Outer Disk Radius Research Articles

The effect of thermal winds on the outbursts evolution of LMXB systems

ABSTRACT Theoretical models of accretion discs and observational data indicate that the X-ray emission from the inner parts of an accretion disc can irradiate its outer regions and induce a thermal wind, which carries away the mass and angular momentum from the disc. Our aim is to investigate the influence of the thermal wind on the outburst light curves of black hole X-ray binary systems. We carry out numerical simulations of a non-stationary disc accretion with wind using upgraded open code freddi. We assume that the wind launches only from the ionized part of the disc and may turn off if the latter shrinks fast enough. Our estimates of the viscosity parameter α are shifted downward compared to a scenario without a wind. Generally, correction of α depends on the spectral hardness of central X-rays and the disc outer radius, but unlikely to exceed a factor of 10 in the case of a black hole low-mass X-ray binary (BH LMXB). We fit 2002 outburst of BH LMXB 4U 1543 − 47 taking into account the thermal wind. The mass-loss in the thermal wind is of order of the accretion rate on the central object at the peak of the outburst. New estimate of the viscosity parameter α for the accretion disc in this system is about two times lower than the previous one. Additionally, we calculate evolution of the number of hydrogen atoms towards 4U 1543 − 47 due to the thermal wind from the hot disc.

An Optically Discovered Outburst from XTE J1859+226

Using the Zwicky Transient Facility, in 2021 February we identified the first known outburst of the black hole X-ray transient XTE J1859+226 since its discovery in 1999. The outburst was visible at X-ray, UV, and optical wavelengths for less than 20 days, substantially shorter than its full outburst of 320 days in 1999, and the observed peak luminosity was 2 orders of magnitude lower. Its peak bolometric luminosity was only 2 × 1035 erg s−1, implying an Eddington fraction of about 3 × 10−4. The source remained in the hard spectral state throughout the outburst. From optical spectroscopy measurements we estimate an outer disk radius of 1011 cm. The low observed X-ray luminosity is not sufficient to irradiate the entire disk, but we observe a surprising exponential decline in the X-ray light curve. These observations highlight the potential of optical and infrared synoptic surveys to discover low-luminosity activity from X-ray transients.

Large Myr-old Disks Are Not Severely Depleted of Gas-phase CO or Carbon

We present an ACA search for [C i] 1–0 emission at 492 GHz toward large T Tauri disks (gas radii ≳ 200 au) in the ∼1–3 Myr-old Lupus star-forming region. Combined with Atacama Large Millimeter/submillimeter Array 12 m archival data for IM Lup, we report [C i] 1–0 detections in six out of 10 sources, thus doubling the known detections toward T Tauri disks. We also identify four Keplerian double-peaked profiles and demonstrate that the [C i] 1–0 fluxes correlate with 13CO, C18O, and 12CO(2–1) fluxes, as well as with the gas disk outer radius measured from the latter transition. These findings are in line with the expectation that atomic carbon traces the disk surface. In addition, we compare the carbon and carbon monoxide (CO) line luminosities of a Lupus and literature sample with [C i] 1–0 detections with predictions from the self-consistent disk thermo-chemical models of Ruaud et al. These models adopt interstellar medium carbon and oxygen elemental abundances as input parameters. With the exception of the disk around Sz 98, we find that these models reproduce all the available line luminosities and upper limits, with gas masses comparable to or higher than the minimum-mass solar nebula and gas-to-dust mass ratios ≥10. Thus, we conclude that the majority of large Myr-old disks conform to the simple expectation that they are not significantly depleted in gas, CO, or carbon.

Optical/UV emission in the Tidal Disruption Event ASASSN-14li: implications of disc modelling

