Discoveries

This webpage concerns results from peer-reviewed papers in journals, which I consider my best work. Three scientific publications are from 2022. Monographs from 2020 and 2019 are described under “Books/Special Issues.”

For a list of heat transfer papers, see https://sites.wustl.edu/hofmeister/files/2022/09/HeatTransPubs.pdf. For a list of publications regarding gravitation, see https://sites.wustl.edu/hofmeister/files/2022/09/GravitationPubs.pdf. Two papers of general interest are accessible publicly: “Has Axial Spin Decline Affected Earth’s Geologic and Tectonic History?” at https://rdcu.be/cXQkM and “Can Modern Science Answer the Great Questions?” at https://rdcu.be/cXQtA.

Making Connections

Comparison of hurricanes to spiral galaxies. (a) Image of Category 3 Fran off the Florida coast from the National Oceanic and Atmospheric Administration (http://www.noaa.gov/. (b) Image of the Whirlpool galaxy from NASA/IPAC Extragalactic Database (NED: https://ned.ipac.caltech.edu/). Both images are in the visible and are public domain. Scale bars are approximate. (c) Rotation curves of four large hurricanes, similar to Fran, near their centers from flight data http://www.air-worldwide.com/…/AIRCurrents-Wind-Profiles-in-Parametric-Hurricane-Models. (d) Rotation curves of complex spirals, far from galactic centers: data from NED. From: Hofmeister, A.M. and Criss, R.E. 2017, Implications of Geometry and the Theorem of Gauss on Newtonian Gravitational Systems and a Caveat Regarding Poisson’s Equation. Galaxies 5, 89-100.  http://www.mdpi.com/207

Hurricane data focus on the central regions. As is well-known the EYE is calm. Likewise, the center of a spiral galaxy has null velocity. Every spinning object has a stationary center, like a record if the object is flat. If the object is round, like a planet or star, then the polar axis is stationary.

That spiral galaxies behave similar to records, although spirals are not solid like a record, has puzzled many. Our analogy to hurricanes, which are made of gas and suspended water droplets solves this dilemma: Spin does not require a solid body. Galaxies likewise have droplets (stars) and gas (hydrogen), and exhibit arms and complex patterns. Galaxies spin.

See https://www.weather.gov/ilm/HurricaneFran for time-lapse images and maps of wind speed. Because data on spirals focus on the outer, more complex regions, information on hurricanes is useful to understand evolutionary behavior: see Criss, R.E. and A.M. Hofmeister. Galactic density and evolution based on the Virial theorem, energy minimization, and conservation of angular momentum. Galaxies 6,115-135; https://doi.org/10.3390/galaxies6040115 .

New Ideas in Earth and Planetary Science

A New Mechanism for Plate Tectonics: Apply Force!

Hofmeister, Criss, and Criss (2022) have provided a new mechanism for plate tectonics. This mechanism centers on gravitation forces and torques experienced by the Earth in the three-body Sun-Earth-Moon system, and the effect of spin on the layered Earth.  Large scale motions arise from forces (see figures below). Importantly, plate tectonics (lateral forces) differs from volcanism (vertical buoyancy).  Our proposal does not require large internal heating, other than the small leakage of heat that produces melting at the mid-ocean ridges. Our idea greatly differs from the mantle convection hypothesis which requires heat source exceeding anything plausible plus deformation and buoyancy of rocks made strong by immense hydrostatic compression (for a theoretical analysis see https://doi.org/10.1007/s12583-017-0819-4.

Two figures from a lecture given virtually for China University of Geosciences, Wuhan, on April 19, 2022.

To hear more on our mechanism, please listen to this podcast on CBC “Quirks and Quarks” radio show.

Is the Moon Approaching Earth?

It is popularly believed that the Moon is slowly receding from the Earth, and this motion is attributed to conversion of Earth’s axial spin to lunar orbital momentum. The belief is untrue and the mechanism is impossible.

Considering conservation of energy and angular momentum and that the orbit is constrained by two parameters (eccentricity e and the semi-major axis a) , Hofmeister, Criss, and Criss show (https://www.mdpi.com/2674-0346/1/2/7) shows that torque in the Earth-Moon-Sun system increases eccentricity (elongation) of the lunar orbit. Consequently, the average lunar orbital radius is decreasing and so the Moon is approaching the Earth on average (see figure).

