We don’t really know yet. String Theory says that it is a twisted-up 10-dimensional space called a Calabi-Yau manifold like the one shown here (Credit:Wikipedia).
When you are asking what something looks like, you are asking to describe it in relation to its surrounding space. The problem with asking this question of a fundamental particle in the quantum worldis that we do not have a working understanding of what space[time] is like at these scales. Without THAT, we cannot stand even begin to answer this question. There are some indirect clues that could help us constrain the answer in a meaningful way.
Take the simplest particle we know about, the electron. Experiments show that it is just a knot in the electromagnetic field with no solid surface or internal structure. As you scatter more and more energetic particles off of it, all you see is a region of increasingly higher electric field strength. One thing is for sure, it cannot be a tiny sphrical ball with its electric charge on its surface. Such a system would explode instantly from the enormous electrostatic repulsive forces. Also, because they obey special relativity, they cannot have an extended shape because as they travel through space, their shape would change and no two observers would agree who was observing the correct shape for an elementry particle. Any physical extension would violate
A theory called quantum electrodynamics has been tested to a phenomenal number of decimal places, 10 or more, and still agrees with experimental data on the electron and how it behaves quantum mechanically. The theory stipulates that the electron is a pure point particle with absolutely no internal structure. If you added internal structure, the theory would violate special relativity.
Recently, experimenters have found that in certain kinds of experiments, there MAY be a weak departure from the predictions of QED and experiment, but the scale at which this happens is about 10-20 centimeters or so, and at energies above 100 GeV. Future experiments will check this.
Also, some recent results at Fermilab seem to indicate that quarks may have some internal structure making them less than fundamental, however, these experiments are far less conclusive and under considerable controversy given the statistical significance of the results.
On the theoretical side, it has been widely expected for decades now that at a scale of 10-33 centimeters, the structure of space-time will cease to be the implacable, smooth ‘surface’ we use in modern quantum theory. It will become something quite bizarre like the figure at the top of this page. Perhaps an unimaginable froth of mini-worm holes, quantum loops or strings wiggling about in some strange kind of hyperspace with a dozen or more dimensions. At these scales, all particles loose their point line character. All quantum fields reduce to some more complex topological structure as the superstring theorists advocate.
Conceptually, I have not the slightest idea how to interpret the mathematics, however, if the mathematics lead to testable and verifiable predictions, how we think of the mathematics becomes a moot issue. Just like we cannot accept quantum indeterminacy, duality, and special relativity as ‘common sense’ ideas, yet this is the way nature seems to work.
The current, seemingly comprehensive theory of the origin and evolution of our universe is called inflationary, big bang cosmology. This theory explains how our universe emerged from a singularity of high temperature and density and expanded and cooled. As it did so, space dilated at a faster-than-light pace and cooled so that the familiar elementary particles and forces could produce the cosmic background radiation and a variety of complex particles as the universe expanded. By about three minutes after the Big Bang, the ratio of hydrogen to helium had been cosmologically established, and so had the amount of cosmic background photons and neutrinos. In addition, during the rapid ‘inflation’ period, irregularities in the cosmic density of matter were established as well as a background of gravitational radiation.
The spectacular results of the WMAP satellite , along with the results from COBE and a number of other high-precision studies of the cosmic background radiation have now established that the universe is 13.7 billion years old, and that its space geometry is exactly ‘flat’. Also, according to the most recent WMAP ‘pie graph’ above, the amount of the gravitating matter in ordinary stars and gas is only 4.6%. The rest of the gravitating ‘mass’ of the observable universe is in the form of Dark Matter (24%), with an additional contribution of Dark Energy (71.4%). It is this Dark Energy that is causing our universe to expand at an accelerated pace, which has been observationally detected using distant supernovae.
There have been over the years several potential rivals to Big Bang cosmology, but with the exception of Steady State theory, none have attracted more than a handful of interested supporters. The reason is that they failed to predict, or offer explanations for, some basic observations that are widely agreed upon to be key tests of any cosmological theory, even by the rivals to Big Bang theory!
