized the results in Table 1.

Table 1: Power-law index of the electron injection spectrum ($p$) resulted from the fitting of the HAWC gamma-ray spectrum, under the assumption that the slow-diffusion zone size is $30$ pc. Values lower than expectations, namely $p<1$, are denoted in italics. $\delta$ | $p$
---|---
maximum likelihood |  $2\sigma$ upper limit |  $3\sigma$ upper limit
$0$ | $1.19$ | $1.85$ | $2.17$
$1/3$ | 0.81 | $1.36$ | $1.82$
$1/2$ | 0.59 | $1.17$ | $1.65$

On the other hand, we fit the spectral data under the assumption of varying
$r_{*}$ values while maintaining $\delta=0$. The derived $3\sigma$ confidence
upper limits of $p$ from these fits are summarized in Table 2, indicating a
decrease in the upper limit of $p$ as $r_{*}$ increases. Numerical simulations
suggest that the spectral index of electrons within a bow-shock PWN could
approach $1$ at the colliding flow between the stellar-wind termination shock
and the bow shock Bykov  _et al._ (2017). However, existing theories or
observations do not support spectral indices to be smaller. Moreover, the
x-ray PWN of Geminga exhibits two components with distinct spectral indices,
with the harder and softer components corresponding to $p=1.08$ and $2.26$,
respectively Posselt  _et al._ (2017). As discussed in Ref. Fang and Bi
(2022), these components may indicate two disparate modes of electron escape,
suggesting that the total injection spectrum could be a composite of both
components. Consequently, for the Geminga halo, an injection spectral index of
$1$ can be regarded as the lower limit. This allows us to estimate the upper
limit of $r_{*}$. At $r_{*}=70$ pc, the upper limit of $p$ falls below $1$,
suggesting that models with $r_{*}\geq 70$ pc are disfavored. Combined with
the lower limit obtained in Sec. III, we are able to provide a comprehensive
constraint on the slow-diffusion zone size, which lies between $30$ and $70$
pc.

Table 2: $3\sigma$ confidence upper limits of the power-law index of the electron injection spectrum ($p$) resulted from fitting the HAWC gamma-ray spectrum, assuming $\delta=0$. Values lower than expectations, namely $p<1$, are denoted in italics. $r_{*}$ [pc] | $30$ | $40$ | $50$ | $60$ | $70$
---|---|---|---|---|---
UL for $p$ | $2.17$ | $1.60$ | $1.32$ | $1.03$ | 0.87

Recently, the x-ray observation by eROSITA provides an upper limit of $1.4$
$\mu$G for the ISM magnetic field within $1^{\circ}$ around the Geminga pulsar
Khokhriakova  _et al._ (2023). This is lower than the $3$ $\mu$G adopted in
the initial HAWC study and our above calculations. Thus, we also consider a
scenario with $B=1$ $\mu$G, where the synchrotron energy-loss rate for
electrons is significantly reduced. This change notably affects the lifetime
of higher-energy electrons, thereby influencing the estimation of the cutoff
term in the electron injection spectrum. When $B$ shifts from $3$ $\mu$G to
$1$ $\mu$G in the case of $r_{*}=30$ pc and $\delta=0$, the best-fit $E_{c}$
varies from $94$ TeV to $83$ TeV. However, this adjustment in magnetic field
strength has a negligible effect on $p$, with its $3\sigma$ confidence upper
limit merely changing from $2.17$ to $2.19$, leaving our conclusion
unaffected.

##  V Implication on the positron excess

The positron excess is one of the most intriguing phenomena in cosmic-ray
studies. As measured by space experiments Adriani  _et al._ (2009); Ackermann
_et al._ (2012); Aguilar  _et al._ (2013), the high-energy positron spectrum
significantly exceeds the predicted secondary positron flux produced by
cosmic-ray nuclei spallation during Galactic propagation. Among astrophysical
accelerators, pulsars (or their PWNe) are the most plausible sources of this
excess Hooper  _et al._ (2009); Yuksel  _et al._ (2009); Yin  _et al._ (2013),
given their capacity to generate high-energy electron/positron pairs.

The positron spectrum measured by AMS-02 reveals a high-energy cutoff at
$\approx 800$ GeV with a significance of $4.7\sigma$ Aguilar  _et al._ (2019);
Kounine (2023). As discussed in our previous studies, this cutoff is likely
attributed to the cooling effect of the positrons generated by a dominant
source rather than the superposition of multiple sources Fang  _et al._
(2019b). This is due to the fact that pulsars or PWNe are unlikely to share a
common acceleration limit at $\sim 1$ TeV. Among observed pulsars, the most
plausible dominant sources of the measured high-energy positrons are Geminga,
Monogem, and PSR B1055-52 Fang  _et al._ (2019b); Bykov  _et al._ (2019);
Martin  _et al._ (2022), all of which are nearby and middle-aged pulsars.

Within the framework of the two-zone diffusion model, the contribution of
nearby pulsars to the positron spectrum at Earth is primarily affected by
parameters such as the diffusion coefficient, the slow-diffusion zone size,
and the positron injection spectrum. We use parameters determined from fitting
the HAWC data (as detailed in Sec. III and IV) to predict the contribution of
Geminga to the positron spectrum at Earth. As shown in Fig. 2 of Ref. Fang
_et al._ (2019b), the positron spectrum produced by Geminga, under assumptions
of $r_{*}=50$ pc and $p=1.8$, is too hard compared to the AMS-02 measurement.
Reducing $r_{*}$ or increasing $p$ can yield a softer positron spectrum.
Intriguingly, as indicated in Table 2, an inverse correlation exists between
the required $p$ and $r_{*}$ when interpreting the gamma-ray spectrum of the
Geminga halo. Consequently, by setting $r_{*}$ to its lower limit of $30$ pc,
we can achieve the softest possible positron spectrum at Earth. Furthermore, a
smaller $\delta$ corresponds to faster diffusion of low-energy positrons,
enabling more low-energy positrons to reach Earth and further soften the
spectrum.

