arXiv subjects

W. R. Webber

Publications and source records attributed to W. R. Webber.

At least 37 records · Page 2Linked to original sources

arXiv2015

A Galactic Cosmic Ray Electron Spectrum from 2 MeV to 2 TeV That Fits Voyager 5-60 MeV Data at Low Energies and PAMELA and AMS-2 Data at 10 GeV Using an Electron Source Spectrum E^-2.25 A Calculation Using a Monte Carlo Diffusion Model

In this paper we fit the observed galactic cosmic ray electron spectrum from a few MeV to 1 TeV. New data from Voyager from 5-60 MeV beyond the heliopause is used along with high energy data from the PAMELA, FERMI and AMS-2 instruments in Earth orbit. Using a Monte Carlo diffusion model for galactic propagation we obtain a source rigidity spectrum with a spectral index =-2.25 independent of energy below 10 GeV, possibly steepening above 10 GeV to 2.40 at the highest energies. This spectrum will fit the electron data over 5 orders of magnitude to within + 10% at both low and high energies. This steepening of the electron source spectrum could be an important feature of the acceleration process, e.g., synchrotron loss during acceleration could steepen the source spectrum. This fit requires only a single break in the rigidity dependence of the diffusion coefficient by ~1.0 power in the exponent at about 1.0 GV. The calculations also predict the distribution of electrons perpendicular to the galactic disk. The galaxy does act as a calorimeter for electrons in the energy range 0.1-10 GeV where ~80% of the electrons are trapped. At higher energies the electrons escape the galactic disk rapidly. At energies ~1 TEV the electrons loose most of their energy by synchrotron and inverse Compton emission within < 0.5 Kpc of their origin near the galactic equator. At energies < 0.1 GeV, because of the increase in the diffusion coefficient below ~1 GV, and the absence of other significant loss processes, the electrons escape and form a difficult to detect, but important galactic halo population.

physics.space-ph↗

arXiv2015

Interstellar Spectra of Primary Cosmic Ray Nuclei from H through Fe Measured at Voyager and a Comparison with Higher Energy Measurements, An Interpretation of the Spectra from 10 MeVnuc to Over 100 GeVnuc Using a Leaky Box Model

Utilizing new Voyager measurements at lower energies and higher energy spacecraft measurements near the Earth, the interstellar spectra of primary cosmic ray nuclei from H to Fe have now been determined from ~10 MeV/nuc to > 100 GeV/nuc. These measurements are compared with the predictions from a Leaky Box propagation model. It is found that above ~50-100 MeV/nuc the spectra of all the nuclei, H, He, C, O, Ne, Mg, Si and Fe are well connected between 100 MeV/nuc and 10 GeV/nuc and above by simple source rigidity spectra proportional to P-2.28, with the exponent independent of rigidity and using a rigidity dependent diffusion coefficient ~P0.50 above ~1.0 GV. This leads to intensities and spectra ~P-2.78 at ~100 GeV/nuc and above, which are consistent with new AMS-2 and PAMELA measurements of H, He and C to within + 10%. Below 50-100 MeV the spectra of these primary charges fall into two groups. The spectra of primary nuclei with Z > 6 fall more rapidly at low energies than those of H and He and more rapidly than would be expected from a pure diffusive LBM, where energy loss by ionization is important at low energies. Increasing the fraction of ionized hydrogen in the interstellar medium from 15% to 30% does not satisfactorily explain the systematics of the intensity decreases of the heavier primary nuclei at low energies. Other processes beyond those assumed in a simple leaky box model, such as a lack of cosmic ray sources near the Sun are being investigated as the source of this more rapid intensity decrease of these nuclei. The spectra of H and He are very similar to each other but both contain more low energy particles than would be expected from a correspondence with the heavier nuclei spectra. We believe that this may indicate the presence of a local (heliospheric) component of these nuclei in addition to the galactic component.

physics.space-ph↗

arXiv2015

The Cosmic Ray Helium and Carbon Nuclei Spectra Measured by Voyager 1 at Low Energies and Earth Based Measurements of these Nuclei up to 200 GeV nuc Concordance at High Energies with a Leaky Box Propagation Model