Polar plot of of the lunar orbit. Heavy black shows the current semi-major axis and eccentricity. The central dot shows the actual size of Earth. Black dotted curve is a circular orbit with the same energy. Orbits calculated using r = a(1-e2)/(1+e cos(theta)) are shown. Red long dashes show e = 0.206 which also describes Mercury’s orbit, the most eccentric among the planets. Green medium dashes = high e seen in orbits of some distant prograde moons. Figures from Hofmeister, A.M., Criss,  R.E., and  Criss, E.M.   Theoretical and Observational Constraints on Lunar Orbital Evolution in the Three-body Earth-Moon-Sun System. Astronomy, 1. https://www.mdpi.com/2674-0346/1/2/7. Creative commons licence.
Physically, elongation occurs because the Sun pulls the Moon (image at top from NASA, public domain) either towards itself and away from the Earth (pink moon position) or towards the earth and towards itself (green moon position). Several accelerations exist. Inclination also changes.

Because the Earth is at the focus of the reduced 2-body orbit, the Moon is getting closer at perigee and further at apogee (see figure above). Extreme elongation will eventually result in the Sun capturing our Moon. Venus and Mercury have no moons, and Earth lacks distant satellites, suggesting that the Sun earlier robbed close-in rocky planets of their companions.

Lunar drift is too small to be constrained through lunar laser ranging measurements, mainly because atmospheric refraction corrections are larger than drift can possibly be. Our findings support co-accretion.

Jupiter Jerks Everybody Around

Hofmeister and E. M. Criss (2021; https://doi.org/10.3390/sym13050846) showed that each planet is tugged by the other eight via point-mass interactions, which elongates the orbits, particularly that of Mercury. The very long repeat pattern of perturbations (see figures) is not captured by a few hundred years of measurements, especially as the Earth bobbles along with the Moon (see above sections) and is likewise affected by interplanetary interactions. The oddities of Mercury’s orbit can be explained by Newtonian physics, rather than by relativistic acceleration by the Sun. Besides orbital elongation, massive Jupiter is pulling the rocky planets towards its plane due to this particular torque being unopposed.

Figure 9d from Hofmeister and Criss (2021). Sums of the accelerations (relative to Solar) on Mercury calculated numerically. The dominating influence of Jupiter’s 88-day cycle is evident. Neither the tangential nor the radial accelerations return to their starting configurations, and so the “average” over even 5 Earth-years does not accurately describe the motions of Mercury.
Figure 5a from Hofmeister and Criss (2021). Effect of the directional changes of outboard Jupiter’s pull on Earth. In the reference frame of a stationary Earth, Jupiter moves retrograde (heavy black arrow). The pull to Jupiter is strongest at the closest approach. Inside the stippled sector, the force vector between Earth and Jupiter has a component towards the Sun, but the distance between Jupiter and the Earth is greater so the pull of Jupiter is weaker than in white sector. The net effect of Jupiter over the circuit is to oppose the pull of the Sun. All planets affect Mercury in the same manner.
Figure 2a from Hofmeister and Criss (2021). Eccentricity and inclination relative to Jupiter’s orbit; data from NASA. Mercury, which is jerked by all other planets, has the most elliptical and tilted orbit
Figure 6 from Hofmeister and Criss (2021). Schematics of out-of-plane perturbations, with reference to the orbital plane of Jupiter Not to scale. Dark and medium grey arrows indicate that the gravitational and centrifugal forces from the Sun are nearly equal. Light grey arrows depict the pull to Jupiter. White arrow = the component of Jupiter’s pull perpendicular to the orbit of the POI. Black arrow = the radial component of Jupiter’s action: (a)Action on the inner planets when close to Jupiter; (b) Action on planets outboard of Jupiter, when close. In diverse configurations, Jupiter pulls (torques) any given POI towards Jupiter’s orbital plan.

Basic Physics – Some like it Hot

New and Improved Thermodynamic Equations for Solids

Classical thermodynamics treats solids and gases as behaving in the same manner, which is not true. Solids have the property of rigidity (resistance to shear). This immense energy reservoir was previously neglected:

Summary and comparison of the characteristics of solids and gases most relevant to heat and its flow. The shear modulus, G, describes a special type of stored energy in solids, which is part of the elastic energy, the main reservoir. Atoms are shown as balls, with dotted arrows indicating direction of long-distance motions. Sine waves without arrowheads indicate local, back-and-forth, microscopic motions. From Hofmeister et al. https://doi.org/10.3390/ma15072638

Further more, Fourier’s law was not previously incorporated into thermodynamics of solids, and so the classical model neither describes the flow of heat through matter nor that solids sustain a thermal gradient, up to melting.