Figure 5: The predicted positron spectrum from Geminga under different
assumptions of $r_{*}$ and $\delta$, along with the corresponding $D_{0}$,
$p$, and normalization determined by the fits in Sec. III and IV, where $p$ is
adopted as the $3\sigma$ upper limits as presented in Table 1 and 2. The
scenario with $r_{*}=30$ pc and $\delta=0$ provides the most optimistic
prediction to the AMS-02 measurements among all the models consistent with the
HAWC measurements of the Geminga halo.

As an optimistic estimate, we adopt $r_{*}=30$ pc and $\delta=0$, along with
the upper limit of the injection spectrum index $p=2.17$ and the corresponding
best-fit normalization derived in Sec. IV. The predicted positron energy
spectrum, $I(E_{e})=n(E_{e})c/4\pi$, depicted by the red solid line in Fig. 5,
is consistent with the measurements of both absolute flux and spectral shape
below $\approx 500$ GeV. However, the predicted spectral cutoff occurs at a
higher energy than observed. One possible explanation is that the positrons
traverse regions with higher magnetic field strength than assumed, resulting
in a spectral cutoff at lower energies. Alternatively, an older pulsar, such
as PSR B1055-52, could be the main contributor to the high-energy positrons
instead of Geminga. Fig. 5 also shows two additional scenarios with a larger
$r_{*}$ or a larger $\delta$, where $p$ is set as the corresponding $3\sigma$
confidence limits presented in Tables 1 and 2. Under these conditions, the
positron flux produced by Geminga is insufficient to interpret the
observation.

It should be noted that the calculations above are based on extrapolating the
parameters derived by the HAWC measurements to lower energies. As the positron
excess primarily occurs between $10$ GeV to $1$ TeV, the GeV gamma-ray
measurements of the Geminga halo would impose more direct constraints on the
positron injection spectrum and the diffusion coefficient within this energy
range Xi  _et al._ (2019); Di Mauro  _et al._ (2019); Zhou  _et al._ (2022).
However, there is no definitive conclusion on the GeV measurements of the
Geminga halo at present.

##  VI Conclusion

Based on the latest SBP and gamma-ray spectrum measurements of the Geminga
halo by HAWC, we have studied the slow-diffusion zone size ($r_{*}$) around
Geminga and the parameters of the electron injection spectrum. Current
theories expect slow diffusion to be confined within tens of parsecs around
Geminga, leading us to employ a two-zone diffusion model to characterize
electron propagation. Our results suggest that $r_{*}$ is within the range of
$30-70$ pc. The lower end of this range is constrained by the goodness of fit
when fitting the SBP with the model. Models with $r_{*}\lesssim 30$ pc are
disfavored by the chi-square test with a confidence level higher than $95\%$.
The SBP derived from these models declines too quickly with increasing
$\theta$ compared to the data, owing to a significant drop in the electron
number density beyond $r_{*}$.

The upper limit of $r_{*}$ is estimated by fitting the gamma-ray spectrum of
the Geminga halo, along with the expectations for the power-law index ($p$) of
the electron injection spectrum based on simulations and PWN observations,
where $p\gtrsim 1$. Under the two-zone diffusion model, the derived $r_{*}$
exhibits an inverse correlation with $p$ when fitting the angle-integrated
gamma spectrum within $10^{\circ}$ around the pulsar. For $r_{*}\gtrsim 70$
pc, the upper limit of $p$ at a $3\sigma$ confidence level falls below $1$,
implying that those models are disfavored.

Conversely, when $r_{*}$ is set to its lower limit of $30$ pc, we obtain the
maximum $p$ permitted by the HAWC spectrum measurement, with an upper limit of
$2.17$ at a $3\sigma$ confidence level. We also find that when $r_{*}=30$ pc
and $p=2.17$, the positron flux generated by Geminga at Earth fits well with
the positron spectrum measured by AMS-02 in the $50-500$ GeV range, in terms
of both the spectral shape and the absolute flux. Notably, the upper limit of
$p$ coincides with that required to interpret the diffuse gamma-ray excess
using gamma-ray pulsar halos, where $p\approx 2.2$ Yan  _et al._ (2023).
Besides, the cutoff energy of the electron injection spectrum is determined to
be $\approx 100$ TeV.

In interpreting the gamma-ray spectrum of the halo, the energy index of the
diffusion coefficient ($\delta$) also exhibits degeneracy with $p$ and
$r_{*}$. We have considered several typical values for $\delta$ ($0$, $1/3$,
and $1/2$). The above upper limits for $r_{*}$ and $p$ are derived under the
condition of $\delta=0$. If $\delta$ increases, these constraints would be
stricter. For example, when $\delta=1/2$, the upper limit for $p$ would
decrease to $1.65$. In the near future, the Large High Altitude Air Shower
Observatory (LHAASO) is expected to provide energy-dependent morphology
measurements for the Geminga halo, which could break the degeneracy among the
parameters.

###### Acknowledgements.

The author would like to thank H. Zhou and R. Torres for sharing the energy
range of the Geminga halo morphology measurements and the point spread
function size of HAWC. This work is supported by the National Natural Science
Foundation of China under grants No. 12105292 and No. U2031110.

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# Overview and Analysis of Seismic Resilient Structures with Modular Rocking
Shear Walls

Mehrdad Aghagholizadeh<EMAIL_ADDRESS>[ Assistant Professor,
Department of Civil and Environmental Engineering, Loyola Marymount University
Los Angeles, CA 90045

###### Abstract

The high occupancy rates in urban multi-story buildings, combined with present
safety concerns, necessarily prompt a reassessment of performance goals. Given
the notable seismic damage and instances of weak-story failures that have been
documented after major earthquakes, this paper studies the use of modular
shear walls that are free to rock above their foundation. This paper first
provides a comprehensive background in analysis of rocking elements such as
columns and shear-walls. Then discusses different configurations of rocking-
shear-walls. Next, the paper provides two numerical case studies on 9-story
and 20-story moment-resisting frames using OpenSees. The floor displacement
and interstory drifts under various earthquake excitations for both structures
compared for the cases of with and without modular rocking walls. The result
shows that the addition of rocking-shear-wall, makes the first mode of the
frame becomes dominant which enforces a uniform distribution of interstory
drifts that would avoid a soft-story failure.