A comparison of the Helium and Carbon interstellar spectra measured at Voyager in the local interstellar medium leads to a different interpretation than a comparison of the Hydrogen to Helium spectra. This is because the He/C ratio is observed to increase rapidly with energy below 40 MeV/nuc in contrast to an almost constant H/He ratio at these low energies. Both the He and C spectra that are observed at Voyager above 40 MeV/nuc and much higher energy spectra from the PAMELA measurements of these two components up to 100 GeV/nuc can be accurately fit to within 10% assuming galactic propagation in a leaky box type of diffusion model in the galaxy with identical source spectra P-2.28 for He and C using a diffusion coefficient P0.50 above 1 GV rigidity. These same exponents also fit the H spectrum from 40 MeV to over 100 GeV. At low energies an excess of He relative to C is observed that would amount to about 20% of the modeled galactic component at 10 MeV/nuc.

physics.space-ph↗

arXiv2015

A Fit to the Galactic Cosmic Ray Hydrogen and Helium Spectra at Voyager 1 at Low Energies and Earth Based Measurements at Much Higher Energies with Identical Rigidity Independent Source Spectra for the Hydrogen and Helium Nuclei

Voyager 1 data from beyond the heliopause provide the first direct measurements of the interstellar cosmic ray spectra below 1 GeVnuc. In this paper we combine these Voyager measurements of H and He nuclei from 3-600 MeVnuc with higher energy measurements at 1 AU from the BESS and PAMELA experiments up to 100 GeVnuc. Using a Weighted Leaky Box Model for propagation in the galaxy, we obtain an excellent fit to these new Voyager observations and the much higher energy spectra up to 100 GeVnuc by using source spectra which are P-2.28, with the exponent independent of rigidity from low to high rigidities; along with a rigidity dependence of the diffusion path length which is P-0.5 at rigidities 1.00 GV, and possibly changing to P1.0 at lower rigidities.

physics.space-ph↗

arXiv2015

A Comparison of Radial Intensity Profiles of Termination Shock Particles and Anomalous Cosmic Rays in the Outer North-South Heliosheaths Using CRS data from V1 and V2

Using data from the Voyager 1 and 2 CRS telescopes available on the web through the end of 2014 we have studied the intensity variations of termination shock H nuclei and anomalous cosmic ray H and He nuclei as a function of radial distance. In contrast to the inner part of the heliosheath where the intensity vs. radius profiles in the North and South heliosheaths are much different, these intensity vs. radius profiles, as well as the intensities themselves, are more similar in the outer North and South heliosheaths as measured by V1 and V2 respectively. In the N heliosheath, taken to be 27.6 AU thick beyond the HTS crossing distance of 94 AU, the intensities of termination shock particles and anomalous cosmic rays reach a maximum at between 110-112 AU or at a location ~halfway between the termination shock and the heliopause. They then decrease more or less continuously to an intensity ~2/3 of the maximum for each component just before the final dropout at ~121 AU, just inside the heliopause. This intensity-time behavior favors theories in which the acceleration and loss effects lead to a maximum near the middle of the heliosheath.

physics.space-ph↗

arXiv2014

A Forecast for a South Heliopause Crossing by Voyager 2 in Late 2014 Using Intensity-time Features of Energetic Particles Observed by V1 and V2 in the North and South Heliosheaths

We have used corresponding marker features in the intensity vs. time profile of termination shock particles coming from within the heliosheath and also 5-12 MeV electrons which are coming from beyond the heliopause that are both observed at both V1 and V2 in the outer N and S heliosheaths, to estimate the thickness of the S heliosheath. These features in the N heliosheath have been observed on the CRS instrument on V1 in the N heliosheath beyond 110 AU or within 12 AU of the heliopause. Although the intensity-time features in the S heliosheath are quite different overall, certain features stand out at both V1 and V2. The correspondence of these features in the N and S heliosheath allows us to estimate that V2 will reach the S heliopause near the end of 2014.

physics.space-ph↗

arXiv2014

Extraordinary Solar Modulation Effects On Galactic Cosmic Rays Observed By V1 Near The Heliopause

