Astro2020 Science White Paper

Physics of cosmic plasmas from high angular resolution X-ray imaging of galaxy clusters

Thematic Areas:                    Planetary Systems     Star and Planet Formation        Formation and Evolution of Compact Objects            Cosmology and Fundamental Physics Stars and Stellar Evolution   Resolved Stellar Populations and their Environments              Galaxy Evolution                Multi-Messenger Astronomy and Astrophysics

Principal Author:

Name: Maxim Markevitch Institution: NASA GSFC Email: Phone: 301-286-5947

Co-authors: Esra Bulbul (CfA), Eugene Churazov (MPA, IKI), Simona Giacintucci (NRL), Ralph Kraft (CfA), Matthew Kunz (Princeton), Daisuke Nagai (Yale), Elke Roediger (Hull), Mateusz Ruszkowski (Michigan), Alex Schekochihin (Oxford), Reinout van Weeren (Leiden), Alexey Vikhlinin (CfA), Stephen A. Walker (NASA GSFC), Qian Wang (Maryland), Norbert Werner (ELTE, Masaryk), Daniel Wik (Utah), Irina Zhuravleva (Chicago), John ZuHone (CfA)


Galaxy clusters are massive dark matter-dominated systems filled with X-ray emitting, optically thin plasma. Their large size and relative simplicity (at least as astrophysical objects go) make them a unique laboratory to measure some of the interesting plasma properties that are inaccessible by other means but fundamentally important for understanding and modeling many astrophysical phenomena — from solar flares to black hole accretion to galaxy formation and the emergence of the cosmological Large Scale Structure. While every cluster astrophysicist is eagerly anticipating the direct gas velocity measurements from the forthcoming microcalorimeters onboard XRISM, Athena and future missions such as Lynx, a number of those plasma properties can best be probed by high-resolution X-ray imaging of galaxy clusters. Chandra has obtained some trailblazing results, but only grazed the surface of such studies. In this white paper, we discuss why we need arcsecond-resolution, high collecting area, low relative background X-ray imagers (with modest spectral resolution), such as the proposed AXIS and the imaging detector of Lynx.

March 11, 2019  v3

Modern astrophysics relies on computer simulations to help us understand complex phenomena in the Universe, from solar flares to supernova explosions, black hole accretion, galaxy formation and the emergence of Large Scale Structure. As supercomputers advance, the benefits of numeric simulations will grow. However, for systems that include plasma, there is a fundamental limitation — we can’t simultaneously model all the relevant linear scales from first principles. For example, turbulence in the cosmological volume is driven by structure formation on the galaxy cluster scale ( cm), but can cascade down to scales as small as the ion gyroradius ( cm), a dynamic range that is impossible to implement in codes. To model such systems, we have to rely on observed plasma properties and encode them at the “subgrid” level. However, many properties that affect large-scale phenomena — viscosity, heat conductivity, energy exchange between the particle populations and the magnetic field — are still unmeasured and their theoretical estimates uncertain by orders of magnitude because of the complexity of the plasma physics. Of course, apart from being “under the hood” of many astrophysical systems, plasma physics is interesting on its own.

Mircoscale phenomena in plasmas (where is the ratio of thermal to magnetic pressure) can be studied in situ in our space neighborhood. Larger scales, including the transition from “kinetic” to “fluid” regime, can be probed in another natural laboratory that is galaxy clusters. Clusters are Megaparsec-size clouds of X-ray emitting, optically thin plasma (ICM), permeated by tangled magnetic fields and ultrarelativistic particles, with typical . This regime is directly relevant for many astrophysical systems, among them SNR, accretion disks and the intergalactic medium.

Several phenomena observed in clusters are sensitive to plasma physics. Turbulence is one, and it will be characterized by the future microcalorimeters (XRISM and Athena) using Doppler shifts of the X-ray emission lines. Several important measurements can be done using high-resolution X-ray imaging. Shock fronts, discovered by Chandra thanks to its sharp mirror, let us study heat conductivity, the electron-ion temperature equilibration and the physics of cosmic ray acceleration1. Another interesting plasma probe is provided by the ubiquitous, sharp contact discontinuities, or “cold fronts”1. While Chandra has obtained tantalizing results, it has only scratched the surface of what can be learned from detailed imaging of these and some other cluster phenomena.