###### keywords:

rocking wall, seismic resiliency , finite element analysis , earthquake
response modification , modular structures

††journal: Journal of Engineering Structures

link]https://sehm.lmu.build/

##  1 Introduction

To address significant seismic damage in moment-resisting frames, which
sometimes led to weak-story failures, the idea of a rigid core system became
increasingly popular. [1, 2, 3, 4, 5]. Presence of large axial loads on the
shear walls in a tall building reduces its ductility significantly; while the
ductility demands are appreciable under long-duration pulse motions [6, 7].
Moreover, the base of the core wall may suffer from cyclic degradation under
prolonged shaking which usually results to permanent inelastic deformations.
Such inelastic response may result to permanent drifts and lead to large
repair costs; therefore, the entire design becomes unsustainable. An example
of such failure after an earthquake is shown in Figure 1 (left) for a
fourteen-story moment-resisting frame with a fixed shear-wall building during
1964 Anchorage, Alaska earthquake [8].

Figure 1: Left: A fourteen-story reinforced concrete apartment building in
Anchorage, Alaska, was severely damaged during the 1964 Alaska earthquake. One
of the main exterior shear walls, shown in this figure, failed at the second
floor, exposing the steel reinforced bars within the concrete (Image courtesy
of the U.S. Geological Survey, https://library.usgs.gov/ [9]). Right:
Schematic of the first mode deformation of a tall moment-resisting-frame with
a rocking-shear-wall.

After 1994 Northridge, California earthquake followed by 1995 Kobe, Japan
earthquake, coherent acceleration pulses (0.8-1.5 sec duration at that time)
which result in large monotonic velocity, received revived attention. Makris
(1996) [10], Alavi and Krawinkler (2004) [11] studied the destructive
potential of pulse-like ground motions recorded near the causative fault of
earthquakes. In particular, several tall moment-resisting frames that had been
designed in accordance with the existing seismic-code provisions exhibited a
weak-story failure–in some cases several stories above the ground (see Figure
2).

Figure 2: Weak-story failure at the higher stories of the buildings after the
1995 Kobe, Japan Earthquake. (Image courtesy of the National Oceanic and
Atmospheric Administration (NOAA) [8]).

Over the past few decades, there has been an increasing effort to highlight
the unique advantages with allowing main vertical structural elements (such as
piers in bridges or shear wall in buildings) to uplift and rotate above their
foundation to mobilize a lower failure mechanism by design. The advantages of
this approach is that the failures associated with cyclic degradation are
essentially avoided; while, permanent displacements remain small due to the
inherent recentering tendency of the rocking mechanism [12, 13, 14, 15, 16,
17, 18, 19, 20].

Furthermore, with advances in construction methods and need of higher quality
materials in complex engineering projects, interest in use of prefabricated
structural elements is increasing. The advantages of this type of construction
can be summarized as, higher quality structural members, decreasing the cost
of (on-site) construction, and at the same time reducing construction time
[21, 22, 23]. Hence, use of vertical structural members that are not fixed at
the their foundation have a great potential in prefabricated construction
method.

Part of the reason for this articulated or semi-articulated seismic design
alternative is that on several occasions, the further strengthening of the
building with fixed-based shear walls leads to the attraction of larger
seismic forces and the entire approach reaches an impasse given that the
resulting forces that develop cannot be accommodated by cost-effective
foundations [20]. Another major issue that is a concern in multistory
buildings in which their earthquake performance relies on ductile behavior is
that after severe shaking the multi-story building may end up with appreciable
permanent displacements and there is a need for re-centering which in most
cases leads to demolition (an outcome against the emerging trends of
functional recovery) as happened after the 2011 Christchurch, New Zealand
earthquake [24].

This paper first provides a comprehensive literature review of rocking
elements such as columns and shear walls. Then discusses different
configurations of rocking shear walls. Lastly, the paper provides a numerical
study for two steel moment-resisting frames (9 and 20 story) when they are
coupled with a modular rocking shear wall under various seismic loading and
compares their floor displacements and story drifts.

##  2 Overview of the Literature

Articulated construction of structural members dates back to ancient Greek and
Roman buildings. Examples of such buildings are Acropolis of Athens, columns
of remaining structures in the Roman Forum, and columns of the Pantheon in
Rome (shown in the Figure 3). The articulated construction of columns in these
ancient buildings are either comprise of a monolithic column or multi-block
columns [25]. What makes these ancient articulated construction interesting is
that these structures survived several seismic motions during their life
cycle. The seismic performance of these structures are caused by their unique
properties, that is; re-centering is accomplished by their self-weight,
negative stiffness nature of the rocking blocks, and the large size of the
structures develops enough resistance to overturning mobilizing its rotational
inertia [26].

Figure 3: Left: A view of columns in the Roman Forum, Rome, Italy. Right: The
monolithic columns of Pantheon’s portico in Rome, Italy.

In contemporary structural engineering, the concept of allowing a tall,
slender structure to uplift and rock was first advanced and implemented in
modern civil engineering in the late 1960s in New Zealand with the design and
construction of the stepping piers of the South Rangitikei bridge [27, 13, 28,
29]. This distinctive design was developed out of necessity because the piers
of the South Rangitikei Bridge were over 75 meters tall, resulting in
excessively large overturning moments at the pier foundations when using a
conventional capacity design. The design of the South Rangitikei Bridge was
developed by the Physics and Engineering Laboratory of Department of
Scientific and Industrial Research (DSIR) in New Zealand [27, 13, 28, 29, 30].
Their efforts resulted to the development of the torsionally yielding steel
dampers that was used to enhance the energy dissipation of the base of the
stepping piers of the South Rangitikei Rail Bridge [12, 31, 32].

Another early investigation into the concept of shear walls capable of
uplifting and rocking was conducted by Meek in 1978 [33]. Inspired by the
seminal work on rocking blocks by Housner (1963) [34] this study used a
simplified analysis of the core rocking wall and a frame when the wall and
footing rock on the soil. This study demonstrated that rocking walls
significantly reduce the base shear and moment at the base of the wall
compared to conventional fixed-base shear walls. Around the same time Clough
and Huckelbridge explored the idea of steel frames that were free to uplift
and rock above their foundation with shake table tests [35]. The test consists
of a three-story single-bay steel frame with and without column uplift. The
rocking steel frame of this study was pinned at the bottom, allowing free body
rotation when the column has been uplifted. The experiment revealed that the
presence of rocking reduced the seismic demand relative to a fixed base frame.