DeSitter Cosmology ca 1917 – The universe is presumed to be entirely empty of matter, but expands exponentially in time because of the presence of a non-zero cosmological constant. This refutes nearly every observational fact now available, including the actual rate of expansion which follows a linear, Hubble law, not an exponential expansion law with time. Also, the density of matter in the universe is not zero because we are here, and so are a lot of other stars and galaxies!
Einstein Static Cosmology ca 1917-The universe does not expand, and is static in time. The cosmological constant is precisely tuned to balance the attractive tendency of matter. Like DeSitter cosmology, it also fails to agree with modern observations, because the universe is expanding linearly with time. It is also unstable to the slightest perturbation in the value of this constant.
Lemaitre Cosmology ca 1924 – The universe started with a ‘big bang’ with no cosmological constant. The initial state was a giant radioactive atom containing all the matter in the universe near absolute zero. This theory agrees with the observed expansion, but fails to explain the existence of the cosmic background radiation and the universal abundances of hydrogen, helium and deuterium because it requires the decay of a single massive ‘super atom’ at the instant of the Big Bang.
Steady State Cosmology ca 1950 – Developed by Fred Hoyle and Thomas Gold, it proposes that the universe has been expanding for eternity and that new galaxies are created, atom by atom, in intergalactic space ‘out of empty space’. This theory had its heyday in the 1950’s and 1960’s, but was never able to explain convincingly where the cosmic background radiation came from, why it is so isotropic, and why its temperature is fixed at 2.7 K. It also provided no clues as to why there ought to be a universal abundance ratio for hydrogen, helium and deuterium.
Cold Big Bang Cosmology ca 1965 – Developed by David Layzer at Harvard, it proposes that the Big Bang occurred, but that the initial state was at absolute zero and consisted of a pure solid of hydrogen. This fragmented into galaxy-sized clouds as the universe expanded. It provided no explanation for where the cosmic background field came from and why it is isotropic and at a temperature of 2.7 K.
Hagedorn Cosmology ca 1968 – Physicist Robert Hagedorn proposed that all of the details of big bang theory are probably true, except that the early history of the universe had a limiting temperature of about 1 trillion degrees because the structure of matter has an infinite ladder of ‘fundamental particles’ out of which electrons, protons and neutrons are constructed. This has been refuted with the discovery that quarks exist which place any limiting temperature for the early history of the universe at temperatures well above 1000 trillion degrees.
Brans-Dicke Cosmology ca 1955 – Einstein’s equation for gravity in general relativity is modified to include a ‘scalar’ field. This field causes the value of the constant of gravity to change slowly over billions of years. This also leads to a modification of the early history of the universe. Experimental searches for a change in the constant of gravity show that it has not changed to within experimental error during the last 2 – 3 billion years. It would cause the evolution of the Earth-Moon system to be significantly altered, and the evolution of the Sun to be severly modified. Neither of these effects have been observed.
Old Inflationary, Big Bang Cosmology ca 1980 – Developed as a ‘toy’ model by Alan Guth in 1980. The Inflationary era which ended 10^-34 seconds after the Big Bang caused the nucleation of innumerable ‘bubbles’ of true vacuum which merged together to form a patina of matter and radiation in a very lumpy configuration. The cosmic background radiation, however, shows that the universe is very smooth to 1 part in at least 10,000 since about 300,000 years after the Big Bang. There is no evidence for such a turbulent and lumpy transition era.
Oscillatory Big Bang Cosmology ca 1930 – This a a possible modification to Big Bang cosmology that differs only in that the current expansion will be replaced by a collapse phase and then an expansion phase etc etc. There is no evidence that there was ever a prior expansion-collapse phase. The universe also does not seem to have enough matter to make it a ‘closed’ universe destined to recollapse in the future; an important requirement for any future oscillation cycle.
Big Bang Cosmology with added Neutrino Families ca 1970 – Big Bang cosmology is largely correct, except that to solve the ‘missing’ or ‘dark’ matter problem, new families of neutrinos have to be added to the universe. This would change the cosmological abundance ratios of helium and deuterium relative to hydrogen so that the current observed values no longer are possible. There is also no experimental evidence that more than 3 types of neutrinos exist; and these are already consistent with the measured cosmological abundances.