We discuss two extraordinary increases of cosmic ray intensity observed by Voyager 1 in the last 1.1 AU before heliopause crossing, Aug 2012 at 121.7 AU. The two increases are roughly similar in amplitude and result in a total increase of cosmic ray nuclei around 1 GV of over 50 percent and of 0.01 GV electrons of a factor about 2. During the first increase, the changes in the magnetic, B, field are small. After the first increase, the B field variations and cosmic ray changes become large and during the second increase the B field variations and the cosmic ray changes are correlated to within a day. The intensity variations of H and He nuclei during these time intervals are measured from 0.1 to over 1 GV. The total GCR increse over the two events resemble those expected from a simple force-field 'like' solar modulation model with a modulation potential of about 80MV. This is nearly one third of the total modulation potential of about 250 GV required to produce the modulation of these particles observed at the earth at the 2009 sunspot minimum and adds a new aspect to ideas about heliospheric modulation.

astro-ph.SR↗

arXiv2014

Transient Heliosheath Modulation

Voyager 1 has explored the solar wind-interstellar medium interaction region between the terminal shock and heliopause following the intensity distribution of galactic cosmic ray protons above 200 MeV energy. Before this component reached the galactic level at 121.7 AU, 4 episodes of rapid intensity change occured similar to the Forbush Decreases found near the sun, rather than the expected result of models related to those describing Long Term Modulation in the inner solar system. Because the mean solar wind flow is both expected and observed to be perpendicular to the radial direction close to the heliopause, explanation is given in terms of transient radial flows related to possible heliopause boundary flapping. It is necessary that radial flows are at the sound speed found for conditions downstream of the teminal shock and that the relevant perpendicular cosmic ray diffusion is controlled by 'slab' field fluctuations accounting for 20 percent or less of the total power in field variance. However, additional radial drift motion related to possible north to south gradients in the magnetic field may allow the inclusion of some diffusion according to 2-D turbulence theory. The required field gradients may arise due to variation in the field carried by the solar plasma deflected away from the solar equatorial plane. Modulation amounting to a total 30 percent drop in galactic intensity requires explanation by a combination of several transient episodes.

astro-ph.SR↗

arXiv2014

Voyager 1 and 2 Observations of Cosmic Ray Intensities in the North-South Heliosheaths-Implications for the Latitude Extent of the Heliospheric Current Sheet and Radial Structure in the Heliosheath

The paper describes differences in the intensity as a function of radial distance of anomalous and galactic cosmic rays in the N-S heliosheaths as observed by the Voyager 1 and 2 spacecraft respectively. The anomalous cosmic ray (ACR) intensities above a few MeV in the N heliosheath reach a maximum at ~2009.5, about 16 AU beyond the heliospheric current sheet (HTS), thus indicating a possible source location for these higher energy particles. The galactic cosmic ray electron and nuclei intensities continue to increase rapidly throughout the entire N-heliosheath out to 121.7 AU. Two sudden increases of these electrons at 16.6 and 22.2 AU beyond the HTS suggest that the V1 spacecraft passed significant heliosheath structures at these times. Later, at 121.7 AU, which is a distance of 27.6 AU beyond the initial HTS crossing distance, V1 in the N heliosheath appears to pass beyond the ACR trapping region as the anomalous particles disappear suddenly and completely. At V2 in the S heliosheath, large 42-21 day periodic intensity variations were observed for a few MeV particles just beyond the HTS crossing distance at 83.7 AU and again about a year later when V2 was ~4 AU beyond this shock. These are the dominant features of the energetic particle populations in the inner part of the S-heliosheath. The ACR and electrons later decreased to almost background levels until a sudden increase at 2011.25, at a distance of about 10 AU beyond the HTS. The features observed by V2 in the S heliosheath relative to those at V1 indicate a much more structured S heliosheath in which the latitude of V2 relative to the HCS maximum latitude (sector zone) may play an important role. Estimates of a S heliosheath trapping/modulation boundary for energetic particles in the range 105-110 AU are obtained in this paper from a comparison with those at 121.7 AU observed in the N hemisphere.

physics.space-ph↗

arXiv2013

The Structure of The Heliopause

Voyager 1 has explored the solar wind-interstellar medium interaction region between the Terminal Shock and the Heliopause, following the intensity distribution of the shock accelerated anomalous component of cosmic rays in the MeV energy range. The sudden disappearance of this component at 121.7 AU from the sun is discussed in terms of three models for the transition into the interstellar plasma flow. Particles trapped flowing parallel to the boundary may penetrate up to one Larmour radius beyond. If the boundary is stationary, Voyager 1 directly samples this distance. The boundary could flap, depending on Heliosheath pressure changes and Voyager 1 then samples the extent of this motion. Finally, a turbulent boundary layer is considered in which the MeV particle distribution falls off with distance, thus measuring diffusion within the layer.