Fig. 1: (a) X-ray image of the Bullet cluster, the textbook example of a bow shock. The shock is driven by a moving subcluster, whose front boundary is a “cold front.” (b) Expected electron and ion temperature profiles across a shock front. Temperatures are unequal immediately after the shock and then equalize. If electron heat conduction is not suppressed, a temperature precursor is also expected. (c) Chandra deprojected electron temperature profile immediately behind the Bullet shock (crosses; errors are ) with models for Coulomb collisional and instant equipartition2. This measurement favors fast electron-proton equilibration, but uncertainties are large.

Plasma Equipartition Times

The common assumption that all particles in a plasma have the same local temperature may not be true if the electron-ion equilibration timescale is longer than heating timescales3, 4. This timescale is fundamental for such processes as accretion onto black holes and X-ray emission from the intergalactic medium. It can be directly measured using cluster shocks.

At a low-Mach shock, ions are dissipatively heated to a temperature , while electrons are adiabatically compressed to a lower . The two species then equilibrate to the mean post-shock temperature5 (Fig. 1). From the X-ray brightness and spectra, we can measure the plasma density and across the shock (this requires only a modest spectral resolution). For the typical low sonic Mach numbers in clusters (), the mean post-shock temperature can be accurately predicted from the shock density jump. If the equilibration is via Coulomb collisions, the region over which the electron temperature increases is tens of kpc wide — resolvable with a Chandra-like telescope at distances of . This direct test is unique to cluster shocks because of the fortuitous combination of the linear scales and relatively low Mach numbers; it cannot be done for the solar wind or SNR shocks.

A Chandra measurement for the Bullet cluster shock (Fig. 1) suggests that equilibration is quicker than Coulomb2, although with a systematic uncertainty that arises from the assumption of symmetry and requires averaging over a sample of shocks. With Chandra, this measurement is limited to only three shocks, and the results are contradictory2, 6, 7. A more sensitive imager is needed to find many more shocks (most of them in the cluster outskirts), select a sample of suitable ones, and robustly determine this basic plasma property.

Fig. 2: Plasma viscosity determines how the gas is stripped from the infalling groups and galaxies. Left: If viscosity is not strongly suppressed, galaxies falling into clusters should exhibit prominent tails of stripped gas8. Middle, right: An infalling galaxy (NGC1404), which appears not to have such a tail9, and a much larger infalling group in the outskirts of a cluster10, which does.

Heat Conductivity

Heat conduction erases temperature gradients and competes with radiative cooling, and is of utmost importance for galaxy and cluster formation. The effective heat conductivity in a plasma with tangled magnetic fields is unknown, with a large uncertainty for the component parallel to the field, which recent theoretical works predict to be reduced11, 12, 13, 14, 15. The existence of cold fronts in clusters confirms that conduction across the field lines is very low16, 17, 18, but constraints for the average or parallel conductivity are poor19, 18. Shock fronts are locations where the parallel component can be constrained, because the field lines should connect the post-shock and pre-shock regions (unlike for the magnetically-insulated cold fronts), though the field structure in the narrow shock layer can be chaotic. Electron-dominated conduction may result in an observable precursor (Fig. 1).

The magnetic field can be stretched and untangled in a predictable way in the cluster sloshing cool cores. The characteristic spiral temperature structure that forms there20 can also be used to constrain parallel conductivity. A telescope with a bigger mirror than Chandra’s could look for temperature precursors in shocks and obtain detailed maps of temperature gradients along the field filaments in many cluster cores to measure the conductivity.


Plasma viscosity is a fundamental quantity that governs damping of turbulence and sound waves, suppression of hydrodynamic instabilities and mixing of different gas phases, and thus relevant to such important processes as heating the gas, spreading metals ejected from galaxies, and amplification of magnetic fields. At present it is largely unknown. Isotropic viscosity can be determined from the dissipation scale of the power spectrum of turbulence. XRISM and Athena will pursue that via the velocity measurements in the ICM, though it is unclear if the dissipation scale will be reachable21. The turbulence spectrum can also be constrained by observing the gas density fluctuations22, 23. However, the plasma viscosity should be anisotropic and may affect turbulence and other phenomena differently. It is thus useful to approach it from several angles. Two subtle phenomena in galaxy cluster images can help us probe the viscosity through its effect on gasdynamic instabilities.