Chronometric Cosmology ca 1970 – Developed by I. Segal at MIT, it proposes that space-time has a different mathematical structure than the one that forms the basis for Big Bang cosmology. So far as we can tell, the major disagreement is in the rate of the expansion of the universe which comes out as a quadratic law between distance and expansion speed, rather than the linear Hubble Law. This proposal seems to be inconsistent with what has been observed for distant galaxies during the last 3-4 decades. There may be other disagreements with Big Bang cosmology, but Chronometric cosmology has not been explored deep enough to make testable predictions in these other areas.
Alfven Cosmology ca 1960 – Developed by physicist Hans Alfven, it proposes that the universe contains equal parts of matter and anti-matter. No explanation is made for many of the other observational facts in cosmology. If there were equal parts of matter and anti-matter, there ought to be regions in the universe where these were in contact to produce X-rays or gamma rays due to the annihilation process. No such large-scale background has ever been detected that can be attributed to proton or electron annihilation.
Plasma Cosmology ca 1970 – The matter in the universe, on the largest scales, is not neutral, but has a very weak net charge which is virtually undetectable. This causes electromagnetic forces to dominate over gravitational forces in the universe so that all of the phenomena we observe are not the products of gravitation alone. This is an intriguing theory, but other then denying their importance, it cannot easily explain the origin of the cosmic background radiation, its isotropy and temperature, and the abundances of helium and deuterium.
The basic observations that are agreed to me cosmological tests for any theory are:
1…. The universe is expanding. – This is a large-scale observation which spans the entire observable universe so it must be ‘cosmological’
2…. There exists a cosmic background radiation field detectable at microwave frequencies. – Why doesn’t it occur at other frequencies and only seen in the microwave region, covering every direction of the sky?
3…. The cosmic microwave background field is measurably isotropic to better than a few parts in 100,000 after compensation is made for the relativistic Doppler effect caused by Earth/Sun/Milky Way motion. – This is a large-scale property of this phenomenon that has nothning to do with the Milky Way or other galaxies so it must be a cosmological feature.
4…. The cosmic microwave background radiation field is precisely that of a black body. – Many other kinds of radiation are known, but NONE have exactly a black body spectrum. Only the cosmic background radiation is a perfect black body to the limits of our ability to measure its spectrum.
5…. The cosmic microwave background radiation field has a temperature of 2.7 K. – Why 2.7 K? Why not 5.019723 K. Only Big Bang cosmology predicts a relic radiation at a temperature near 3 degrees and not some other value.
6…. There does exist a universal abundance ratio of helium to hydrogen consistent with the current expansion rate and cosmic background temperature. – Whether we look at the compsition of stars, planets or even gas clouds in distant galaxies, we always seem to come upon a ‘universal’ constant ratio of helium to hydrogen and deuterium to hydrogen. There must be an explanation for this that has nothing to do with just our solar system or Milky Way.
7…. The cosmological abundance of deuterium relative to hydrogen and helium is consistent with the levels expected given the current expansion rate and density. – If the universe expanded faster, then there would be less time for heavier elements such as helium and deuterium to form.
8…. There are only three families of neutrinos. – Although we have not confirmed this to be true in the vicinity of distant galaxies, we do see the same kinds of elements and physics occurring out there, especially supernovae whose physics depend very sensitively upon the numbers of distinct types of neutrinos, and the constancy of the underlying ‘weak interaction’ physics.
9…. The night sky is not as bright as the surface of the Sun. – A simple but profound observation which can only be resolved by the correct distribution of stars in the universe, their ages, and the expansion of the universe.
10… The cosmic background radiation field is slightly lumpy at a level of one part in 100,000 to 1,000,000. – Why is this? And why by this amount?
11… There are no objects that have ages indisputably greater than the expansion age of the universe. – Our universe nearby does not seem to have very old stars older than 20 billion years even though their properties should be easily recognizable and a simple extension of the physics and evolution of the oldest stars we do see.