astro-ph.SR↗

arXiv2013

Calculations of the Propagated LIS Electron Spectrum Which Describe the Cosmic Ray Electron Spectrum below ~100 MeV Measured Beyond 122 AU at Voyager 1 and its Relationship to the PAMELA Electron Spectrum above 200 MeV

The new Voyager measurements of cosmic ray electrons between 6-60 MeV beyond 122 AU are very sensitive indicators of cosmic ray propagation and acceleration in the galaxy at a very low modulation level. Using a Monte Carlo diffusion model with a source spectrum with a single spectral index of -2.2 at all energies we are able to fit this observed Voyager spectrum and the contemporary PAMELA electron spectrum over an energy range from 6 MeV to ~200 GeV. This spectrum has a break in it but this break is due to propagation effects, not changes in the primary spectrum. This break is gradual, starting at > 2 GeV where the spectrum is ~E^-3.2 and continuing down to ~100 MeV or below where the spectrum becomes ~E^-1.5. At the higher energies the loss terms due to synchrotron radiation and inverse Compton effects which are ~E^2.0 steepen the exponent of the source spectrum by 1.0. At lower energies, these terms become unimportant and the loss is governed by diffusion and escape from the galaxy. A diffusion term which is proportional to beta^-1 below ~0.32 GV (which also fit the H and He spectra measured at Voyager) and has a value = 3x1028 cm^2 x s^-1 at 1 GV and a boundary at +/-1 Kpc will fit the Voyager or other similar spectra at low energies.

physics.space-ph↗

arXiv2013

A Comparison of the Galactic Cosmic Ray H, He and C/O Nuclei Spectra Measured Between ~5 and 500 MeV/nuc Beyond 122 AU at Voyager 1 with the Predictions of a Diffusion Model for Propagation in the Galaxy

After the disappearance of lower energy heliospheric particles at Voyager 1 starting on August 25th, 2012, spectra of H, He and C/O nuclei were revealed that resembled those to be expected for galactic cosmic rays. These spectra had intensity peaks in the range of 30-60 MeV, decreasing at both lower energies down to a few MeV and at higher energies up to several hundred MeV. We have modeled the propagation of these particles in the galaxy using an updated Leaky Box Diffusion model which determines the spectra of these components from ~2 MeV to >200 GeV. The key parameters used in the model are a galactic input spectrum ~P^-2.24, the same for all components and independent of rigidity, and a diffusion coefficient that is ~P^0.5 above a lower rigidity and increases ~beta^-1.0 below a lower rigidity ~0.56 GV. These same parameters also fit the high energy H and He data from ~10-200 GeV/nuc from the PAMELA and BESS experiments. The new Voyager spectra for all three nuclei are thus consistent with rigidity spectra ~P^-2.24 from the lowest energies to at least 100 GeV. Deviations from this spectrum can reasonably be attributed to propagation effects. Some deviations between the calculated and newly observed spectra are noted, however, below ~30 MeV/nuc, particularly for C/O nuclei, that could be significant regarding the propagation and sources of these particles.

physics.space-ph↗

arXiv2013

The Unfolding of the Spectra of Low Energy Galactic Cosmic Ray H and He Nuclei as the Voyager 1 Spacecraft Exits the Region of Heliospheric Modulation

This paper describes the unfolding of the solar modulated galactic cosmic ray H and He nuclei spectra beyond ~105 AU in the heliosheath. Between 2008.0 and 2012.3 when Voyager 1 went from about 105 to 120.5 AU the spectral intensities of these two components between about 30 and 500 MeV/nuc unfolded (increased) in a manner consistent with an average modulation potential decrease ~5 MV per AU as described by a Parker like cosmic ray transport in the heliosphere where the overall modulation is described by a modulation potential in MV. Between 120.5 and 121.7 AU, however, as a result of two sudden intensity increases starting on May 8th and August 25th, 2012, this modulation potential decreased by ~80 MV and spectra resembling possible local interstellar spectra for H and He were revealed. Considering these spectra to be the local interstellar spectra would imply that almost 1/3 of the total modulation potential of about 270 MV required to explain the spectra of these components observed at the Earth must occur in just a 1 AU radial interval in the outer heliosheath. As a result about ~80% of the total modulation potential observed at the Earth at this time occurs in the heliosheath itself. The remaining 20% of the total modulation occurs inside the heliospheric termination shock. The details of these intensity changes and their description by a simple modulation model are discussed.