By accounting for steady-state heat flow and rigidity of solids Hofmeister, A.M., Criss,  E.M., and  Criss, R.E.  (2022, Thermodynamic relationships for perfectly elastic solids undergoing steady-state heat flow. Materials 15, 2638; https://doi.org/10.3390/ma15072638 ) provide a new set of equations (see Table) and confirmed these equations using data from ~100 different studies, with a focus on isotropic solids: 

Summary of new equations from Hofmeister, A.M., Criss,  E.M., and  Criss, R.E.  (2022)  https://doi.org/10.3390/ma15072638 . Note: alpha is thermal expansivity; cp is specific heat, B is bulk modulus, rho is density. 

Steady-state conditions are adiabatic since heat content (Q) is constant. Because average temperature is also constant and the thermal gradient is fixed in space, conditions are simultaneously isothermal in time: Under these dual restrictions, thermal transport properties do not enter into our analysis.

We show pressure-volume work (PdV) in a perfectly elastic solid arises from internal interatomic forces, which are linked to Young’s modulus (X) and a constant (n) accounting for cation coordination. We find that adiabatic and isothermal bulk moduli (B) are equal. Also, Q/V depends on temperature only. Distinguishing deformation from volume changes elucidates how solids thermally expand.  These findings lead to simple descriptions of key physical properties, listed in the table above.

The figures below using all the data we could find on isotropic solids, and some on hexagonal structures.

Response of heat capacity and storativity (density times cP to pressure: (a) Dependence on the inverse of bulk modulus. (b) Direct dependence on B.  Grey diamonds and grey dashed line = directly determined storativity from dynamic experiments: The error bar is from Gerlich and Andersson [ J. Phys. C: Solid State Phys. 1982, 15, 5211-5222.]. Open cross and red dotted line = metal cP directly measured by calorimetry. Aqua triangles = heat capacity obtained by difference. Green short dashed line = ideal correspondence. Figure 11 from Hofmeister et al. (https://doi.org/10.3390/ma15072638).
Dependence of density times Young’s modulus on alpha/cP. Fits are least squares and are labeled with the number of solids in each category: (a) Insulators and cubic fcc metals. Lead strongly influences the slope due to its softness, as shown by the two fits. Iridium has little influence as it is near a cluster of points. Orthorhombic Fe2SiO4 has a shearing transition whereas alpha for orthorhombic Mn2SiO4 is unconfirmed; (b) Cubic bcc and hexagonal hcp metals. Outliers Li and Be have very small cations and few valance electrons. Figure 14 from Hofmeister et al. (2022)

Heat transport depending on length-scale points to radiative diffusion

Please see “Current Reseach” for a figure and discussion. For experiments and a theoretical discussion of metals, see Criss, E.M. and Hofmeister, A.M. Isolating lattice from electronic contributions in thermal transport measurements of metals and alloys and a new model. International Journal of Modern Physics: B 31 (75 pp) (http://www.worldscientific.com/doi/pdf/10.1142/S0217979217502058)

Basic Physics – Push me/Pull you

Newtonian gravitational attraction is not simply radial

Jupiter, taken by NASA’s Hubble Space Telescope on Aug. 25, 2020. Credits: NASA, ESA, STScI, A. Simon (Goddard Space Flight Center), M.H. Wong (University of California, Berkeley), and the OPAL team

Gravitational attraction to an oblate body is not central (i.e., pull is only directed towards the center along the special directions and in all cases does not inversely depends on the radial distance). Non-centrality went unrecognized, probably because Gauss’s description of the axially symmetric field is counterintuitive. Hofmeister, Criss and Criss (2018, open access: https://www.sciencedirect.com/science/article/pii/S003206331730257X) provide simpler, but exact, formulae for attraction to oblate bodies and some convenient close approximations.

Findings of this paper pertain to mass of the giant planets and precession of their satellites. Orbits of dwarf galaxies about Andromeda and the Milky Way are also explained.

New Inverse Models for Geophysics and Astronomy

Inverse problems are fundamentally different than forward (fitting) problems.