12… There are about 10,000,000,000 photons in the cosmic background radiation field for every proton and neutron of matter. – This is an important ‘thermodynamic’ number which tells us how the universe has evolved up to the present time. Why is its entropy so huge?
13… The degree of galaxy clustering observed is consistent with an expanding universe with a finite age less than 20 billion years. – A direct observation which again tells us that gravity has not had a long time to act to build up large complex structures in today’s universe.
14… There are no elements heavier than lithium which have a universal abundance ratio. – What process created these heavier elements?
15… The universe was once opaque to its own radiation. – This must follow from the black body shape of the cosmic background radiation.
16… The universe is now dominated exclusively by matter and not a mixture of matter and anti-matter. – Only a few contenders to Big Bang cosmology make any attempt at explaining this direct observation.
So there you have it. This is not a game of billiards where the cue ball (data) is carefully lined up so that Big Bang theory comes out looking inevitable. Any of these other theories have been repeatedly invited to take their best shot too, and the results are always the same. The proponents have to intervene to even get their theories to pony up a simple prediction for any of these cosmological data.
The biggest prediction of Big Bang cosmology lies in its very foundations. It is based on the inerrancy of general relativity and how this theory accounts for gravity under extreme conditions. Its basic predictions have been tested many times, and new exotic phenomena such as the Lenze-Thirring effect and gravity waves have also been predicted and confirmed by the theory. It seems to be a flawless explanation for how grfavity workd, but if it is accurate, then we need lots of Dark Matter and Dark Energy in the universe in addition to the 5% of stars and gas that we can see. That is the big problem.
Dark matter is found not only on the cosmic scale but in regions as small as galaxies. In fact it was discovered in galaxies long before WMAP made its first studies. Our own Milky Way seems to have six times more gravitating stuff orbiting its center than in all the luminous matter and gas clouds we can detect. In fact, any large systems of matter we have studied have this Dark MAtter problem. Some physicists have interpreted this as an actual breakdown in General Relativity itself, but their proponents cannot find an extension or replacement for general relativity that makes Dark Matter go away. Meanwhile, physicists have not detected any of the particle candidates for Dark matter at the Large Hadron Collider or other labs around the world.
So Dark Matter can be added to Big Bang cosmology, but we don’t yet know what kind of physical material it is, or whether there might be something subtly wrong with General Relativity itself.
The Big Bang did not happen inside the 3-d space of our universe, at least that’s what our best understanding of physics seems to be telling us during the last 100 years! That means it did not happen ‘over there’ a billion light years beyond Antares.
The only guide we have for answering this question is Big Bang cosmology and Einstein’s general theory of relativity. These make very specific predictions for what happened to space and time during the Big Bang.
The figure above (Credit: Martin Kornmesser, Luis Calcada, NASA, ESA/Hubble) shows the expansion of the universe predicted by general relativity. Note that the latitude and longitude positions of the ‘star’ galaxies remains the same but the distance between them is dilating as the radius of the sphere increases. This is a representation of how our universe is expanding. From this geometric analogy, if the surface of the sphere represented the 3-d volume of our universe at a specific time since the big bang, you see that the volume of space is increasing but space isnt being added in from some where else! This is a sphere in 4-dimensions with a 3-d surface, in geometric analog to a 3-d basket ball with a 2-d surface. Also, as you shrink the sphere’s radius, the volume of 3-d space decreases steadily until it approaches the condition where the radius is zero. You can do this with a mathematical ‘balloon’ but not a real one. At zero radius we also have the condition where the 3-d volume of space also vanishes.
Now suppose that this spherical surface was filled with atoms. As you shrink the volume of space, the density of this matter increases steadily. When you get close to the time where the radius is zero, the average density of matter in the 3-d space of this sphere has grown enormously. When the radius becomes zero, the density becomes infinite and we have what physicists call a singularity.
So, the best, non-mathematical description that any cosmologist can create for describing the Big Bang is that it occurred in every cubic centimeter of space in the universe with no unique starting point. In fact, it was an event which our mathematics indicate, actually brought space and time into existence. It did not occur IN space at a particular location, because it created space ( and time itself) as it went along. There may have existed some state ‘prior’ to the Big Bang, but it is a state not described by its location in time or space. This state preceded the existence of our time and space.
