The analysis of some braking mechanisms for neutron stars was carried out to determine the sign of the second derivative of the pulsar period. This quantity is the important parameter for calculations of the braking index n. It is shown that this derivative can be positive and lead to decreasing of n. It is necessary to correct the methods of calculations of n used this moment because they are based as a rule on the suggestion on the constancy of pulsar parameters (magnetic fields, angles between some axes and so on). The estimations of corrections to braking indices are obtained. It is shown that these corrections can be marked for pulsars with long periods and their small derivatives.
The coupled evolution of pulsar rotation and inclination angle in the wind braking model is calculated. The oblique pulsar tends to align. The pulsar alignment will affect its spin-down behavior. As a pulsar evolves from the magneto-dipole radiation dominated case to the particle wind dominated case, the braking index will first increase and then decrease. In the early time, the braking index may be larger than 3. And during the following long time, the braking index will be always smaller than 3. The minimum braking index is about one. This can explain the existence of high braking index larg…
Pulsars are good clocks in the universe. One fundamental question is that why they are good clocks? This is related to the braking mechanism of pulsars. Nowadays pulsar timing is done with unprecedented accuracy. More pulsars have braking indices measured. The period derivative of intermittent pulsars and magnetars can vary by a factor of several. However, during pulsar studies, the magnetic dipole braking in vacuum is still often assumed. It is shown that the fundamental assumption of magnetic dipole braking (vacuum condition) does not exist and it is not consistent with the observations. The…
Eight pulsars have low braking indices, which challenge the magnetic dipole braking of pulsars. 222 pulsars and 15 magnetars have abnormal distribution of frequency second derivatives, which also make contradiction with classical understanding. How neutron star magnetospheric activities affect these two phenomena are investigated by using the wind braking model of pulsars. It is based on the observational evidence that pulsar timing is correlated with emission and both aspects reflect the magnetospheric activities. Fluctuations are unavoidable for a physical neutron star magnetosphere. Young p…
Braking indices are used to describe the evolution of pulsars rotation, and can offer insights into the braking mechanism that dominates the slow down. Here we discuss the main difficulties associated with measuring braking indices and the complexity of interpreting these measurements. Considering recent braking index measurements on pulsars with large and regular glitches, we comment on the significant effects that the loosely coupled superfluid inside pulsars might have on their spin evolution.
The pulsar wind model is updated by considering the effect of particle density and pulsar death. It can describe both the short term and long term rotational evolution of pulsars consistently. It is applied to model the rotational evolution of the Crab pulsar. The pulsar is spun down by a combination of magnetic dipole radiation and particle wind. The parameters of the Crab pulsar, including magnetic field, inclination angle, and particle density are calculated. The primary particle density in acceleration region is about 10^3 times the Goldreich-Julian charge density. The lower braking index …
Braking index measurements offer the opportunity to explore the processes affecting the long-term spin evolution of pulsars and possible evolutionary connections between the various pulsar populations. For young pulsars the long-term trends are generally obscured by short term phenomena such as timing noise and the recoveries form large glitches. Here we present a new method to overcome the latter and report on braking index measurements for the Vela-like pulsars PSR B1800-21 and PSR B1823-13, an updated measurement for Vela and new estimates for four more glitching pulsars observed at Jodrell…
The recently discovered rotationally powered pulsar PSR J1640-4631 is the first to have a braking index measured, with high enough precision, that is greater than three. An inclined magnetic rotator in vacuum or plasma would be subject not only to spin-down but also to an alignment torque. The vacuum model can address the braking index only for an almost orthogonal rotator that is incompatible with the single peaked pulse profile. The magnetic dipole model with the corotating plasma predicts braking indices between $3-3.25$. We find that the braking index of $3.15$ is consistent with two diffe…
Isolated pulsars are rotating neutron stars with accurately measured angular velocities $Ω$, and their time derivatives which show unambiguously that the pulsars are slowing down. Although the exact mechanism of the spin-down is a question of debate, the commonly accepted view is that it arises either through emission of magnetic dipole radiation (MDR) from a rotating magnetized body, through emission of a relativistic particle wind, or via higher order magnetic multipole or gravitational quadrupole radiation. The calculated energy loss by a rotating pulsar is model dependent and leads to the …