ABSTRACT We predict late-time optical/UV emission from tidal disruption events (TDEs) from our slim accretion disc model (Wen et al. 2020) and explore the impact of the black hole mass M•, black hole spin a•, and accretion disc size. We use these synthetic spectra to successfully fit the multiband Swift observations of ASASSN-14li at >350 d, setting only the host galaxy extinction and outer disc radius as free parameters and employing the M•, a•, disc inclination, and disc accretion rates derived from fitting 10 epochs of ASASSN-14li’s X-ray spectra with the slim disc. To address the nature of the early-time optical/UV emission, we consider two models: shock dissipation and reprocessing. We find that (1) the predicted late-time optical/UV colour (e.g. u − w2) is insensitive to black hole and disc parameters unless the disc spreads quickly; (2) a starburst galaxy extinction model is required to fit the data, consistent with ASASSN-14li’s post-starburst host; (3) surprisingly, the outer disc radius is ≈2 × the tidal radius and ∼constant at late times, showing that viscous spreading is slow or non-existent; (4) the shock model can be self-consistent if M• ≲ 106.75 M⊙, i.e. on the low end of ASASSN-14li’s M• range (106.5–7.1 M⊙; 1σ CL); larger black hole masses require disruption of an unrealistically massive progenitor star; (5) the gas mass needed for reprocessing, whether by a quasi-static or an outflowing layer, can be <0.5 M⊙, consistent with a (plausible) disruption of a solar-mass star.

Modified models of radiation pressure instability applied to 10, 105, and 107 M⊙ accreting black holes

Context. Some accreting black holes exhibit much stronger variability patterns than the usual stochastic variations. Radiation pressure instability is one of the proposed mechanisms that might account for this effect. Aims. We model luminosity changes for objects with a black hole mass of 10, 105, and 107 solar masses, using the time-dependent evolution of an accretion disk that is unstable as a result of the dominant radiation pressure. We concentrate on the outburst timescales. We explore the influence of the hot coronal flow above the cold disk, the inner purely hot flow, and the effect of the magnetic field on the time evolution of the disk-corona system. For intermediate-mass black holes and active galactic nuclei, we also explore the role of the disk outer radius because a disk that is fed by tidal disruption events (TDE) can be quite small. Methods. We used a 1D vertically integrated time-dependent numerical scheme that models the simultaneous evolution of the disk and corona, which is coupled by the vertical mass exchange. We parameterized the strength of the large-scale toroidal magnetic fields according to a local accretion rate. We also discuss a possible inner optically thin flow, the advection-dominated accretion flow (ADAF). This flow would require modification of the inner boundary condition of the cold disk flow. For the set of the global parameters, we calculated the variability timescales and outburst amplitudes of the disk and the corona. Results. We found that the role of the inner ADAF and the accreting corona are relatively unimportant, but the outburst character strongly depends on the magnetic field and on the outer radius of the disk if this radius is smaller (due to the TDE phenomenon) than the size of the instability zone in a stationary disk with infinite radius. For microquasars, the dependence on the magnetic field is monotonic, and the period decreases with the field strength. For higher black hole masses, the dependence is nonmonotonic, and an initial rise of the period is later replaced with a relatively rapid decrease as the magnetic field continues to rise. A still stronger magnetic field stabilizes the disk. When we assumed a smaller disk outer radiusfor 105 and 107 M⊙, the outbursts were shorter and led to complex multiscale outbursts for some parameters, thus approaching the behavior of deterministic chaos. Conclusions. Our computations confirm that the radiation pressure instability model can account for heartbeat states in microquasars. The rapid variability detected in intermediate-mass black holes in the form of quasi-periodic eruptions can be consistent with the model, but only when it is combined with the TDE phenomenon. The yearly repeating variability in changing-look active galactic nuclei in our model also requires a small outer radius either due to the recent TDE or due to the gap in the disk that is related to a secondary black hole.

A Complex Dust Morphology in the High-luminosity AGN Mrk 876

Recent models for the inner structures of active galactic nuclei (AGNs) advocate the presence of a radiatively accelerated dusty outflow launched from the outer regions of the accretion disk. Here, we present the first near-IR variable (rms) spectrum for the high-luminosity nearby AGN Mrk 876. We find that it tracks the accretion disk spectrum out to longer wavelengths than the mean spectrum, due to a reduced dust emission. The implied outer accretion disk radius is consistent with the IR results predicted by a contemporaneous optical accretion disk reverberation mapping campaign, and much larger than the self-gravity radius. The reduced flux variability of the hot dust could either be due to the presence of a secondary constant dust component in the mean spectrum or be introduced by the destructive superposition of the dust and accretion disk variability signals, or be some combination of the two. Assuming thermal equilibrium for optically thin dust, we derive the luminosity-based dust radii for different grain properties, using our measurement of the temperature. We find that in all the cases considered, the values are significantly larger than the dust response time measured by IR photometric monitoring campaigns, with the least discrepancy present relative to the result for a wavelength-independent dust emissivity law, i.e., a blackbody, which is appropriate for large grain sizes. This result can be well explained by assuming a flared disk-like structure for the hot dust.