It has been over a year since I published my book on how to build DIY magnetometers that can detect geomagnetic storms. The $8.00 B/W book ‘Exploring Space Weather with DIY Magnetometers‘, is available at Amazon by clicking [HERE]. It contains 146 pages with 116 illustrations and figures that describe six different magnetometers that you can build step-by-step for under $50.00.
Over the last year I have posted at my Astronomy Cafe blog, and at various LinkedIn groups, the magnetograms from my most sensitive magnetometers to show how well they capture the rapid changes of Earth’s magnetic field during a geomagnetic storm. Since I started posting these DIY magnetograms on Linkedin, they have received over 13,500 views so there seem to be a lot of teachers, amateur astronomers and space weather enthusiasts interested in my DIY technology.
On the NOAA Scale, these storms are stronger than G2 and are currently happening every month or so during the sunspot-maximum period. This blog is my Gallery of the storms I have detected so far. I also show the data for each storm event observed from the Fredericksburg Magnetic Observatory (FRD) located about 200 miles south of my suburban Maryland location. This will give you a sense of just how accurate my designs are compared to the far more expensive, professional-grade systems.
By the way, the July 14, 2023 magnetogram ‘spots’ shows what can be accomplished by a simple $5.00 soda bottle magnetometer if you follow a design with a laser pointer and a 7-meter projection distance as described in my book.
Ok…so here are the magnetograms in reverse chronological order starting from the most recent storms and working down the list to the earlier ones towards the end of 2023. I am only presenting the magnetograms and not a lot of supporting information about the circumstances of the storms themselves. For this information, visit the Spaceweather.com website and in the upper right corner of the webpage in the Archive area enter the date of the storm and you will be able to see a lot of info and even amateur photos of the resulting aurora themselves.
October 10-12, 2024, Major Kp=8-9, Great Aurora. FRD magnetic observatory plot (red) versus the RM3100 (black). Single-digit Kp index numbers on top row (from 2 to 9). The features that look like sudden ‘glitches’ at Kp=9, 8 and 7 seem to be very real and rapid changes in the D-component (angular displacement). 1000 units on the vertical axis corresponds to 200 arcseconds or 3.3-arcminutes variation.
August 12, 2024. Major Kp=8 storm. Green arrows are the Sq current variations. FRD (red) and RM3100 (black).
August 2, 2024 Kp=7 storm event. FRD(black) and RM3100 (blue).
June 23, 2024, Kp=8. Sq minima (arrows). Storm event (blue bar).
May 12, 2024, Kp=6 storm. FRD(red), RM3100 (black), Photo (blue), Hall (green).
May 10, 2024 Kp=8-9 major geomagnetic storm.
May 5, 2024 storm Kp=4-5. FRD(red), RM3100 (black)
March 23, 2024. Diurnal Sq dips (arrows) and a strong geomagnetic storm (hour 40). FRD data (red) and RM3100 data (black).
November 27, 2023. Minor Kp=6 storm at running UT of 83-86 . FRD data (yellow), RM3100 (grey), photoelectric magnetometer (orange), Hall sensor (blue)
November 5, 2023. A significant Kp=7 geomagnetic storm superposed on a few wobbles due to Sq current effects.
October 30, 2023. Three days of Sq variations and no geomagnetic storms. FRD data (red), RM3100 magnetometer (black), photoelectric magnetometer (blue), Hall effect magnetometer (orange).
July 14, 2023. Kp=4 geomagnetic storm (blue bar) with three cycles (yellow) of the diurnal Sq current. Red line = FRD data. Spots = soda bottle magnetometer.
We had a minor geomagnetic storm on Monday just after the major storm on Saturday that everyone saw. This minor storm launched a CME caused by an X-5.8 solar flare on Friday, but despite early estimates it might rival the major storm, it was a glancing blow to Earth’s magnetic field and caused no aurora over much of the Lower-48 States. Many had hoped they would get to see an aurora in Maryland and other mid-latitude locations but the storm was too week to be seen in most states that had enjoyed the Great Storm of May 10-11.