physics.space-ph↗

arXiv2012

At Voyager 1 Starting on about August 25, 2012 at a Distance of 121.7 AU From the Sun, a Sudden Disappearance of Anomalous Cosmic Rays and an Unusually Large Sudden Increase of Galactic Cosmic Ray H and He Nuclei and Electron Occurred

At the Voyager 1 spacecraft in the outer heliosphere, after a series of complex intensity changes starting at about May 8th, the intensities of both anomalous cosmic rays (ACR) and galactic cosmic rays (GCR) changed suddenly and decisively on August 25th (121.7 AU from the Sun). The ACR started the intensity decrease with an initial e-folding rate of intensity decrease of ~1 day. Within a matter of a few days, the intensity of 1.9-2.7 MeV protons and helium nuclei had decreased to less than 0.1 of their previous value and after a few weeks, corresponding to the outward movement of V1 by ~0.1 AU, these intensities had decreased by factors of at least 300-500 and are now lower than most estimates of the GCR spectrum for these lower energies and also at higher energies. The decrease was accompanied by large rigidity dependent anisotropies in addition to the extraordinary rapidity of the intensity changes. Also on August 25th the GCR protons, helium and heavier nuclei as well as electrons increased suddenly with the intensities of electrons reaching levels ~30-50% higher than observed just one day earlier. This increase for GCR occurred over ~1 day for the lowest rigidity electrons, and several days for the higher rigidity nuclei of rigidity ~0.5-1.0 GV. After reaching these higher levels the intensities of the GCR of all energies from 2 to 400 MeV have remained essentially constant with intensity levels and spectra that may represent the local GCR. These intensity changes will be presented in more detail in this, and future articles, as this story unfolds.

physics.geo-ph↗

arXiv2012

Large periodic time variations of termination shock particles between ~0.5-20 mev and 6-14 mev electrons measured by the crs experiment on Voyager 2 as it crossed into the heliosheath in 2007: An example of freshly accelerated cosmic rays?

We have examined features in the structure of the heliosheath using the fine scale time variations of termination shock particles (TSP) between ~0.5 - 20 MeV and electrons between 2.5-14 MeV measured by the CRS instrument as the V2 spacecraft crossed the heliospheric termination shock in 2007. The very disturbed heliosheath at V2 is particularly noteworthy for strong periodic intensity variations of the TSP just after V2 crossed the termination shock (2007.66) reaching a maximum between 2007.75 and 2008.0. A series of 42/21 day periodicities was observed at V2 along with spectral changes of low energy TSP and the acceleration of 6-14 MeV electrons. Evidence is presented for the acceleration of TSP and electrons at the times of the 42/21 day periodicities just after V2 crossed the HTS. Spectra for TSP between 2-20 MeV and electrons between 2.5-14 MeV are derived for three time periods including the time of the HTS crossing. The energy spectra of TSP and electrons at these times of intensity peaks are very similar above ~3 MeV, with exponents of a power law spectrum between -3.0 and -3.6. The ratio of TSP intensities to electron intensities at the same energy is ~500. The electron intensity peaks and minima are generally out of phase with those of nuclei by ~1/2 of a 42 day cycle. These charge dependent intensity differences and the large periodic intensity changes could provide new clues as to a possible acceleration mechanism.

physics.geo-ph↗

arXiv2012

20-125 mev/nuc cosmic ray carbon nuclei intensities between 2004-2010 in solar cycle #23 as measured near the earth, at voyager 2 and also in the heliosheath at voyager 1 - modulation in a two zone heliospehre