On rapid binary mass transfer – I. Physical model

ABSTRACT In some semidetached binary systems, the donor star may transfer mass to the companion at a very high rate. We propose that, at sufficiently high mass-transfer rates such that the accretion disc around the companion becomes geometrically thick (or advection-dominated) near the disc outer radius, a large fraction of the transferred mass may be lost through the outer Lagrangian (L2) point, as a result of the excessive energy generated by viscous heating that cannot be efficiently radiated away. A physical model is constructed where the L2 mass-loss fraction is given by the requirement that the remaining material in the disc has Bernoulli number equal to the L2 potential energy. Our model predicts significant L2 mass-loss at mass transfer rates exceeding $\mbox{a few}\, 10^{-4}\, {\mathrm{ M}_\odot \, \mathrm{yr}^{-1}}$. An equatorial circumbinary outflow (CBO) is formed in these systems. Implications for the orbital evolution and the observational appearance of the system are discussed. In particular, (1) rapid angular momentum loss from the system tends to shrink the orbit, and hence may increase the formation rate of mergers and gravitational-wave sources; and (2) photons from the hot disc wind are reprocessed by the CBO into longer wavelength emission in the infrared bands, consistent with Spitzer observations of some ultra-luminous X-ray sources.

Bifurcation buckling of isotropic annular disc using conforming and non-conforming finite element

Non-conforming and conforming polynomial is used to develop sector finite element for analysing the isothermal bifurcation buckling of isotropic annular disc. The sector finite element has three degrees of freedom for non-conforming and four degrees of freedom for conforming element respectively. To obtain the shape function for the sector finite element, the displacement polynomial is chosen from the Pascal's triangle, the displacement polynomial is used to obtain the polynomial corresponding to the nodal degree of freedom for the element and evaluated at each node of the sector finite element using the nodal coordinates. The kinematics, strain displacement relations and the stress strain relations is based on the Kirchhoff’s plate theory. The stiffness matrix and geometric stiffness matrix are evaluated in MATHEMATICA and then imported in the FORTRAN complier. A FORTRAN CODE is developed to solve the eigenvalue problem for bifurcation buckling of clamped-clamped isotropic annular disc with uniform temperature rise. ORIGIN software is used to plot the buckled mode shape for non-conforming and conforming sector finite element for isotropic annular disc. The number of circumferential waves at the onset of bifurcation buckling increase as the radius ratio increases. The critical buckling temperature increases with increase in thickness of the annular disc, so is the case when the inner radius increases for a given outer radius and thickness of the annular disc.

Eccentric Dust-ring Formation in Kozai–Lidov Gas Disks

Abstract A highly misaligned gas disk around one component of a binary star system can undergo global Kozai–Lidov (KL) oscillations for which the disk inclination and eccentricity are exchanged. With hydrodynamical simulations of a gas and dust disk we explore the effects of these oscillations on the dust density distribution. For dust that is marginally coupled to the gas (St ≈ 1), we find that the dust undergoes dynamical behavior similar to that of the gas disk but the radial distribution of dust may be very different from that of the gas. The inward radial drift of the dust is faster in an eccentric disk leading to a smaller outer dust disk radius. The dust breaks into multiple narrow eccentric rings during the highly eccentric disk phase. Eccentric dust-ring formation may have significant implications for the formation of planets in misaligned disks. We suggest that multiple dust rings may generally occur within gas disks that have sufficiently strong eccentricity peaks at intermediate radii.