  Galaxy stripping tails.

Figure 2a shows a striking difference in the simulated X-ray appearance of the tail of the cool stripped gas behind a galaxy as it flies through the ICM8. In an inviscid plasma, the gas promptly mixes with the ambient ICM, but a modest viscosity suppresses the mixing and makes the long tail visible. Deep Chandra images of such infalling galaxies NGC1404 (Fig. 2b) and M89 favor efficient mixing and a reduced viscosity9, 24. Other infalling groups in the cluster periphery do exhibit unmixed tails (e.g., Fig. 2c). This points to a possibility of a systematic study to constrain effective viscosity — and directly observe its effect on gas mixing — in various ICM regimes. However, a more sensitive instrument with lower background is required to study these subtle, low-contrast extended features, most of which will be found in the low-brightness cluster outskirts.

Fig. 3: MHD simulation of a sloshing cluster core with viscosity (isotropic) and magnetic field25. X-ray brightness gradients are shown. Initial values are given; sloshing amplifies the magnetic field and produces lower , which result in plasma depletion regions. The appearance of cold fronts can be used to constrain the effective plasma viscosity and magnetic field strength.

  Instabilities in cold fronts.

Cold fronts — contact discontinuities in the ICM that separate regions of different density and temperature in pressure equilibrium1 — are ubiquitous in merging subclusters, where they are seen as sharp X-ray brightness edges (e.g., the “bullet” boundary in the Bullet cluster, Fig. 1a). They are also found in most cool cores, where they emerge as the dense gas of the core “sloshes” in the cluster gravitational well26. Sloshing produces velocity shear across the cold front, which should generate Kelvin-Helmholtz instabilities (Fig. 3a). If the ICM is viscous, K-H instabilities are suppressed27, 28, 29 (Fig. 3b). Chandra has discovered K-H instabilities in a few cold fronts and placed an upper limit on the effective isotropic viscosity of Spitzer30, 31, 32 (or, equivalently in this context, a full Braginskii anisotropic viscosity29). To constrain the viscosity from below requires finding instabilities for a range of density contrasts. These subtle wiggles can be seen only with high resolution and lots of photons, and a systematic study requires a larger-area telescope.

Plasma Depletion Layers

The velocity shear at cold fronts (and elsewhere in the cluster) should stretch and amplify the magnetic fields, forming magnetic layers parallel to the front. Such layers can suppress the instabilities even without the viscosity33, although a certain initial field strength is required (compare Figs. 3a,c). A distinguishing feature between these two suppression mechanisms is seen in Fig. 3c. Wherever the field is amplified, thermal plasma is squeezed out, forming plasma depletion layers (PDL, like the ones in the solar wind around planets34) that can become visible in the X-ray image1.

Fig. 4: Plasma depletion layers in a cluster core. (a) MHD simulation of a sloshing core35; color shows the field strength. As the gas swirls in the core, it forms filaments of stretched and amplified field. (b) Pressure profiles across two filaments, extracted along the line in panel a. While total pressure is monotonic, thermal pressure shows dips (both the density and the temperature dip). (c) Possible observation of such “feathery” structure in the Perseus core36. Subtle X-ray “channels,” possibly of similar origin, have also been seen by Chandra in A52018 and A214232.

In Fig. 4, we show how PDL can form in a sloshing core. Chandra has reported hints of this new phenomenon — low-contrast “channels” in A520 and A214218, 32 and “feathery” structures in Virgo and Perseus37, 36, Fig. 4c). Apart from disentangling the effects of viscosity and magnetic fields on cold front stability, observing PDL in clusters would have a more general significance — it allows us literally to see the structure of the intracluster magnetic field. Combined with the radio images, this can map the distribution of cosmic ray electrons in the ICM. Observing these subtle image features requires many more photons than Chandra can collect for most clusters. A future imager with a much bigger mirror can give us this novel tool for cluster plasma studies.

Cosmic Ray Acceleration

Across the universe, shocks accelerate particles to very high energies via the first-order Fermi mechanism38. Microscopic details of this fundamental process remain poorly known for astrophysical plasmas, and particle-in-cell simulations are still far from covering realistic plasma parameters.