Much greater focus is given to the forward (fitting) approach, now more than ever, because computers are widely available. Forward modelling is a plug-and-chug approach. Specifically, either known or assumed, inputs (e.g., source characteristics) are inserted into a standard equation, formula or program, that returns a result.  Nice fits can be obtained by changing the free parameters. But as discussed by Transtrum et al. (2015) a match can be obtained using the wrong physics, particularly when parameters are lumped (multiplied).

In the less familiar inverse problem, the nature of a remote source is deduced from its output or response (Groetsch, C.W. Inverse Problems: Activities for Undergraduates; Cambridge University Press, Cambridge, UK, 1999). This approach is rarely taught, as analytical mathematics are essential, but use of analytical mathematics is declining.

Analytical solutions to inverse problems are unique and therefore are extremely important. Groetsch (1999) teaches by example, and so several examples are given below.

Mantle temperatures from PREM without specifying Mineralogy

Temperatures in the lower mantle were inferred from seismologic models of bulk modulus (B) and pressure (P) vs depth (z) by using a new identity d ln(k)/dP = 7⅓/BT , where k is thermal conductivity (Hofmeister, A.M.  Lower mantle geotherms, flux, and power from incorporating new experimental and theoretical constraints on heat transport properties in an inverse model.  European J. Mineralogy 34, 149-165: https://doi.org/10.5194/ejm-34-149-2022).

Measured pressure derivatives of thermal conductivity as a function of sample compressibility (the inverse of bulk modulus). All data collected from mm sized samples below 2 GPa. The box lists parameters for a linear fit with no intercept. Metals and Si (red crosses) are included in the fit. KBr exemplifies hydroscopic alkali halides which are soft (bulk moduli < 16 GPa). Details on the 24 heat transport studies of this figure are given in Hofmeister (2021;https://www.mdpi.com/1996-1944/14/2/449) ), table 3. Bulk modulus has been measured many times for the samples and is listed in compilation of Bass (1995) and several others. Figure 2 from Hofmeister (https://doi.org/10.5194/ejm-34-149-2022)

The model is based on B responding oppositely to P and T:

fig. 5 (Hofmeister, 2022) showing the LM and core

 

Construction of possible curves for the depth response of the P component of mantle dB/dP at 871 km and deeper. The edges of the Transition Zone and Inner Core define the x-axis. Discontinuities shallower than 871 km and in D” are not addressed in our approach.  If d2B/dP2 is small, near 0, then PREM indicates that warming first occurs as z increases but at greater depths, temperatures decrease.

See below for implications for the core.

Because cross derivatives are negliglible, the LM gradient, dT/dz, depends on d2B/dP2 and dB/dT. These properties which vary little among dense phases, an so the resulting temperatures do not depend on the details of mineralogy.  However, to calculate flux and power from dT/dz, cases of high (oxide) and low (silicate) k need considering. Geotherm calculations are independent of κ, and thus of mineralogy, but require specifying a reference temperature at some depth: a wide range is considered at 670 km.

Figure 8 from Hofmeister (2022) for one particular starting temperature. Other temperatures just shift the curves up or down.

Families of geotherms for possible values of 2nd P derivatives of LM bulk modulus. All curves use the T derivative of B for silicate perovskite from Aizawa et al. (2004). For the dry peridotite solidus, results of Zhang and Herzberg (1994) are merged with high P data or Fiquet (2010).  Heavy black line is the eutectic melting curve of the Fe-S system (e.g. Morard et al., 2017).

A thermal maximum in the LM exists due to the shape of PREM curves (see above). Except for an oxide composition with miniscule d2B/dP2, the LM heats the core, causing it to melt. Deep heating is attributed to cyclical stresses from >1000 km daily and monthly fluctuations of the barycenter inside the LM (see “Plate tectonics” section, above).

Ages of Stars Extracted from Histograms of their Spin

M39: Open Cluster in Cygnus. Credit: Heidi Schweiker, WIYN, NOAO, AURA, NSF. From https://apod.nasa.gov/apod/ap090412.html

Star ages are very important to astronomy as they are intimately linked to stellar processes and evolution. Only our Sun has a known age, based on consistent, isotopic data on meteorites.