Linking Outer Disk Pebble Dynamics and Gaps to Inner Disk Water Enrichment

Abstract Millimeter continuum imaging of protoplanetary disks reveals the distribution of solid particles and the presence of substructures (gaps and rings) beyond 5–10 au, while infrared (IR) spectra provide access to abundances of gaseous species at smaller disk radii. Building on recent observational findings of an anti-correlation between the inner disk water luminosity and outer dust disk radius, we aim here at investigating the dynamics of icy solids that drift from the outer disk and sublimate their ice inside the snow line, enriching the water vapor that is observed in the IR. We use a volatile-inclusive disk evolution model to explore a range of conditions (gap location, particle size, disk mass, and α viscosity) under which gaps in the outer disk efficiently block the inward drift of icy solids. We find that inner disk vapor enrichment is highly sensitive to the location of a disk gap, yielding for each particle size a radial “sweet spot” that reduces the inner disk vapor enrichment to a minimum. For pebbles of 1–10 mm in size, which carry the most mass, this sweet spot is at 7–15 au, suggesting that inner gaps may have a key role in reducing ice delivery to the inner disk and may not allow the formation of Earths and super-Earths. This highlights the importance of observationally determining the presence and properties of inner gaps in disks. Finally, we argue that the inner water vapor abundance can be used as a proxy for estimating the pebble drift efficiency and mass flux entering the inner disk.

The Be Star 66 Ophiuchi: 60 Years of Disk Evolution

Abstract We use a time-dependent hydrodynamic code and a non-LTE Monte Carlo code to model disk dissipation for the Be star 66 Ophiuchi. We compiled 63 years of observations from 1957 to 2020 to encompass the complete history of the growth and subsequent dissipation of the star’s disk. Our models are constrained by new and archival photometry, spectroscopy, and polarization observations, allowing us to model the disk dissipation event. Using Markov Chain Monte Carlo methods, we find that the properties of 66 Oph are consistent with those of a standard B2Ve star. We computed a grid of 61,568 Be star disk models to constrain the density profile of the disk before dissipation using observations of the Hα line profile and spectral energy distribution. We find at the onset of dissipation the disk has a base density of 2.5 × 10−11 g cm−3 with a radial power-law index of n = 2.6. Our models indicate that after 21 yr of disk dissipation 66 Oph’s outer disk remained present and bright in the radio. We find an isothermal disk with constant viscosity with an α = 0.4 and an outer disk radius of ∼115 stellar radii best reproduces the rate of 66 Oph’s disk dissipation. We determined the interstellar polarization in the direction of the star in the V band is p = 0.63 ± 0.02% with a polarization position angle of θ IS ≈ 857 ± 07. Using the Stokes QU diagram, we find the intrinsic polarization position angle of 66 Oph’s disk is θ int ≈ 98° ± 3°.

X-ray Dips in AGN and Microquasars – Collapse Timescales of Inner Accretion Disc

Abstract The temporal behaviour of X-rays from some AGN and microquasars is thought to arise from the rapid collapse of the hot, inner parts of their accretion discs. The collapse can occur over the radial infall timescale of the inner accretion disc. However, estimates of this timescale are hindered by a lack of knowledge of the operative viscosity in the collisionless plasma comprising the inner disc. We use published simulation results for cosmic ray diffusion through turbulent magnetic fields to arrive at a viscosity prescription appropriate to hot accretion discs. We construct simplified disc models using this viscosity prescription and estimate disc collapse timescales for 3C 120, 3C 111, and GRS 1915+105. The Shakura-Sunyaev α parameter resulting from our model ranges from 0.02 to 0.08. Our inner disc collapse timescale estimates agree well with those of the observed X-ray dips. We find that the collapse timescale is most sensitive to the outer radius of the hot accretion disc.

Stress intensity factor solutions and equations for radial surface cracks contained in rotating hollow disks

This study aims at introducing generalized fracture solutions for 3-D radial semi-elliptical surface cracks contained in rotating hollow disks to the literature. Normalized mode-I stress intensity factors (SIFs) are presented for a wide range of cases that involve different values of crack aspect ratio, depth, radial coordinate and geometric parameters of the disk. By leveraging the obtained numerical solutions, regression-based empirical equations are also generated as functions of the governing parameters of the problem. It is shown that for a specific crack shape and normalized depth, normalized SIFs decrease with increasing radial position and increase with increasing ratio of the inner and outer disk radii. A tangential stress-based procedure is also introduced and verified that can be used to determine the normalized SIFs for a crack of a radial position by employing the normalized SIFs of another crack of same features and located at a different radial coordinate. The usage and accuracy of the developed empirical equations are also verified on radial surface cracks in flat rotating disks and disks with practically used geometric details, such as outer rim and inner bore.