Many galaxy clusters exhibit striking “radio relics” in their outskirts39. These Mpc-long, arc-like structures are synchrotron signatures of ultrarelativistic () electrons. Some relics, as well as sharp edges of giant radio halos, coincide with ICM shocks40, 41, 7, suggesting that shocks have something to do with those electrons. However, the shock Mach numbers are low () and it is unclear how they reach the acceleration efficiency needed to produce the relics42, 43. Other puzzles include similar-Mach shocks that produce very different radio features41 and a relic for which the shock is ruled out44. Particle acceleration in the ICM appears more complex than a classical Fermi picture. Proposed solutions involve re-acceleration of aged relativistic particles43 as well as modifications to the Fermi mechanism in a magnetized plasma. To gain insight into these universal processes, we need a systematic comparison of shocks in the X-ray and radio. However, most radio relics are found far in the cluster outskirts, where the X-ray emission is too dim for Chandra. A low-background, high-area, high-resolution X-ray imager is needed to discover and study shocks there.

Finding Most Powerful Agn Outbursts

AGN that reside in many cluster cores eject copious amounts of energy into the ICM, preventing runaway radiative cooling of the gas at the cluster centers45. They inflate X-ray cavities in the ICM; radio observations show that these cavities are filled with relativistic plasma. A recent discovery of a giant ghost bubble outside the core in Ophiuchus46 suggest that the AGN effects may extend far beyond the cluster cool cores, and that AGN can produce far more powerful outbursts than we infer from the energetics of the cavities in the cluster cores47. If this phenomenon is widespread, as hinted at by recent low-freqency radio surveys by LOFAR and MWA, clusters can be affected more strongly by the AGN feedback than previously thought. Forensic evidence for that can be provided by large, low-contrast ghost cavities outside cluster cores48. Their detection requires a low-background, high-area X-ray imager.

What Kind of Instrument We Need

All the above studies require a much greater collecting area and much lower background than the current X-ray instruments can provide. Critically, they also require high angular resolution — at least Chandra-like — both to resolve the sharp spatial features and to remove the faint point sources of the Cosmic X-ray Background that dominate the flux in the cluster outskirts, where most of those features will be found. AXIS, a proposed Probe, and the imaging detector of Lynx, a proposed Flagship, will have the requisite resolution and photon-collecting capabilities. They will also enable unsurpassed low-background imaging for keV (where the soft diffuse Galactic background becomes insignificant), as shown in the accompanying white paper49.