Stars loose spin with age. To quantify the rate of loss, and thus ages, we developed an analytical inverse model that uses histogram data on stars in open clusters to unequivocally determine the physical law governing how dwarf star spin depends on time (t) and mass (M). [Criss, R.E. and Hofmeister A.M. 2021 Quantification of Sub-solar Star Ages from the Symmetry of Conjugate Histograms of Spin Period and Angular Velocity. Symmetry  13, Paper 1519; https://doi.org/10.3390/sym13081519 ]:

Summary of our model and its application to a well known open cluster.
Analysis of red dwarfs in various clusters. Really fast stars and really slow stars are not seen, one is not ignited and the other may explode.
Stars in any open cluster have many ages: (a) Open clusters considered to be old, plus 29 stars in Coma Berenices (red-brown dots) and 30 slow rotators in M67 (light pink dots). Bigger stars die/transition sooner than small. (b) Clusters with moderate model ages. Most stars are ~3 Ga old, which a substantial portion of young stars, and some evidence for stellar demise in NGC 2516, which lacks fast rotators. (c) Clusters with young ages, plus data on 20 fast rotators in the young cluster IC4665 (pale blue-gray triangles). The average age is near 4 Ga, and no evidence exists for stellar demise.
Note that the last panel explores a stronger dependence on mass, given the lack of data on star ages. This shows uncertainty of our model.

Our analysis points to the Sun being the oldes type-G0 star in the dozen clusters studied.

Spinning Galaxies are entirely composed of Real Matter

Edge-On NGC 891. Image Credit & Copyright: Adam Block, Mt. Lemmon SkyCenter, U. Arizona. From https://science.nasa.gov/edge-ngc-891. This photo of a spiral galaxy and many other images show that the shape is that of a oblate sphereoid. Long ago, Newton and MacLaurin showed that the oblate is the stable gravitational shape for a spinning object.

Motions in a spiral galaxy are coherent, as can be seen in face-on views which depict regular patterns which earned the name spiral.

Face-on view of Grand Spiral Galaxy NGC 1232. Basically, the symmetry is close to a 4-fold rotation axis. The smaller barred spiral to the left has a 2-fold axis. Image Credit: FORS, 8.2-meter VLT Antu, ESO. Public domain.

For a description of the physics of spinning flat spiral galaxies, see Hofmeister and Criss (2017, Canadian Journal of Physics 95, 156-166: https://cdnsciencepub.com/doi/abs/10.1139/cjp-2016-0625). The mathematics of spinning oblate shapes is summarized in the figure below:

Summary of our inverse model. (top left) Schematics of the oblate, which consists of nested homeiods. (top right) Mathematics of our inverse model, which calculates density and mass as a function of radius. The z distance is simply related to r, as shown. (bottom) Results for the Milky Way. Although elipticity e is not known, assuming different values (c/a ratios) just shift the curves. Many spirals have c/a near 0.1. In part (a), upper box describes line patterns associated with v(r) from five different sources. Lower box lists aspect ratios assumed in analyzing each dataset for mass, shown on the right-axis. Error bars on v are shown. Downturns in Min and r indicate that the visual “edge” of the Galaxy is gradual and between 18-30 kpc. In (b), density above r = 0.1 kpc was fit to a power law, as shown. From Criss, R.E. and A.M. Hofmeister (2020). Density Profiles of 51 Galaxies from Parameter-Free Inverse Models of Their Measured Rotation Curves. Galaxies 8, no. 19, https://doi.org/10.3390/galaxies8010019.)

These equations are simple and easy to use and were applied to a wide variety of spiral, elliptical, and dwarf galaxies (https://doi.org/10.3390/galaxies8010019). The basis of the model is that v(r) data depict an average condition at an instant of time, i.e. the galaxy is spinning stably. Stability requires that each homeoidal shell (top left) has constant density, after Newton’s theorem. The resulting equations (above) from Criss and Hofmeister (2020) describe the galaxies as differentially spinning, which allows probing their mass from their interior motions. Records do not differentially spin because they have rigidity, a key property of solids (e.g., https://doi.org/10.3390/ma15072638).

See “Books/Editing” section for discussion of two additional inverse solutions for galactic rotation, assuming other geometries (the equatorial plane and the thin disk). Models by Sipols and Pavlovich and Feng also use Newtonian physics and also show that dark matter is not needed.

In a nutshell, all forward models use Newton’s law in a matter that requires galaxies to be spherical. But since galaxies are flat (see photo) the main-stream approach requires a sphere of surrounding dark matter to compensate for incorrect use of Newton’s established force law.