Despite the remarkable originality of these early works and their technical
merit, these papers did not receive the attention it deserved, and it was some
two decades later that the PRESSS (PREcast Seismic Structural Systems) Program
[14, 36] reintroduced the concept of uplifting and rocking of the joint shear
wall system [37, 38].

Kurama et al. (1999, 2002) investigated behavior of unbonded post-tensioned
precast concrete walls [15, 39, 40]. These studies propose a design method
based on an idealized trilinear relationship between base shear and roof
drift. The trilinear relationship comprises four stages: the decompression
state (when the wall begins to uplift), the softening state (the linear limit
determined either by the gap opening of the walls or the nonlinear behavior of
concrete under compression), the yielding state (where the strain in the post-
tensioning steel first reaches the linear limit), and the failure state (where
the wall ultimately fails). This study [15] validates the analytical model
using test results from the National Institute of Standards and Technology
(NIST) [41].The verification of the analytical model compared with test
results shows that the analytical model is reasonably agreed with the test in
loading; however, the model is not accurate in unloading phase. Additionally,
in Kurama et al. [39] the drift results of the analytical model, based on the
equal displacement assumption, do not accurately predict the experimental
results.

Holden et al. (2003) compared behavior of monolithic reinforced concrete walls
with prestressed concrete walls using reversed cyclic quasi-static lateral
loading [42]. In this study two geometrically identical (half-scale) concrete
wall were tested under quasi-static loading. One wall was conventional
reinforced fixed-end shear wall, while the other wall was a partially
prestressed precast wall which was free to uplift and rock on the pivoting
points. For the rocking wall in this study assumed a bilinear elastic behavior
which is inspired by the damage avoidance design (DAD) paradigm [43] and it
ignores effects of wall’s mass inertia. Results from this study shows that
partially prestressed rocking wall with energy dissipating devices achieved
drift level on excess of 3% with no visible damage. This study primarily
focused on the advantages of self-centering walls compared to monolithic shear
walls, particularly regarding wall damage after lateral loading and the
comparison of residual displacements.

Ajrab et al. (2004) analyzed a rocking wall-reinforced concrete frame system
with additional tendon system and dampers [17]. To calculate the lateral
capacity demand of the rocking wall-frame system, this study employed a
capacity design approach. In the pre-rocking stage, the system behavior
governed by structural flexibility, when uplifting of the wall initiated,
based on the equilibrium of internal and external works, the base shear
capacity of the system is calculated without considering effect of rotational
inertia of the rocking wall. Then overall performance of the structure under
MCE (Maximum Considered Earthquake) and MAE (Maximum Assumed Earthquake) is
compared with the maximum displacement of the structure, calculated using
time-history analysis. The results of designed structure under different
ground motions showed that the adopted capacity-demand method predicts larger
displacements in comparison to what was obtained from time-history analysis.
Additionally, inter-story drifts are also reduced and became more uniformly
distributed through the height of the building.

To strengthen moment-resisting frames to near-fault ground motion effects,
Alavi and Krawinkler (2004) introduced pinned rocking wall system similar to
the one shown in Figure 4 (right) [16]. Near-fault ground motions cause a
highly non-uniform distribution of story ductility demand. Coupling the
moment-resisting frame with the pinned wall in this study showed that this is
an effective strategy that reduces drift demands of structures with a wide
range of periods and various performance levels.

Figure 4: Conventional Fixed shear wall (left) compared to stepping rocking
wall (center) and Pinned rocking wall (right).

Resilience and serviceability of self-centering structural systems after major
earthquakes were studied by Filiatrault et al. [44]. The paper points the main
advantages of these systems as: their large lateral displacement capacity, the
lack of structural damage associated with large displacements and their
ability to return to the original position upon unloading [44].

Lu (2005) studied behavior of rocking wall-frame system considering its 3D
effect [45]. Purpose of the study was modeling wall’s neural axis migration
and showing its significance and assessment of 3D effect of the wall in order
to control it. This study showed that the uncontrolled wall rocking can cause
beam-wall connection failures. This study conducted an experiment on a six
story high wall-frame system when it is excited with different ground motions.
The main goal of the experimental work was to investigate the coupled system’s
maximum displacement. This paper concluded that most of the first-story
lateral drift was attributed to the wall rigid body rotation about its
pivoting points during inelastic response.

Restrepo and Rahman (2007) studied the effect of energy dissipators in self-
centering walls [46] in an experimental work. The study showed that even wall
with no dampers will have no residual displacement after experiments.

Effect of post-tensioning a self-centering wall was investigated by Erkmen and
Schultz (2009) [47]. Results of the experiments showed that the rocking wall
is capable of recentering, even if the post-tensioning force is died out after
cycling loading.

Tozano et al. (2009) conducted an experimental work on a confined masonry wall
that is free to rock [48]. To provide damping, the wall was equipped with
energy dissipators. The mockup wall was build on 40% scale. The structure
showed a good performance under different levels of ground motions and met the
design criteria. Results from this study showed that the rocking masonry wall
was able to recenter after series of seismic loading and showed no visible
damages even after maximum roof drift ratio of 2.%.

Wada et al. (2011) [49, 18] retrofitted an existing eleven-story reinforced
concrete moment resisting frame with a damped pinned-rocking wall. This
building was located in the campus of Tokyo Institute of Technology in Japan.
The analytical model of the building with wall is studied using ABAQUS
software. In the building in order to take advantage of vertical displacement
in the pinned-rocking-wall sets of vertical hysteretic dampers were used. The
results showed that the retrofitted building experiences less drifts and the
drifts are evenly distributed along the height of the building.

An experimental and numerical study on a 1:3 scaled structure with a rocking
wall called propped rocking wall (PRW) conducted by Nicknam and Filiatrault
(2014) [50]. The experimental model’s measured fundamental period was
significantly larger than the numerical model. The displacement results were
close but the numerical model damps out quicker than the experimental test.