  • 1 Markevitch, M. & Vikhlinin, A. Shocks and cold fronts in galaxy clusters. Phys. Rep. 443, 1–53 (2007). astro-ph/0701821.
  • 2 Markevitch, M. Chandra Observation of the Most Interesting Cluster in the Universe. In Wilson, A. (ed.) The X-ray Universe 2005, vol. 604 of ESA Special Publication, 723 (2006). astro-ph/0511345.
  • 3 Takizawa, M. Two-Temperature Intracluster Medium in Merging Clusters of Galaxies. ApJ 520, 514–528 (1999).
  • 4 Kawazura, Y., Barnes, M. & Schekochihin, A. A. Thermal disequilibration of ions and electrons by collisionless plasma turbulence. Proc. Nat. Acad. Sci. 116, 771 (2019). arxiv:1807.07702.
  • 5 Zeldovich, Y. B. & Raizer, Y. P. Elements of gasdynamics and the classical theory of shock waves (Academic Press, New York, NY, ed. W.D. Hayes & R.F. Probstein, 1966).
  • 6 Russell, H. R. et al. Shock fronts, electron-ion equilibration and intracluster medium transport processes in the merging cluster Abell 2146. MNRAS 423, 236–255 (2012).
  • 7 Wang, Q. H. S., Giacintucci, S. & Markevitch, M. Bow Shock in Merging Cluster A520: The Edge of the Radio Halo and the Electron-Proton Equilibration Timescale. ApJ 856, 162 (2018).
  • 8 Roediger, E. et al. Stripped Elliptical Galaxies as Probes of ICM Physics: II. Stirred, but Mixed? Viscous and Inviscid Gas Stripping of the Virgo Elliptical M89. ApJ 806, 104 (2015).
  • 9 Su, Y. et al. Deep Chandra Observations of NGC 1404: Cluster Plasma Physics Revealed by an Infalling Early-type Galaxy. ApJ 834, 74 (2017).
  • 10 Eckert, D. et al. The stripping of a galaxy group diving into the massive cluster A2142. A&A 570, A119 (2014).
  • 11 Schekochihin, A. A., Cowley, S. C., Kulsrud, R. M., Rosin, M. S. & Heinemann, T. Nonlinear Growth of Firehose and Mirror Fluctuations in Astrophysical Plasmas. Physical Review Letters 100, 081301 (2008).
  • 12 Kunz, M. W., Schekochihin, A. A. & Stone, J. M. Firehose and Mirror Instabilities in a Collisionless Shearing Plasma. Physical Review Letters 112, 205003 (2014).
  • 13 Komarov, S. V., Churazov, E. M., Kunz, M. W. & Schekochihin, A. A. Thermal conduction in a mirror-unstable plasma. MNRAS 460, 467–477 (2016).
  • 14 Komarov, S., Schekochihin, A. A., Churazov, E. & Spitkovsky, A. Self-inhibiting thermal conduction in a high- , whistler-unstable plasma. Journal of Plasma Physics 84, 905840305 (2018).
  • 15 Roberg-Clark, G. T., Drake, J. F., Swisdak, M. & Reynolds, C. S. Wave Generation and Heat Flux Suppression in Astrophysical Plasma Systems. ApJ 867, 154 (2018).
  • 16 Ettori, S. & Fabian, A. C. Chandra constraints on the thermal conduction in the intracluster plasma of A2142. MNRAS 317, L57–L59 (2000).
  • 17 Vikhlinin, A., Markevitch, M. & Murray, S. S. A Moving Cold Front in the Intergalactic Medium of A3667. ApJ 551, 160–171 (2001).
  • 18 Wang, Q. H. S., Markevitch, M. & Giacintucci, S. The Merging Galaxy Cluster A520—A Broken-up Cool Core, A Dark Subcluster, and an X-Ray Channel. ApJ 833, 99 (2016).
  • 19 Markevitch, M. et al. Chandra Temperature Map of A754 and Constraints on Thermal Conduction. ApJ 586, L19–L23 (2003).
  • 20 ZuHone, J. A., Markevitch, M., Ruszkowski, M. & Lee, D. Cold Fronts and Gas Sloshing in Galaxy Clusters with Anisotropic Thermal Conduction. ApJ 762, 69 (2013).
  • 21 ZuHone, J. A., Markevitch, M. & Zhuravleva, I. Mapping the Gas Turbulence in the Coma Cluster: Predictions for Astro-H. ApJ 817, 110 (2016).
  • 22 Schuecker, P., Finoguenov, A., Miniati, F., Böhringer, H. & Briel, U. G. Probing turbulence in the Coma galaxy cluster. A&A 426, 387–397 (2004).
  • 23 Zhuravleva, I. et al. Gas density fluctuations in the Perseus Cluster: clumping factor and velocity power spectrum. MNRAS 450, 4184–4197 (2015).
  • 24 Kraft, R. P. et al. Stripped Elliptical Galaxies as Probes of ICM Physics. III. Deep Chandra Observations of NGC 4552: Measuring the Viscosity of the Intracluster Medium. ApJ 848, 27 (2017).
  • 25 Bellomi, E., ZuHone, J. A., Ntampaka, M. & Forman, W. in prep. (2019).
  • 26 Ascasibar, Y. & Markevitch, M. The Origin of Cold Fronts in the Cores of Relaxed Galaxy Clusters. ApJ 650, 102–127 (2006).