We present a phase-coherent timing solution for PSR J1640-4631, a young 206 ms pulsar using X-ray timing observations taken with NuSTAR. Over this timing campaign, we have measured the braking index of PSR J1640-4631 to be n = 3.15+/-0.03. Using a series of simulations, we argue that this unusually high braking index is not due to timing noise, but is intrinsic to the pulsar's spin-down. We cannot, however, rule out contamination due to an unseen glitch recovery, although the recovery timescale would have to be longer than most yet observed. If this braking index is eventually proven to be sta…
PSR B0540-69 has braking index measurement in its persistent state: n=2.129 \pm 0.012. Recently, it is reported to have spin-down state changes: a suddenly 36 % increase in the spin-down rate. Combining the persistent state braking index measurement and different spin-down states, PSR B0540-69 is more powerful than intermittent pulsars in constraining pulsar spin-down models. The pulsar wind model is applied to explain the variable timing behavior of PSR B0540-69. The persistent state braking index of PSR B0540-69 is the combined effect of magnetic dipole radiation and particle wind. The parti…
Recently, Parthsarathy et al. analysed long-term timing observations of 85 young radio pulsars. They found that 11 objects have braking indices ranging $\sim 10-100$, far from the classical value $n=3$. They also noted a mild correlation between measured value of $n$ and characteristic age of a radio pulsar. In this article we systematically analyse possible physical origin of large braking indices. We find that a small fraction of these measurements could be caused by gravitational acceleration from an unseen ultra-wide companion of a pulsar or by precession. Remaining braking indices cannot …
We report the results of a 16-month monitoring campaign using the Swift satellite of PSR B0540-69, a young pulsar in the Large Magellanic Cloud. Phase connection was maintained throughout the campaign so that a reliable ephemeris could be determined, and the length of the campaign is adequate to accurately determine the spin frequency and its first and second derivatives. The braking index is 0.031 +/- 0.013 (90% confidence), a value much lower than previously reported for B0540-69 and almost all other young pulsars. We use data from the extensive monitoring campaign with RXTE to show that tim…
PSR J1846-0258 is an object which straddles the boundary between magnetars and rotation powered pulsars. Though behaving for many years as a rotation-powered pulsar, in 2006, it exhibited distinctly magnetar-like behavior - emitting several short hard X-ray bursts, and a flux increase. Here we report on 7 years of post-outburst timing observations of PSR J1846-0258 using the Rossi X-ray Timing Explorer and the Swift X-ray Telescope. We measure the braking index over the post-magnetar outburst period to be $n=2.19\pm0.03$. This represents a change of $Δn=-0.46\pm0.03$ or a 14.5$\;σ$ difference …
Recent timing observation constrained the braking index of the X-ray pulsar PSR J1640-4631 to be $n=3.15\pm0.03$, which is the highest value of all pulsars with measured braking indices so far. In this Letter, we investigate whether pulsar braking by combined between the magnetic dipole emission and the gravitational radiation might have a braking index greater than three. For conventional neutron star and low mass quark star candidates, the inferred ellipticities derived by the observed braking index are obviously much larger than the theoretical estimated maximum value. If PSR J1640-4631 is …
An isolated pulsar is a rotating neutron star possessing a high magnetic dipole moment that generally makes a finite angle with its rotation axis. As a consequence, the emission of magnetic dipole radiation (MDR) continuously takes away its rotational energy. This process leads to a time decreasing angular velocity of the star that is usually quantified in terms of its braking index. While this simple mechanism is indeed the main reason for the spin evolution of isolated pulsars, it may not be the only cause of this effect. Most of young isolated pulsars present braking index values that are c…
The magnetic dipole radiation (MDR) model is currently the best approach we have to explain pulsar radiation. However a most characteristic parameter of the observed radiation, the braking index n$_{\rm obs}$ shows deviations for all the eight best studied isolated pulsars, from the simple model prediction n$_{\rm dip}$ = 3. The index depends upon the rotational frequency and its first and second time derivatives, but also on the assumption of that the magnetic dipole moment and inclination angle, and the moment of inertia of the pulsar are constant in time. In a recent paper [Phys. Rev. D 91,…