The recovery of cosmic ray Carbon nuclei of energy ~20-125 MeV/nuc in solar cycle #23 from 2004 to 2010 has been followed at three locations, near the Earth using ACE data and at V2 between 74-92 AU and also at V1 beyond the heliospheric termination shock at between 91-113 AU. To describe the observed intensity changes and to predict the absolute intensities measured at all three locations we have used a simple spherically symmetric (no drift) two-zone heliospheric transport model with specific values for the diffusion coefficient in both the inner and outer zones. The diffusion coefficient in the outer zone is determined to be ~5-10 times smaller than that in the inner zone out to 90 AU. For both V1 and V2 the calculated C nuclei intensities agree within an average of \pm 10% with the observed intensities. Because of this agreement between V1 and V2 observations and predictions there is no need to invoke an asymmetrical squashed heliosphere or other effects to explain the V2 intensities relative to V1 as is the case for He nuclei. The combination of the diffusion parameters used in this model and the interstellar spectrum give an unusually low overall solar modulation parameter ϕ= 250 MV to describe the Carbon intensities observed at the Earth in 2009. At all times both the observed and calculated spectra are very closely ~ E1.0 as would be expected in the adiabatic energy loss regime of solar modulation.

physics.geo-ph↗

arXiv2011

Recovery of 150-250 MeV Cosmic Ray Proton Intensities Between 2004-2010 as Measured Near the Earth, at Voyager 2 and also in the Heliosheath at Voyager 1 - A Two Zone Heliosphere

The recovery of cosmic ray protons of energy ~150-250 MeV/nuc in solar cycle #23 from 2004 to 2010 has been followed at the Earth using IMP, ACE and balloon data and also at V2 between 74-92 AU and at V1 beyond the heliospheric termination shock (91-113 AU). The correlation coefficient between the intensities the Earth and V1 during this time period, is 0.936, allowing for a ~0.9 year delay due to the solar wind propagation time from the Earth to the outer heliosphere. To describe these intensity changes and to predict the absolute intensities measured at all three locations we have used a simple spherically symmetric (no drift) two-zone heliospheric transport model with specific values for the diffusion coefficient in both the inner and outer zones. The diffusion coefficient in the outer zone, from about 90 to 120 (130) AU, is determined to be ~5-10 times smaller than that in the inner zone out to 90 AU. This means that the outer zone acts much like a diffusing barrier in this model. The absolute magnitude of the intensities and the intensity changes at V1 and the Earth are described to within a few percent by a diffusion coefficient that varies with time by a factor ~4 in the inner zone and only ~1.8 (1.25) in the outer zone over the time period from 2004-2010. These diffusion coefficients and their variations are essentially the same as those derived earlier from a similar study using He nuclei of the same energy. This model and the diffusion coefficients used provide a total modulation potential at the Earth ~250 MV in 2009. The difference ~10-20% between calculated and observed intensities at V2 can be explained if the heliosphere is squashed by ~10% in distance (non-spherical) so that the HTS is closer to the Sun in the direction of V2 compared to V1.

physics.geo-ph↗

arXiv2011

Recovery of 150-250 MeV/nuc Cosmic Ray Helium Nuclei Intensities Between 2004-2010 Near the Earth, at Voyager 2 and Voyager 1 in the Heliosheath - A Two Zone Helioshpere

The recovery of cosmic ray He nuclei of energy ~150-250 MeV/nuc in solar cycle #23 from 2004 to 2010 has been followed at the Earth using IMP and ACE data and at V2 between 74-92 AU and also at V1 beyond the heliospheric termination shock (91-113 AU). The correlation coefficient between the intensities at the Earth and at V1 during this time period is remarkable (0.921), after allowing for a ~0.9 year delay due to the solar wind propagation time from the Earth to the outer heliosphere. To describe the intensity changes and to predict the absolute intensities measured at all three locations we have used a simple spherically symmetric (no drift) two-zone heliospheric transport model with specific values for the diffusion coefficient in both the inner and outer zones. The diffusion coefficient in the outer zone, assumed to be the heliosheath from about 90 to 120 (130) AU, is determined to be ~5 times smaller than that in the inner zone out to 90 AU. This means the Heliosheath acts much like a diffusing barrier in this model. The absolute magnitude of the intensities and the intensity changes at V1 and the Earth are described to within a few percent by a diffusion coefficient that varies with time by a factor ~4 in the inner zone and only a factor of ~1.5 in the outer zone over the time period from 2004-2010. For V2 the observed intensities follow a curve that is as much as 25% higher than the calculated intensities at the V2 radius and at times the observed V2 intensities are equal to those at V1. At least one-half of the difference between the calculated and observed intensities between V1 and V2 can be explained if the heliosphere is squashed by ~10% in distance (non-spherical) so that the HTS location is closer to the Sun in the direction of V2 compared to V1.

physics.geo-ph↗