Wiebel and Christopoulos studied the controlled rocking steel braced frames
(CRSBFs) for seismic resisting systems [51, 52]. In their studies they
developed a framework to design these systems. In their study they showed how
to design the base rocking joint to achieve a targeted response based on the
results of series of analyses of single-degree-of-freedom systems with flag-
shaped hystereses [51].

Grigorian and Grigorian recommended a design procedure called design-led
analysis for the rocking-wall-moment-frame (RWMF) systems and studied few
examples of generic structures [53, 54]. The rocking-wall that was used in
these analysis was a pinned-rocking wall. Their study was based mostly on
capacity design which did not consider the rotational inertia of the wall
under dynamic loading.

Nazari et al. (2017) investigated different precast rocking walls with various
prestressing and tendon area configurations [55, 56, 57]. In terms of
performance under different seismic loadings rocking walls performed
satisfactorily and sustained negligible damages.

Gioiella et al. studied the implementation of rocking frames as a viable
method in retrofitting of an existing building [58, 59]. The retrofit of the
existing building was conducted coupling the building with turss pinned
rocking towers equipped with dampers. The study concluded that use of the
proposed rocking system helped to reduce the horizontal displacements and
accelerations.

Di Egidio et el. investigated seismic response of frames coupled with an
external rocking wall and idea of adding inertes to the coupled system [60,
61]. In their study they showed that rocking rigid block improves the seismic
response of frame structure [60], and it showed that use of inerters increases
the effectiveness of coupling with rocking wall [61].

In series of publications Aghagholizadeh and Makris studied dynamics of a
structural frame when it is coupled with rocking walls [62, 63, 64, 65, 66,
67, 68, 20, 69, 70, 71, 72]. In their studies they investigated a single-
degree-of-freedom idealization of a moment-frame coupled with a rocking wall.
They compared the results of different rocking wall configuration and
confirmed the single-degree-of-freedom model against a multi-degree of freedom
moment frame when it is coupled with a rocking wall. Some of the results
presented in the next section of this paper is derived from the studies that
has been conducted within these work [62, 63, 64, 65, 66, 67, 68, 20].

Most of these aforementioned studies introduce the unique advantages of
rocking action by referencing the seminal paper by Housner [34], who noticed
that tall, slender, free-standing columns, while they can easily uplift even
when subjected to a moderate ground acceleration (uplifting initiate when
$\ddot{u}_{g}>g\times(base/height)$); they exhibit remarkable seismic
stability due to a size-frequency scale effect. In his 1963 paper Housner
shows that there is a safety margin between uplifting and overturning and that
as the size of the free-standing column increases or the frequency of the
excitation pulse increases, this safety margin increases appreciably to the
extent that large free-standing columns enjoy ample seismic stability. Makris
[26, 73] explained that as the size of the free-standing rocking column
increases, the enhanced seismic stability primarily originates from the
difficulty to mobilize the rotational inertia of the column (wall) which
increases with the square of the column (wall) size. Further studies by Makris
and Vassiliou [74, 75] showed that as the size of the column (wall) increases,
the resistance to mobilize the rotational inertia increases to such an extent,
that the effect of vertical tendons becomes increasingly marginal.

##  3 Comparison of Rocking Shear Walls for Different Configurations

###  3.1 Stepping Rocking Wall and Pinned Rocking Wall

Two of the major configurations for rocking walls are pinned rocking wall
(shown in Figure 4 right), and stepping rocking wall (shown in Figure 4
center). Depending of the objective both have advantages and dis-advantages.
Implementing the pinned rocking wall helps to avoid one of the weak points of
a stepping rocking wall which is damage in its pivoting points caused by
impact to the foundation. But having a stepping rocking wall provides some
energy dissipation when the wall changes its pivoting corner from one side to
the other which is caused by the impact.

With the assumption that the rocking wall behaves similar to a rigid wall
(since the wall is not fixed in the bottom and for low to mid-rise buildings,
this is a reasonable assumption) and the wall-spring connecting arm is long
enough, the system shown in Figure 5 is a single-degree-of-freedom system
where the lateral translation of the mass, $u$, is related to the rotation of
the stepping rocking wall, $\theta$ [63].

For bilinear force-displacement behavior of the nonlinear spring Bouc-Wen
model is selected [76, 77]. Force that develops in the nonlinear spring can be
shown as:

Figure 5: Top row: Single-degree-of-freedom idealization of a yielding frame
when it is coupled to a stepping rocking wall (left) and pinned rocking wall
(right). Bottom row: Bilinear idealization with its control parameters for the
nonlinear spring model and force-displacement diagram of the stepping rocking
wall (left) and pinned rocking wall (right).

$F_{s}(t)=ak_{1}u(t)+(1-a)k_{1}u_{y}z(t),$ (1)

where $a=k_{2}/k_{1}$ is the post–to-pre yielding stiffness ratio and
$-1\leqslant z(t)\leqslant 1$ is a dimensionless internal variable described
by:

$\dot{z}(t)=\dfrac{1}{u_{y}}[\dot{u}(t)-\beta\dot{u}(t)|z(t)|^{n}-\gamma|\dot{u}|z(t)|z(t)|^{n-1}]$
(2)

In equation (2), constants $\beta$, $\gamma$ and $n$ are model parameters that
dictates the shape of the hysteretic loop [76, 77, 78].

Using equilibrium of forces it can be shown the equation of motion for
positive and negative rotation of the stepping rocking wall as follows (for
detailed calculation of the equations of motion see [63, 79]):