Irregular emission cycles in the Oe star HD 60 848

We analyse the spectroscopic and photometric variability of the Oe star HD 60 848 over the last twenty five years. The spectra reveal recurrent, but irregular cycles of increased circumstellar emission lines. These cycles are highly asymmetric displaying a slow increase over about 6 years, followed by a fast decay within about 6 months. Our analysis focuses on the most recent cycle (2013 – 2020). The equivalent width and velocity separation of the emission peaks indicate variations of the outer disk radius by a factor ~ 2.2, although the variability appears more complex than expected from first principle relations for optically thin Keplerian disks. We observe a time delay between the variations of the strengths of He iλ 5876 on the one hand and Hα and Hβ on the other hand. We interpret this behaviour in a two-step disk growth scenario, where the disk first expands radially before its density increases. A difference in behaviour is also seen between Hα and the H i Paschen lines, with the latter displaying a more symmetric cycle, similar to the photometric variability. The rather fast decays of the Hα emission observed in 2001, 2009 and 2018 – 2019 suggest that the strong radiation field and early spectral type of the star lead to a faster dissipation of the disk than in later-type Be stars, as theoretically expected. We discuss X-ray observations of the star both during a high and a low-emission state. The X-ray spectrum is soft at both epochs, and the X-ray fluxes are only marginally different and remain consistent with the canonical LX/Lbol relation of O-type stars. These results indicate that the circumstellar decretion disk of HD 60 848 has essentially no impact on the star’s X-ray emission, and that the latter most likely arises inside the stellar wind.

The performance of Cu-based friction material in dry clutch engagement

An enhanced Cu-based friction material was prepared by the powder metallurgy techniques and proposed for use in the dry clutch system. The friction characteristics and wear rate of this friction material sliding against 65Mn steel are obtained using Universal Material Tester-5. The friction pairs were subjected to two operating variables, which are sliding speed and temperature. The effect of these variables during the engagement process of the friction pairs is investigated. Knowing the normal applied force and dimension of the clutch disc, the dynamic friction coefficient was translated to friction torque capacity with time. It was found that instability can be excited at low operational conditions when the resulting friction coefficient is high. At 25 ℃, the dynamic friction torque oscillates with time likewise at 400 ℃. Generally, a more stable friction torque is obtained when the sliding speed is varied compared to varying the temperatures. Moreover, the influence of the operating temperatures and sliding speeds on thermal buckling and thermoelastic instability of the friction disc is the second consideration in this work. The onset of thermoelastic instability occurs when the sliding speed exceeded 200 r/min and the results for the growth rate of hot spots were found to agree well with the critical speed of the system. Also, thermal buckling was highly dependent on the temperature difference between the inner and outer radius of the friction disc.

Dual-wavelength ALMA Observations of Dust Rings in Protoplanetary Disks

Abstract We present new Atacama Large Millimeter/submillimeter Array observations for three protoplanetary disks in Taurus at 2.9 mm and comparisons with previous 1.3 mm data both at an angular resolution of ∼0.″1 (15 au for the distance of Taurus). In the single-ring disk DS Tau, double-ring disk GO Tau, and multiring disk DL Tau, the same rings are detected at both wavelengths, with radial locations spanning from 50 to 120 au. To quantify the dust emission morphology, the observed visibilities are modeled with a parametric prescription for the radial intensity profile. The disk outer radii, taken as 95% of the total flux encircled in the model intensity profiles, are consistent at both wavelengths for the three disks. Dust evolution models show that dust trapping in local pressure maxima in the outer disk could explain the observed patterns. Dust rings are mostly unresolved. The marginally resolved ring in DS Tau shows a tentatively narrower ring at the longer wavelength, an observational feature expected from efficient dust trapping. The spectral index (α mm) increases outward and exhibits local minima that correspond to the peaks of dust rings, indicative of the changes in grain properties across the disks. The low optical depths (τ ∼ 0.1–0.2 at 2.9 mm and 0.2–0.4 at 1.3 mm) in the dust rings suggest that grains in the rings may have grown to millimeter sizes. The ubiquitous dust rings in protoplanetary disks modify the overall dynamics and evolution of dust grains, likely paving the way toward the new generation of planet formation.