Nevertheless, my DIY magnetometers did show some life for this Kp=6 event as shown below. This time I had three different magnetometers operating. The top numbers are the 3-hour Kp indices. The red trace is from the Fredricksberg Magnetic Observatory. The black trace is from the RM3100-Arduino system. The blue trace is from the Differential Hall Sensoe system. The green trace os from the Differential Photocell Magnetometer. The two dips marked with ‘Sq’ are the diurnal Sq variations, which were recorded by all magnetometers.
We just passed through the biggest ‘solar storm’ in the last 20 years caused by the massive naked-eye sunspot group called AR-3664. Its size was 15 times the diameter of Earth and rivaled the size of the famous Carrington sunspot of September, 1859. Since it first appeared on May 2, it remained inactive until May 9 when it released an X2.2-class solar flare at 10:10 UT.
This enormous and violent release of energy stimulated the launch of six coronal mass ejections of which three merged to become an intense ‘cannibal CME’ that arrived near Earth on May 10 at 16:45 UT. Its south-directed magnetic field was perfect for imparting the maximum amount of energy to our planet’s magnetosphere. For a transit time of about 24-hours, it was traveling at a speed of about 1,700 km/s when it arrived. It sparked a G5-level extreme geomagnetic disturbance with a Kp index of 9 between May 10, 21:00 UT and May 11, 03:00 UT.
On May 9th at 06:54 UT AR-3664 produced an X-3.9 flare. This was followed on May 11 with a fourth major X-5.8 flare at 1:39 UT, which caused an immediate shortwave radio blackout across the entire Pacific Ocean that lasted for several hours. It is expected that the May 11 flare sparked anoher CME that may arrive near Earth on Monday evening May 13.
The last G5 geomagnetic storm that we experienced was way back in October 28 to November 5, 2003. These Halloween Storms caused power outages in Sweden and damaged transformers in South Africa. Despite many recent cautionary comments in the news media about cellphone and satellite outages and power grid problems, as yet none of these have been identified but perhaps in the next few weeks these technological impacts may start to be mentioned as anecdotes begin to surface.
Unfortunately, many areas on the East Coast were covered by clouds during this three-day period and missed the opportunity to see these major aurorae. However, my DIY magnetometer (see my earlier blog on how to build your own $50 magnetometer (located in Kensington, Maryland (latitude 39o N) was able to keep up with the invisible changes going on, and produced a very respectable record of this entire storm period. As a scientist, I am often working with things I cannot directly see with my eyes, so the fact that I had my trusty magnetometer to reveal these invisible changes around me was pretty cool!
This graph shows a side-by-side comparison of the data recorded by my RM3100 magnetometer (black) and the magnetometer at the Fredericksberg Magnetic Observatory (red). I have shifted and rescaled the plots so you can more easily see how similar they are. This is very satisfying because it shows that even a simple home-made magnetometer can perform very well in keeping up with the minute changes in the geomagnetic field. This plot shows the variation in the so-called D component, which is the local magnetic declination angle. Mathematically is is defined by D = arctan(Bx/By). It’s the angle relative to geographic North that your local compass points.
Below is a slightly different graph of the RM3100 data. As you can see in the first part of the above plot between 36 and 63 UT hours, the smooth change is caused by the diurnal Sq current effect that is correlated with the solar elevation angle. During this storm period, it is assumed to have behaved smoothly during the actual storm, so in the graph below I have subtracted it from the magnetometer data. The result is that I have now isolated the changes due to the storm itself. The top row of numbers are the 3-hour Kp index averages from NOAA. The marked times are for EDT in Maryland. Universal Time is 4 hours ahead of EDT.
This was, indeed, a very powerful storm that lasted about 42 hours. This places it among a handful of exceptional geomagnetic storms that includes the great Carrington Storm of August 28 to Septemer 5, 1859.