$\displaystyle\theta\geq 0$ (3)
$\displaystyle\Big{(}\dfrac{4}{3}+\sigma\cos^{2}(\alpha-\theta)\Big{)}\ddot{\theta}$
$\displaystyle+\sigma\cos(\alpha-\theta)\Big{[}a\omega_{1}^{2}\big{(}\sin\alpha-
sin(\alpha-\theta)\big{)}+2\xi\omega_{1}\dot{\theta}\cos(\alpha-\theta)+\dot{\theta}^{2}\sin(\alpha-\theta)$
$\displaystyle+(1-a)\omega_{1}^{2}\dfrac{u_{y}}{R}z(t)\Big{]}={}-\dfrac{g}{R}\Big{[}(\sigma+1)\dfrac{\ddot{u}_{g}}{g}\cos(\alpha-\theta)+\sin(\alpha-\theta)\Big{]},$

and,

$\displaystyle\theta<0$ (4)
$\displaystyle\Big{(}\dfrac{4}{3}+\sigma\cos^{2}(\alpha+\theta)\Big{)}\ddot{\theta}$
$\displaystyle-\sigma\cos(\alpha+\theta)\Big{[}a\omega_{1}^{2}\big{(}\sin\alpha-
sin(\alpha+\theta)\big{)}-2\xi\omega_{1}\dot{\theta}\cos(\alpha+\theta)+\dot{\theta}^{2}\sin(\alpha+\theta)$
$\displaystyle-(1-a)\omega_{1}^{2}\dfrac{u_{y}}{R}z(t)\Big{]}={}\dfrac{g}{R}\Big{[}-(\sigma+1)\dfrac{\ddot{u}_{g}}{g}\cos(\alpha+\theta)+\sin(\alpha+\theta)\Big{]},$

where wall of size $R=\sqrt{b^{2}+h^{2}}$, slenderness, $\tan\alpha=b/h$,
$\sigma=\frac{spring~{}mass}{wall~{}mass}=m_{s}/m_{w}$ is the mass ratio
parameter, $\omega_{1}=\sqrt{k_{1}/m_{s}}=$ the pre-yielding undamped
frequency and $\xi=\dfrac{c}{2m_{s}\omega_{1}}=$ the pre-yielding viscous
damping ratio of the SDOF oscillator. Equations (3) and (4) are the equation
of motion for positive and negative rotations of the coupled system shown in
Figure 5 (left).

Similarly equation of motion for a pinned rocking wall when it is coupled to a
nonlinear spring can be shown as follows (for detailed calculation of the
equations of motion see [63]):

$\begin{split}&\big{[}\dfrac{1}{3}+(1+\sigma\cos^{2}\theta)\cos^{2}\alpha\big{]}\ddot{\theta}\\\
&+\sigma\cos^{2}\alpha\cos\theta\Big{[}(a\omega^{2}_{1}-\dot{\theta}^{2})\sin\theta+(1-a)\omega^{2}_{1}\dfrac{u_{y}}{R\cos\alpha}z(t)+2\xi\omega_{1}\dot{\theta}\cos\theta\Big{]}\\\
&=-\dfrac{g}{R}\cos\alpha\Big{[}(\sigma+1)\dfrac{\ddot{u}_{g}}{g}\cos\theta-\sin\theta\Big{]},\end{split}$
(5)

where $\omega_{1}=\sqrt{k_{1}/m_{s}}$ = the pre-yielding undamped frequency
and $\xi=\dfrac{c}{2m_{s}\omega_{1}}$ = the viscous damping ratio of the SDOF
oscillator (as in the previous case). Equation (5) is the equation of motion
for both positive and negative rotations of the coupled system shown in Fig. 5
(right).

Comparing equations of motion for stepping (3 and 4) with with pinned wall (5)
it can be observed that the self-weight of the pinned rocking wall works
against the wall’s stability (term, $\frac{g}{R}\sin{\theta}$). This is one of
the benefits of employing stepping rocking wall to pinned rocking wall, this
way the wall have more tendency to recenter to its origin only employing its
self-weight [62]. Hence, the stepping rocking wall will be more effective in
reducing the residual displacement of the structure [63]).

###  3.2 Restrained Rocking Wall

Another configuration of rocking wall is adding prestressing strands to the
rocking wall and anchoring it to the foundation [64]. Some of the main
advantages of using tendons to anchor the wall is that this configuration
prevents possible sliding of the wall on top of its foundation and increases
the vertical stiffness of the rocking-wall.

Figure (6) top presents an idealized single-degree-of-freedom system of a
nonlinear spring and mass system representing a moment-resisting frame coupled
with a vertically restrained stepping rocking wall. Bottom row of Figure (6)
on the left shows bi-linear force-displacement relation assigned to the
spring, and on the right side force-displacement diagram of the restrained
stepping rocking wall.

As it shown in the bottom right of Figure (6), with the increase in the
elasticity (EA) of the restrainer, the lateral stiffness of the vertically
restrained stepping rocking shear wall gradually shifts from negative to
positive [75, 80]. Hence, it should be noted that if using a shear wall with
negative stiffness is one of the objectives in the design, one have to limit
amount of the tendons uses in the stepping rocking wall.

Figure 6: Yielding single‐degree‐of‐freedom oscillator coupled with a
vertically restrained stepping rocking wall.

Using equilibrium of forces the equation of motion for positive and negative
rotation of the vertically restrained stepping rocking wall show as follows
(for detailed calculation of the equations of motion see [64]):

$\displaystyle\theta\geq 0$ (6)
$\displaystyle\Big{(}\dfrac{4}{3}+\sigma\cos^{2}(\alpha-\theta)\Big{)}\ddot{\theta}+\sigma\cos(\alpha-\theta)\Big{[}a\omega_{1}^{2}\big{(}\sin\alpha-
sin(\alpha-\theta)\big{)}+2\xi\omega_{1}\dot{\theta}\cos(\alpha-\theta)$
$\displaystyle+\dot{\theta}^{2}\sin(\alpha-\theta)+(1-a)\omega_{1}^{2}\dfrac{u_{y}}{R}z(t)\Big{]}$
$\displaystyle={}-\dfrac{g}{R}\Big{[}(\sigma+1)\dfrac{\ddot{u}_{g}}{g}\cos(\alpha-\theta)+\sin(\alpha-\theta)+\sin\alpha\sin\theta(\frac{1}{2}\frac{EA}{m_{w}g}\tan\alpha+\frac{P_{o}}{m_{w}g}\frac{1}{\sqrt{2}\sqrt{1-\cos\theta}})\Big{]},$

and,

$\displaystyle\theta<0$ (7)
$\displaystyle\Big{(}\dfrac{4}{3}+\sigma\cos^{2}(\alpha+\theta)\Big{)}\ddot{\theta}-\sigma\cos(\alpha+\theta)\Big{[}a\omega_{1}^{2}\big{(}\sin\alpha-
sin(\alpha+\theta)\big{)}-2\xi\omega_{1}\dot{\theta}\cos(\alpha+\theta)$
$\displaystyle+\dot{\theta}^{2}\sin(\alpha+\theta)-(1-a)\omega_{1}^{2}\dfrac{u_{y}}{R}z(t)\Big{]}$
$\displaystyle={}\dfrac{g}{R}\Big{[}-(\sigma+1)\dfrac{\ddot{u}_{g}}{g}\cos(\alpha+\theta)+\sin(\alpha+\theta)-\sin\alpha\sin\theta(\frac{1}{2}\frac{EA}{m_{w}g}\tan\alpha+\frac{P_{o}}{m_{w}g}\frac{1}{\sqrt{2}\sqrt{1-\cos\theta}})\Big{]},$

in which $EA$ is the axial stiffness of the vertical strand that can be
prestressed with a prestressing force $P_{o}$.