In Paper I of this series, we detected a significant value of the braking index ($n$) for 19 young, high-$\dot{E}$ radio pulsars using $\sim$ 10 years of timing observations from the 64-m Parkes radio telescope. Here we investigate this result in more detail using a Bayesian pulsar timing framework to model timing noise and to perform selection to distinguish between models containing exponential glitch recovery and braking index signatures. We show that consistent values of $n$ are maintained with the addition of substantial archival data, even in the presence of glitches. We provide strong a…
The low braking-index pulsar PSR J1734$-$3333 could be born with superhigh internal magnetic fields $B_{\rm in}\sim10^{15}-10^{16}$ G, and undergo a supercritical accretion soon after its formation in a supernova explosion. The buried multipole magnetic fields will merger into a dipole magnetic field. Since the magnetic flow transfers from the core to the crust of the pulsar, its surface dipole field grows quickly at a power-law form assumed until it saturates at the level of internal dipole field. The increase in surface dipole magnetic field results in the observed low braking index of $n=0.…
The departure of all measured pulsar braking indexes from the "canonical" dipole value 3 has been attributed to several causes in the past. Careful monitoring of the Crab pulsar has revealed permanent changes in the spin-down rate which are most likely the accumulation of small jumps in the angle $α$ between the magnetic and spin axis. Recently, a large permanent change in the braking index of the in the "Crab twin" pulsar B0540-69 has been reported, and an analogous phenomenon seen in the high-field pulsar PSR 1846-0258 has been seen following a glitch, while another similar event (in PSR J11…
Recent timing observation reported that the radio pulsar PSR J1734 - 3333 with a rotating period $P=1.17~\rm s$ is slowing down with a period derivative $\dot{P}=2.28\times 10^{-12}\rm s\,s^{-1}$. Its derived braking index $n=0.9 \pm 0.2$ is the lowest value among young radio pulsars with the measured braking indices. In this Letter, we attempt to investigate the influence of the braking torque caused by the interaction between the fall-back disk and the strong magnetic field of the pulsar on the spin evolution of PSR J1734 - 3333. Analytical result show that this braking torque is obviously f…
In this work, we interpreted the high braking index of PSR J1640$-$4631 with a combination of the magneto-dipole radiation and dipole magnetic field decay models. By introducing a mean rotation energy conversion coefficient $\overlineζ$, the ratio of the total high-energy photon energy to the total rotation energy loss in the whole life of the pulsar, and combining the pulsar's high-energy and timing observations with reliable nuclear equation of state, we estimate the pulsar's initial spin period, $P_{0}\sim (17-44)$ ms, corresponding to the moment of inertia $I\sim (0.8-2.1)\times 10^{45}$ g…
Pulsars are stars that emit electromagnetic radiation in well-defined time intervals. The frequency of such pulses decays with time as is quantified by the {\it braking index} ($n$). In the canonical model $n = 3$ for all pulsars, but observational data show that $n \neq 3$, indicating a limitation of the model. In this work we present a new approach to study the frequency decay of the rotation of a pulsar, based on an adaptation of the canonical one. We consider the pulsar a star that rotates in vacuum and has a strong magnetic field but, differently from the canonical model, we assume that i…
We report the discovery and timing measurements of PSR J1208-6238, a young and highly magnetized gamma-ray pulsar, with a spin period of 440 ms. The pulsar was discovered in gamma-ray photon data from the Fermi Large Area Telescope (LAT) during a blind-search survey of unidentified LAT sources, running on the distributed volunteer computing system Einstein@Home. No radio pulsations were detected in dedicated follow-up searches with the Parkes radio telescope, with a flux density upper limit at 1369 MHz of 30 $μ$Jy. By timing this pulsar's gamma-ray pulsations, we measure its braking index over…
The ~800 yr-old pulsar J1846-0258 is a unique transition object between rotation-powered pulsars and magnetars: though behaving like a rotation-powered pulsar most of the time, in 2006 it exhibited a distinctly magnetar-like outburst accompanied by a large glitch. Here we present X-ray timing observations taken with the Rossi X-ray Timing Explorer over a 2.2-yr period after the X-ray outburst and glitch had recovered. We observe that the braking index of the pulsar, previously measured to be n=2.65+/-0.01, is now n=2.16+/-0.13, a decrease of 18+/-5%. We also note a persistent increase in the t…
Braking indices of pulsars present a scientific challenge as their theoretical calculation is still an open problem. In this paper we report results of a study regarding such calculation which adapts the canonical model (which admits that pulsars are rotating magnetic dipoles) basically by introducing a compensating component in the energy conservation equation of the system. This component would correspond to an effective force that varies with the first power of the tangential velocity of the pulsar's crust. We test the proposed model using data available and predict braking indices values f…