Why is this important? Well, in the grand scheme of things it may not matter much, but as an astronomer it is still a lot of fun to have access to the invisible universe from the comfort of my suburban home. I will let geophysicists have all the fun deciphering all the bumps and wiggles and what they tell us about our magnetic field and solar storms!
Meanwhile, my gear is primed and ready to go to detect this Monday’s next storm. Some predict that it may be even bigger then the one we just experienced. It’s interesting how the Carrington Storm was actually two major storms separated by a few days, with the CME from the first storm also canibalizing several other CMEs that were also enroute.
Like the introduction of hand calculators into the classroom in the 1970s, ChatGPT offers enormous promise but currently suffers from a variety of negative expectations. Some of the arguments against students using calculators in the 1970s classroom are being used today against ChatGPT. I think there are some applications that work very well if you consider ChatGPT to be an intelligent ‘hand calculator’ in the math and physical science classroom. Here is an example I came up with without much effort!
I. Student Pre-Requisite Knowledge
The following example requires students to work with Scientific Notation, calculating the volume of spheres and shells, and working with the mass:density:volume relationship.
II. Exoplanet Interior Modeling
Astronomers have discovered over 5000 exoplanets orbiting other stars. We call these ‘exoplanets’ so that they don’t get confused with the ‘planets’ in our solar system. From a careful study of these exoplanets, astronomers can figure out how long they take to orbit their stars, their distance from the star, their diameters and their masses. How do they use this information to figure out what the insides of these exoplanets look like? This activity will show how a simple knowledge of mass, volume and density provides the clues!
III. Mass, Density and Volume
Mass, volume and density are related to each other. If two things occupy the same volume but have different masses, the less-massive one will have the lower density.
Density = Mass / Volume.
Example 1: A Prospector had his sample weighed to be 20 grams, and its volume calculated by water displacement and found to be 4 cubic centimeters. If pure gold has a density of 19.3 gm/cc, is his sample actually gold or is it iron pyrite (density 5.0 gm/cc)?
Answer: Density = 20 gm/4 cc = 5 gm/cc so it’s iron pyrite or ‘Fools Gold’.
Example 2: A basic principle of physics is that light things of low density float on top of denser things. Why do you have to shake a bottle of salad dressing before you use it?
ChatGPT Query: There are five different liquids mixed together in a bottle. After 10 minutes they sort themselves out. The liquids are: Olive oil ( 0.92 g/cc ), water ( 1.0 g/cc), molassis ( 1.4 g/cc) , vinegar ( 1.0006 g/cc), honey (1.43 g/cc). From bottom to top, how will the liquids separate themselves?
IV. Designing Mercury with a One-Component Interior Model
Mercury was formed close to the sun where only iron and nickel-rich compounds could condense into a planet. Let’s model Mercury and see what we discover. The actual mass of Mercury is 0.055 times Earth.
Step 1 – Use the formula for the volume of a sphere V=4/3 pR3 and with a known radius for Mercury of Rm = 2.43×106 meters to get the volume of the planet of V = 6×1019 m3.
Step 2 – Calculate the mass of Mercury for various choices of density. Give the predicted mass for Mercury in multiples of Earth’s mass of 5.97×1024 kg.
Step 3: Test your knowledge: For a density of 5000 kg/m3 and a radius of 2.43×106 meters, what is the mass of Mercury for these selected values? Give your answer to two significant figures.
Volume = 4/3p (2.43×106meters)3 = 6.0×1019 m3
Mass = 5000 x Volume = 3.0×1023 kg
Mase(Earth units) = 3.0×1023 kg /5.97×1024 kg = 0.05 times Earth
Use ChatGPT to generate data for plotting. Enter this question into the window:
ChatGPT Query: A sphere has a radius of 2.43×10^6 meters. What is the mass of the sphere if its density is 5000 kg per cubic meter? Express your answer in units of Earth’s mass of 5.97 x 10^24 kg. Give your answer to two significant figures.
Repeat the ChatGPT query four times to generate a mass estimate for densities of 4000, 4500, 5000, 5500 and 6500 kg/m3. Plot these points on a graph of mass versus density and draw a line through the values. Which density gives the best match to the observed mass of Mercury of 0.055 Mearth? (Answer: about 5500 kg/m3).