Figure 7: Free-body diagram of a rocking wall with an elastic tendon passing
through its center-line (without considering the spring connection).

Minimum ground acceleration $\ddot{u}_{g~{}min}$ that is required to initiate
a vertically restrained shear wall is increased proportionally with the
magnitude of the prestressing force $P_{o}$ [64]:

$\ddot{u}_{g~{}min}=\dfrac{g~{}tan\alpha}{\sigma+1}\big{(}1+\dfrac{P_{o}}{m_{w}~{}g}\big{)}.$
(8)

Using equilibrium of forces at the pivoting corner of a rocking shear wall,
shown in Figure (7), the normal force, $N$, at the pivoting corner of the
shear wall normalized with respect to the wall mass $m_{w}$, can be calculated
as follows [64],

$\displaystyle\theta\geq 0$ (9)
$\displaystyle\frac{N(t)}{m_{w}g}=1+\frac{R}{g}\big{[}\ddot{\theta}\sin(\alpha-\theta)+\dot{\theta}^{2}\cos(\alpha-\theta)\big{]}+\frac{1}{2}\frac{EA}{m_{w}g}\tan\alpha\sin\theta+\frac{1}{\sqrt{2}}\frac{P_{o}}{m_{w}g}\sqrt{1+\cos\theta}.$

Responses, $u(t)$, and normal forces at the pivoting point,$N(t)$, of a SDOF
system when it is coupled with a rocking wall for the cases of with and
without vertical tendon is shown in Figure (8). The plots on the left are
responses when the system is subjected to the North-South component of NAR
station ground motion recorded during 2023 Kahramanmaras, Turkiye earthquake
and in the right side when the coupled system is subjected to the REHS ground
motion recorded during the 2011 Christchurch, New Zealand earthquake. The
solid heavy black line in the figure represents response of a nonlinear
spring, light black line is the response of the SDOF system when it is coupled
to a stepping rocking wall, and the blue line shows the response of the SDOF
spring when it is coupled to a vertically restrained stepping rocking wall
with pretensioned ($\frac{P_{o}}{m_{w}g}=$0.5) stiff tendons
($\frac{EA}{m_{w}g}=$200). The results shows that the use of a stepping
rocking wall helps the SDOF to recenter to its origin, hence there is no
residual displacement. The maximum displacement of the systems with rocking
walls for both cases are smaller when it is compared to SDOF spring.

Results in Figure (8) shows that by increasing the axial stiffness, $EA$, of
the vertical tendon one increases the lateral stiffness of the entire
structural system; at the same time, vertical tendons increase the vertical
reactions at the pivoting corners (at some cases by more than 50% [64]). This
increases the possibility of additional damage to the pivoting corners of the
shear walls which leads to an unsustainable design. Hence, when prestressing
tendons are used, it is important to further investigate the effect of
additional forces in the pivoting corners and retrofit the pivoting points to
prevent generation of excessive damages.

Figure 8: Time-history analysis of a nonlinear SDOF oscillator coupled with a
vertically restrained stepping rocking wall with preyielding period, T1 = 1.25
seconds, normalized strength $Q/ms$ = 0.08g, wall size ratio, $\omega_{1}/p$ =
10, and structure-to-wall mass ration, $\sigma$ = 5. When subjected to the
North-South component of NAR station ground motion recorded during 2023
Kahramanmaras, Turkiye (left) and REHS ground motion recorded during the 2011
Christchurch, New Zealand (right) earthquakes.

##  4 Numerical Analysis of Multi-Story Moment Resisting Frames When Coupled
with Modular Stepping Rocking Walls

In this section, seismic response of moment-resisting-frames (MRF) coupled
with modular rocking shear walls, using OpenSees [81] is investigated.

The structures that are chosen for this study are moment resisting steel
frames designed for the SAC Phase II Project [82]. These buildings are well-
known benchmark structures to the literature [83, 84, 85, 64, 20]. The
structures designed to meet the seismic code (pre‐Northridge Earthquake) and
represents typical medium‐rise buildings designed for the greater area of Los
Angeles, California. MRFs and their cross sectional information are shown in
Figure (9). The first three natural period of vibrations for the 9-story
building is $2.26$ s, $0.874$ s, and $0.488$ s, and for the 20-story building
is $3.83$ s, $1.33$ s, and $0.769$ s.

Figure 9: Nine‐story (left) and twenty-story (right) moment‐resisting steel
frame designed for the SAC Phase II Project.

Considering the characteristics of rocking shear walls that discussed in
section (3) of this paper, the nine-story and twenty-story frames are
retrofitted with a modular stepping rocking shear wall using axial link
elements between shear wall and the frame. The proposed system (shown in
Figure 10) is comprised of wall segments that are puzzle shaped to provide
mechanical connection between each wall parts. In addition to the mechanical
connections, vertical prestressing strands are used to tie the wall segments
together. These tendons pass through all walls and anchored inside the base
wall above the foundation. This will increases the lateral stiffness of the
wall that helps the wall act as a rigid block (which can be tuned with amount
of prestressing applied) without adding extra stress at the pivoting corners.

Figure 10: Nine‐story (left) and twenty-story (right) moment‐resisting steel
frames coupled with modular stepping rocking shear wall.

In order to model a rocking shear wall in OpenSees, a zero-length fiber
section is define at the base of the wall between the wall and the ground
surface. The cross-section of the zero-length rocking surface is defined,
using of a fiber section with nonlinear, elastic-no-tension (ENT) material
that is available in Opensees [86, 87].