We have investigated the long-term evolutions of the high-magnetic field radio pulsars (HBRPs) with measured braking indices in the same model that was applied earlier to individual anomalous X-ray pulsars (AXPs), soft gamma repeaters (SGRs) and dim isolated neutron stars (XDINs). We have shown that the rotational properties (period, period derivative and braking index) and the X-ray luminosity of individual HBRPs can be acquired simultaneously by the neutron stars evolving with fallback discs. The model sources reach the observed properties of HBRPs in the propeller phases, when pulsed radio …
The very small braking index of PSR J1734-3333, $n=0.9\pm0.2$, challenges the current theories of braking mechanisms in pulsars. We present a possible interpretation that this pulsar is surrounded by a fall-back disk and braked by it. A modified braking torque is proposed based on the competition between the magnetic energy density of a pulsar and the kinetic energy density of a fall-back disk. With this torque, a self-similar disk can fit all the observed parameters of PSR J1734-3333 with natural initial parameters. In this regime, the star will evolve to the region having anomalous X-ray pul…
Isolated pulsars are rotating neutron stars with accurately measured angular velocities $Ω$, and their time derivatives which show unambiguously that the pulsars are slowing down. Although the exact mechanism of the spin-down is a question of debate in detail, the commonly accepted view is that it arises through emission of magnetic dipole radiation (MDR) from a rotating magnetized body. Other processes, including the emission of gravitational radiation, and of relativistic particles (pulsar wind), are also being considered. The calculated energy loss by a rotating pulsar with a constant momen…
Due to the lack of long term pulsed emission in quiescence and the strong timing noise, it is impossible to directly measure the braking index $n$ of a magnetar. Based on the estimated ages of their potentially associated supernova remnants (SNRs), we estimate the values of the mean braking indices of eight magnetars with SNRs, and find that they cluster in a range of $1\sim$42. Five magnetars have smaller mean braking indices of $1<n<3$, and we interpret them within a combination of magneto-dipole radiation and wind aided braking, while the larger mean braking indices of $n>3$ for other three…
We study the putative emission of gravitational waves (GWs) in particular for pulsars with measured braking index. We show that the appropriate combination of both GW emission and magnetic dipole brakes can naturally explain the measured braking index, when the surface magnetic field and the angle between the magnetic dipole and rotation axes are time dependent. Then we discuss the detectability of these very pulsars by aLIGO and the Einstein Telescope. We call attention to the realistic possibility that aLIGO can detect the GWs generated by at least some of these pulsars, such as Vela, for ex…
Recent timing observation provides an intriguing result for the braking index of the X-ray pulsar PSR J1640-4631, which has a measured braking index $n=3.15\pm0.03$. The decrease of the inclination angle between between the spin axis and the magnetic axis can be responsible for such a high braking index. However, the physical mechanisms causing the change of the magnetic inclination angle have not been fully understood. In this Letter, we apply a two-dipole model given by Hamil et al (2016) to explain the decrease of the magnetic inclination angle of PSR J1640-4631. The rotation effect of a ch…
Here we report that the observed braking indices of the 366 pulsars in the sample of Hobbs et al. range from about $-10^8$ to about $+10^8$ and are significantly correlated with their characteristic ages. Using the model of magnetic field evolution we developed previously based on the same data, we derived an analytical expression for the braking index, which agrees with all the observed statistical properties of the braking indices of the pulsars in the sample of Hobbs et al. Our model is, however, incompatible with the previous interpretation that magnetic field growth is responsible for the…
Isolated pulsars are rotating neutron stars with accurately measured angular velocities $Ω$, and their time derivatives that show unambiguously that the pulsars are slowing down. The commonly accepted view is that it arises through emission of magnetic dipole radiation (MDR) from a rotating magnetized body. The calculated energy loss by a rotating pulsar with a constant moment of inertia is assumed proportional to a model dependent power of $Ω$. This relation leads to the power law $\dotΩ$ = -K $Ω^{\rm n}$ where $n$ is called the braking index. The MDR model predicts $n$ exactly equal to 3. Se…