Stress intensity factors and equations for tangential surface cracks in rotating hollow disks

In this study, normalized mode-I stress intensity factors (SIFs) along fronts of three-dimensional tangential surface cracks contained in rotating hollow disks are presented. A wide range of surface cracks with different values of crack aspect ratio, crack depth and radial coordinate along with different geometric parameter settings of the disk are included in the study, yielding a large class of solutions that may be faced with in practical engineering applications. Equations are also developed to allow computations of SIFs for any values of the parameters defining the problem within their practical limits. The results show that for a given crack shape and normalized depth, normalized SIFs decrease with increasing radial coordinate of the crack and with increasing ratio of the inner and outer disk radii. It is also observed that normalized SIFs increase slightly with increasing normalized thickness of the disk for a given crack shape, location and depth. A stress-based procedure is also introduced and validated that can be used to determine the normalized SIFs for a crack located at a radial coordinate based on the normalized SIFs of another crack located at a different radial position.

The hydrogen Balmer lines and jump in absorption in accretion disc modelling – an ultraviolet–optical spectral analysis of the dwarf novae UZ Serpentis and CY Lyrae

ABSTRACT The spectra of disc-dominated cataclysmic variables (CVs) often deviate from the spectra of accretion disc models; in particular, the Balmer jump and absorption lines are found to be shallower in the observations than in the models. We carried out a combined ultraviolet–optical spectral analysis of two dwarf novae (DNe): UZ Ser in outburst, decline, and quiescence, and CY Lyr on the rise to outburst and in outburst. We fit the Balmer jump and absorption lines, the continuum flux level and slope by adjusting the accretion rate, inclination, and disc outer radius. For both systems, we find an accretion rate $\dot{M} \approx 8 \times 10^{-9}\,\mathrm{ M}_\odot\,\mathrm{ yr}^{-1}$ in outburst, and $\dot{M} \approx 2-3 \times 10^{-9}\,\mathrm{ M}_\odot\,\mathrm{ yr}^{-1}$ for the rise and decline phases. The outer disc radius we derive is smaller than expected (Rdisc ≈ 0.2a, where a is the binary separation), except during late rise (for CY Lyr) where Rdisc = 0.3a. UZ Ser also reveals a 60 000 K white dwarf. These results show that during a DN cycle the radius of the disc is the largest just before the peak of the outburst, in qualitative agreement with the disc instability model for DN outbursts. We suspect that an additional emitting component (e.g. disc wind) is also at work to reduce the slope of the continuum and size of the Balmer jump and absorption lines. We stress that both the outer disc radius and disc wind need to be taken into account for more realistic disc modelling of CVs.

Assessment of creep in composite disc having exponential, hyperbolic and uniform thickness profiles

Analysis has been carried out to model second stage creep in a composite (Al-SiCp) disc rotating at high speed under elevated temperature. The model has been developed for various kinds of disc thickness profiles, viz. exponential, hyperbolic and constant, with the same total volume. The impact of varying the disc geometry has been assessed through the stresses and rates of creep induced strains in different discs. The results indicate that the stresses and strain rates in composite disc with exponential thickness variation are the lowest for similar operating conditions. The creep response of exponential disc is observed to further improve when its thickness gradient is increased, achieved by increasing the thickness near the inner and decreasing towards the outer disc radius, without altering the average disc thickness. The use of steeper thickness profile could help in reducing the possibility of creep induced deformations in the rotating disc.

A Novel Axial Vibration Model of Multistage Pump Rotor System with Dynamic Force of Balance Disc

In this paper, a novel axial dynamic model including the transient force of balance disc is proposed to predict the vibration characteristics of multistage pump rotor system. To obtain the dynamic force, the rotating effect of the rotor system is considered and the Navier–Stokes equations are further simplified on the basis of the geometric structure and inner flow characteristic of the balance disc. In addition, based on finite element method and matrix operation, a novel axial motion model of rotor system is established. The efficient Newmark method is applied to describe the dynamic response of the coupled rotor system. The pressure distribution in axial clearance and the corresponding dynamic force present obvious nonlinear reduction as the axial gap increases from 0.2 mm to 1 mm. The inner chamber pressure is more sensitive to the inlet pressure than the rotating speed, especially when the axial gap is 0.2 mm. Moreover, the axial steady amplitude of the rotor system is directly proportional to the rotating speed and initial axial gap, while it is inversely proportional to the outer radius of the balance disc. In addition, the vibration frequencies for axial vibration are multiple even when the motion of the rotor system is regularly reciprocating. The transient force of the balance disc needs to be considered in the calculation of axial rotor dynamics for the multistage pump. The calculated results can provide references for the design of the balance disc and an axial vibration model of the multistage pump rotor system.