In order to consider energy dissipation of the rocking motion during the
impact, similar to [86, 87, 88], dissipative time-stepping integration
procedure of Hilber-Hughes-Taylor [89] (HHT) is being used. HHT damping effect
is function of dissipation factor $a^{\prime}_{d}$ and the time of the
integration. For this study the HHT time step is selected as $10^{-4}$ s and
dissipation factor $a^{\prime}_{d}$ is selected as $-1/3$.

Figure 11: Recorded time histories and elastic response spectra for damping
ratio $\xi=\frac{c}{2m\omega_{o}}=5\%$ and $10\%$ of the four ground motions
used for the response analysis presented in this study.

For the 9-story MRF as it is shown in Figure (10-left) a shear wall with
height of $37.17$ m and width of $6.20$ m is selected (width/height =
$\tan\alpha\approx$ $1/6$). For the case of 20-story building (shown in Figure
(10-right)) a shear wall with height of $80.73$ m and width of $8.0$ m is
selected (width/height = $\tan\alpha\approx$ $1/8$).

Figure 12: Maximum floor displacement (normalized with respect to the total
height of the wall the building) of the 9-story MRF without wall (dark heavy
line in black) and the MRF with a stepping rocking wall (light line in blue)
under various earthquake excitations.

Dynamic time-history analysis of both 9-story and 20-story frames with and
without wall is investigated in this section. This analysis is conducted
applying various seismic records. Figure (11) presents recorded time histories
and elastic response spectra for damping ratio
$\xi=\frac{c}{2m\omega_{o}}=5\%$ and $10\%$. The ground motions used are the
PacoimaDam/164 ground-motion recorded during the 1971 Imperial valley, the
Newhall/360 ground motion recorded during the 1994 Northridge, the REHS ground
motion recorded during the 2011 Christchurch, New Zealand, and the North-South
component of NAR station ground motion recorded during 2023 Kahramanmaras,
Turkiye earthquakes. The selected ground-motions were based on the reasoning
that the distinguishable coherent pulse of these motions has different
duration; therefore, each motion will amplify the inelastic structural
response at different preyielding periods.

Figure 13: Interstory drifts for the 9-story MRF without wall (dark heavy line
in black) and the MRF with a stepping rocking wall (light line in blue) under
various earthquake excitations.

Figures (12) and (13) show the maximum floor displacement, that is normalized
with respect to the above the ground height of the building, and interstory
drifts for the 9-story MRF without wall (dark heavy line in black) and the MRF
with a stepping rocking wall (light line in blue) respectively. Results in the
Figure (12) shows that for all the cases adding a stepping rocking wall
suppress the maximum floor displacement. In addition, it shows that the the
rocking wall enforces the floor displacements to be mostly first-mode
dominant. Most importantly, the main motivation of using rocking walls is to
prevent weak-story failure. In order to investigate this, it is important to
check the interstory drifts. Figure (13) shows the comparison of the
interstory drifts under various earthquakes. The results show that the
addition of rocking wall creates mostly a uniform drift profile when it is
compared to the interstory drift profile of the frame without a wall. This
will prevent generation of weak-stories since the displacements are
distributed evenly along the height of the building.

Figure 14: Maximum floor displacement (normalized with respect to the total
height of the wall the building) of the 20-story MRF without wall (dark heavy
line in black) and the MRF with a stepping rocking wall (light line in blue)
under various earthquake excitations.

Similarly, Figures (14) and (15) show the maximum floor displacement, that is
normalized with respect to the above the ground height of the building, and
interstory drifts for the 20-story MRF without wall (dark heavy line in black)
and the MRF with a stepping rocking wall (light line in blue) respectively.
Floor displacement results (Figure (14)) shows higher mode effects in the
displacement of the 20-story frame when no wall is added. On the other hand,
displacement profile of a 20-story MRF when it is coupled with a rocking wall
is similar to first mode vibration of the building. Since the displacement
profile for the frame with a wall is similar to the first-mode vibration, the
interstory profile results, shown in the Figure (15) is uniform along the
height of the building. This is important since the main reason for the
generation of weak-story is the presence of significant difference in
interstory drifts that is visible for the 20-story frame without a shear wall.

Figure 15: Interstory drifts for the 20-story MRF without wall (dark heavy
line in black) and the MRF with a stepping rocking wall (light line in blue)
under various earthquake excitations.

##  5 Conclusions

The high occupancy rates in urban multi-story buildings, combined with present
safety concerns, necessarily prompt a reassessment of performance goals. Given
the notable seismic damage and instances of weak-story failures that have been
documented after major earthquakes, this paper studied the use of modular
shear walls that are free to rock above their foundation. This paper first
conducted an overview of available literature on rocking elements and provided
nonlinear equations of motions for spring-rocking-wall system for different
configurations. Then a dynamic analysis for two 9-story and 20-story moment-
resisting steel frames when they are coupled with modular rocking shear walls
were investigated to reach the following conclusions.

  * 1.

The self-weight of the pinned rocking wall works against its stability. Hence,
the stepping rocking wall will be more effective in reducing the residual
displacement of the structure.

  * 2.

Time-history analysis of idealized SDOF nonlinear spring with and without
rocking wall shows that the use of stepping rocking wall can effectively
minimize the residual drifts after earthquakes.

  * 3.

The effect of the vertical tendons even when they are stiff
$\frac{EA}{m_{w}g}=200$ and highly prestressed ($P_{o}=m_{w}g$) is marginal.

  * 4.

The vertical prestressing tendons will increase the vertical reactions at the
pivoting corners. Hence, when they are employed, it is important to further
investigate the effect of additional forces in the pivoting corners and
retrofit the pivoting points to prevent generation of excessive damages.

  * 5.

Addition of a modular stepping rocking wall to a moment-resisting frame not
only reduces its maximum floor displacement for most cases, it also enforces a
first-mode dominating response. Hence prevents development of weak-story
failure.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## CRediT authorship contribution statement

Mehrdad Aghagholizadeh: Conceptualization, Formal analysis, Investigation,
Methodology, Software, Validation, Visualization, Writing - review & editing.

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