In Dec. 2011 PSR B0540-69 experienced a spin-down rate transition (SRT), after which the spin-down power of the pulsar increased by ~36%. About 1000 days after the SRT, the X-ray luminosity of the associated pulsar wind nebula (PWN) was found to brighten by 32+/-8%. After the SRT, the braking index n of PSR B0540-69 changes from n=2.12 to n=0.03 and then keeps this value for about five years before rising to n=0.9 in the following years. We find that most of the current models have difficulties in explaining the measured braking index. One exceptive model of the braking index evolution is the …
In this paper, we attribute high braking indices $n>3$ of two magnetars SGR 0501$+$4516 and 1E 2259$+$586 to the decrease in their inclination angles using the double magnetic-dipole model proposed by Hamil et al.(2016). In this model, there are two magnetic moments inside a neutron star, one is generated by the rotation effect of a charged sphere, $M_{1}$, and the other is generated by the magnetization of ferromagnetically ordered material, $M_{2}$. Our calculations indicate that the magnetic moment $M_{2}$ would evolve towards alignment with the spin axis of the two magnetars, and cause the…
A nascent neutron star may be exposed to fallback accretion soon after the proto-neutron star stage. This high accretion episode can submerge the magnetic field deep in the crust. The diffusion of the magnetic field back to the surface will take hundreds to millions of years depending on the amount of mass accreted and the consequent depth the field is buried. Neutron stars with large kick velocities will accrete less amount of fallback material leading to shallower submergence of their fields and shorter time-scales for the growth of their fields. We obtain the relation $τ_{\rm Ohm} \propto v…
Recently, a braking index for the pulsar PSR J1640-4631 has been measured. With a braking index of $n = 3.15 \pm 0.03$, this pulsar has the highest braking index ever measured. As it is well known, a pure magnetic dipole brake yields $n = 3$, whereas a pure gravitational wave brake yields $n = 5$. Therefore, each of these mechanisms alone can not account for the braking index found for PSR J1640-4631. Here we show that such a braking index can be accounted for if the spindown model combines magnetic dipole and gravitational wave brakes. Then, we briefly discuss the detectability of this pulsar…
The location of radio pulsars in the period-period derivative (P-Pdot) plane has been a key diagnostic tool since the early days of pulsar astronomy. Of particular importance is how pulsars evolve through the P-Pdot diagram with time. Here we show that the decay of the inclination angle (alpha-dot) between the magnetic and rotation axes plays a critical role. In particular, alpha-dot strongly impacts on the braking torque, an effect which has been largely ignored in previous work. We carry out simulations which include a negative alpha-dot term, and show that it is possible to reproduce the ob…
Braking index measurements of young radio pulsars are all smaller than the value expected for spin down by magnetic dipole braking. We investigate magnetic field evolution in the neutron star crust due to Hall drift as an explanation for observed braking indices. Using numerical simulations and a semi-analytic model, we show that a $\approx 10^{14}\ {\rm G}$ quadrupolar toroidal field in the neutron star crust at birth leads to growth of the dipole moment at a rate large enough to agree with measured braking indices. A key factor is the density at which the crust yields to magnetic stresses th…
The inter-glitch timing of the Vela pulsar is characterized by a constant second derivative of the rotation rate. This takes over after the post-glitch exponential relaxation, and is completed at about the time of the next glitch. The vortex creep model explains the second derivatives in terms of non-linear response to the glitch. We present inter-glitch timing fits to the present sample covering 16 large glitches, taking into account the possibility that in some glitches part of the step in spin-down rate may involve a "persistent shift", as observed in the Crab pulsar. Modifying the expressi…
We study the role of time dependent magnetic ellipticities ($ε_{B}$) on the calculation of the braking index of pulsars. Moreover, we study the consequences of such a $ε_{B}$ on the amplitude of gravitational waves (GWs) generated by pulsars with measured braking indices. We show that, since the ellipticity generated by the magnetic dipole is extremely small, the corresponding amplitude of GWs is much smaller than the amplitude obtained via the spindown limit.
The current understanding of the spin evolution of young pulsars is reviewed through a compilation of braking index measurements. An immediate conclusion is that the spin evolution of all pulsars with a measured braking index is not purely caused by a constant magnetic dipole. The case of PSR J1734-3333 and its upward movement towards the magnetars is used as a guide to try to understand why pulsars evolve with n < 3. Evolution between different pulsar families, driven by the emergence of a hidden internal magnetic field, appears as one possible picture.