  • 27 Churazov, E. & Inogamov, N. Stability of cold fronts in clusters: is magnetic field necessary? MNRAS 350, L52–L56 (2004).
  • 28 Roediger, E. et al. Viscous Kelvin-Helmholtz instabilities in highly ionized plasmas. MNRAS 436, 1721–1740 (2013).
  • 29 ZuHone, J. A., Kunz, M. W., Markevitch, M., Stone, J. M. & Biffi, V. The Effect of Anisotropic Viscosity on Cold Fronts in Galaxy Clusters. ApJ 798, 90 (2015).
  • 30 Roediger, E., Kraft, R. P., Forman, W. R., Nulsen, P. E. J. & Churazov, E. Kelvin-Helmholtz Instabilities at the Sloshing Cold Fronts in the Virgo Cluster as a Measure for the Effective Intracluster Medium Viscosity. ApJ 764, 60 (2013).
  • 31 Ichinohe, Y., Simionescu, A., Werner, N. & Takahashi, T. An azimuthally resolved study of the cold front in Abell 3667. MNRAS 467, 3662–3676 (2017).
  • 32 Wang, Q. H. S. & Markevitch, M. A Deep X-Ray Look at Abell 2142 – Viscosity Constraints From Kelvin-Helmholtz Eddies, a Displaced Cool Peak That Makes a Warm Core, and A Possible Plasma Depletion Layer. ApJ 868, 45 (2018).
  • 33 Vikhlinin, A., Markevitch, M. & Murray, S. S. Chandra Estimate of the Magnetic Field Strength near the Cold Front in A3667. ApJ 549, L47–L50 (2001).
  • 34 Øieroset, M. et al. The Magnetic Field Pile-up and Density Depletion in the Martian Magnetosheath: A Comparison with the Plasma Depletion Layer Upstream of the Earth’s Magnetopause. Space Sci. Rev. 111, 185–202 (2004).
  • 35 ZuHone, J. A., Markevitch, M. & Lee, D. Sloshing of the Magnetized Cool Gas in the Cores of Galaxy Clusters. ApJ 743, 16 (2011).
  • 36 Ichinohe, Y., Simionescu, A., Werner, N., Fabian, A. C. & Takahashi, T. Substructures associated with the sloshing cold front in the Perseus cluster. MNRAS 483, 1744–1753 (2019).
  • 37 Werner, N. et al. Deep Chandra observation and numerical studies of the nearest cluster cold front in the sky. MNRAS 455, 846–858 (2016).
  • 38 Blandford, R. & Eichler, D. Particle acceleration at astrophysical shocks: A theory of cosmic ray origin. Phys. Rep. 154, 1–75 (1987).
  • 39 van Weeren, R. J., Röttgering, H. J. A., Brüggen, M. & Hoeft, M. Particle Acceleration on Megaparsec Scales in a Merging Galaxy Cluster. Science 330, 347 (2010).
  • 40 Giacintucci, S. et al. Shock acceleration as origin of the radio relic in A521? A&A 486, 347–358 (2008).
  • 41 Shimwell, T. W. et al. Another shock for the Bullet cluster, and the source of seed electrons for radio relics. MNRAS 449, 1486–1494 (2015).
  • 42 Macario, G. et al. A Shock Front in the Merging Galaxy Cluster A754: X-ray and Radio Observations. ApJ 728, 82 (2011).
  • 43 Brunetti, G. & Jones, T. W. Cosmic Rays in Galaxy Clusters and Their Nonthermal Emission. International Journal of Modern Physics D 23, 1430007–98 (2014). arxiv:1401.7519.
  • 44 Markevitch, M., Wik, D. R. & van Weeren, R. In prep.; talk at Diffuse Synchrotron Emission in Clusters of Galaxies, Leiden 2017 (2019).
  • 45 McNamara, B. R. & Nulsen, P. E. J. Heating Hot Atmospheres with Active Galactic Nuclei. ARA&A 45, 117–175 (2007).
  • 46 Giacintucci, S., Markevitch, M., Johnston-Hollitt, M. & Wik, D. R. Discovery of a giant radio fossil in the Ophiuchus galaxy cluster. In prep.; talk at SnowCluster 2018 (2019).
  • 47 McNamara, B. R. et al. The heating of gas in a galaxy cluster by X-ray cavities and large-scale shock fronts. Nature 433, 45–47 (2005).
  • 48 Sanders, J. S., Fabian, A. C. & Taylor, G. B. Giant cavities, cooling and metallicity substructure in Abell 2204. MNRAS 393, 71–82 (2009).
  • 49 Walker, S. A., Nagai, D., Simionescu, A. & Markevitch, M. Unveiling the Galaxy Cluster –- Cosmic Web Connection with X-ray Observations in the Next Decade. White Paper for Astro-2020 Decadal Survey (2019).


SG acknowledges 6.1 Base funding for Basic research in radio astronomy at the Naval Research Laboratory.

Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
Add comment
Loading ...
This is a comment super asjknd jkasnjk adsnkj
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test description