The pulsar timing is observed to be different from predicted by a simple magnetic dipole radiation. We choose eight pulsars whose braking index was reliably determined. Assuming the smaller values of braking index are dominated by the secular evolution of the magnetic inclination, we calculate the increasing rate of the magnetic inclination for each pulsar. We find a possible relation between the rotation frequency of each pulsar and the inferred evolution of the magnetic inclination. Due to the model-dependent fit of the magnetic inclination and other effects, more observational indicators fo…
We report on the timing properties of the `Crab twin' pulsar PSR~B0540$-$69 measured with X-ray data taken with the Swift telescope over a period of 1100\,days. The braking index of the pulsar was estimated to be $n = 0.03\pm0.013$ in a previous study performed in 2015 with 500-day Swift data. This small value of $n$ is unusual for pulsars, and a comparison to old measurements of $n\approx2.1$ for the same target determined $\sim$10 years ago suggests a dramatic change in the braking index. To confirm the small value and large change of $n$, we used 1100-day Swift observations including the da…
PSR J1734-3333 is a radio pulsar rotating with a period P=1.17 s and slowing down with a period derivative Pdot=2.28 x 10^{-12}, the third largest among rotation-powered pulsars. These properties are midway between those of normal rotation-powered pulsars and magnetars, two populations of neutron stars that are notably different in their emission properties. Here we report on the measurement of the second period derivative of the rotation of PSR J1734-3333 and calculate a braking index n=0.9 +- 0.2. This value is well below 3, the value expected for an electromagnetic braking due to a constant…
In the standard scenario for spin evolution of isolated neutron stars, a young pulsar slows down with a surface magnetic field B that does not change. Thus the pulsar follows a constant B trajectory in the phase space of spin period and spin period time derivative. Such an evolution predicts a braking index n = 3 while the field is constant and n > 3 when the field decays. This contrasts with all nine observed values being n < 3. Here we consider a magnetic field that is buried soon after birth and diffuses to the surface. We use a model of a growing surface magnetic field to fit observations …
Braking index of the frequently glitching PSR J0537$-$6910
School of Arts and Sciences, Qingdao Binhai University（青岛滨海学院文理学院）；National Astronomical Observatories, Chinese Academy of Sciences（中国科学院国家天文台）；Department of Astronomy and Space Sciences, Istanbul University（伊斯坦布尔大学天文学与空间科学系）；Bahçeşehir College（巴切希尔学院）；Faculty of Engineering and Natural Sciences, Sabancı University（萨班哲大学工程与自然科学学院）；Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, The University of Manchester（曼彻斯特大学物理与天文学系乔德雷尔银行天体物理学中心）；Departamento de Física, Universidad de Santiago de Chile (USACH)（圣地亚哥智利大学物理系）
The evolutions of a neutron star's rotation and magnetic field (B-field) have remained unsolved puzzles for over half a century. We ascribe the rotational braking torques of pulsar to both components, the standard magnetic dipole radiation (MDR) and particle wind flow ( MDR + Wind, hereafter named MDRW), which we apply to the Crab pulsar (B0531 + 21), the only source with a known age and long-term continuous monitoring by radio telescope. Based on the above presumed simple spin-down torques, we obtain the exact analytic solution on the rotation evolution of the Crab pulsar, together with the r…
Black widows are millisecond pulsars with low-mass companions, a few per cent the mass of the sun, on orbits of several hours. These companions are presumably the remnants of main sequence stars that lost their mass through a combination of Roche-lobe overflow and ablation by the host pulsar's high-energy radiation. While ablation itself is too weak to significantly reduce the mass of the companion star, the ablated wind couples to its magnetic field, removes orbital angular momentum, and thus maintains stable Roche-lobe overflow. We use the MESA stellar evolution code, complemented by analyti…
Black widows are close binary systems in which a millisecond pulsar is orbited by a companion a few per cent the mass of the sun. It has been suggested that the pulsar's rotationally powered $γ$-ray luminosity gradually evaporates the companion, eventually leaving behind an isolated millisecond pulsar. The evaporation efficiency is determined by the temperature $T_{\rm ch}\propto F^{2/3}$ to which the outflow is heated by the flux $F$ on a dynamical time-scale. Evaporation is most efficient for companions that fill their Roche lobes. In this case, the outflow is dominated by a cap around the L…