V. Designing Mars with Two-Component Models
Now we add two components together for planets that have a high density core and a lower density mantle. These would have formed farther out than the orbit of Mercury but with masses lower than than of Earth. The mathematical model consists of a spherical core with a radius of Rc, surrounded by a spherical shell with an inner radius of Rc and an outer radius of Rp, where Rp is the observed planetary radius. Mathematically the model looks like this:
M = Dc x 4/3p Rc3 + Dm x 4/3p ( Rp3 – Rc3)
Draw a diagram of the planet’s interior showing Rc and Rp and confirm that this is the correct formula for the total mass of the planet where Dc is the core density, and Dm is the mantle density.
Test Case: An exoplanet is discovered with a mass of 5.97×10^24 kg and a radius of 6,378 kilometers. If the radius of its core is estimated to be Rc = 3,000 km and its core density is 7000 kg/m3, what is the average density of the mantle material?
Dm = (5.97×1024 – 7000 x 1.1×1020)/9.8×1020 = 5300 kg/m3
Check your answer with ChatGPT using this query. A planet consists of a core region with a radius of Rc and a mantle region extending to the planet’s surface at a radius of Rp. If the planet is a perfect sphere with a radius Rp = 6378 km and Rc = 3000 km, with a total mass of 5.97×10^24 kg, for a core density of 7000 kg/cubic meters, what is the average mantle density? Give the answer to two significant figures.
Now lets use ChatGPT to generate some models and then we can select the best one. We will select a mantle density from three values, 2000, 3000 and 4000 kg/m3. The core density Dc will be fixed at Dc = 9000 kg/m3. We will use the measured radius for Mars of Rp = 3.4×106 meters, and its total mass of Mm = 6.4×1023 kg. We then vary the core radius Rc. We will plot three curves on a graph of Rc versus Mm one for each value of the assumed mantle density. Use this ChatGPT query to generate your data points.
ChatGPT Query: A planet is modeled as a sphere with a radius of Rp=3.4×10^6 meters. It consists of a spherical core region with a radius of Rc surrounded by a spherical shell with an inner radius of Rc and an outer radius of Rp. The core of the planet has a density of 9000 kg/cubic meters. The radius of the core Rc = 30% of the planet’s radius. If the density of the mantle is 2000 kg/cubic meter, what is the total mass of the planet in multiples of the mass of Earth, which is 5.97×10^24 kg? Give your answer to two significant figures?
Repeat this query by changing the mantle density and the core radius values and then plot enough points along each density curve to see the trend clearly. An example of an Excel spreadsheet version of this data is shown in this graph:
This graph shows solutions for a two-component mars model where the mantle has three different densities (2000, 3000 and 4000 kg/m3). The average density of mars is 3900 kg/m3. Which core radius and mantle density combinations seem to be a better match for Mar’s total mass of 0.11 Mearth for the given density of the mantle?
VI: Modeling Terrestrial Planets with a three-component interior.
The most general exoplanet model has three zones; a dense core, a mantle and a low-density crust. This is the expected case for Earth-like worlds. Using our Earth as an example, rocky exoplanets have interiors stratified into three layers: Core, mantle, crust.
Core material is typically iron-nickel with a density of 9000 kg/m3
Mantle material is basaltic rock at a density of 4500 kg/m3
Crust is low-density silicate rich material with a density of 3300 kg/m3
The basic idea in modeling a planet interior is that with the three assumed densities, you vary the volume that they occupy inside the exoplanet until you match the actual mass (Mexo) in kilograms and radius (Rexo) in meters of the exoplanet that is observed. The three zones occupy the radii Rc, Rm, Rp
We will adjust the core and mantle radii until we get a good match to the exoplanet observed total mass and radius. Let’s assume that the measured values for the Super-Earth exoplanet mass is Mp = 2.5xEarth = 1.5×1025 kg, and its radius is Rp = 1.5xEarth = 9.6×106 meters.
Core Volume Vcore = 4/3p Rc3
Mantle Volume Vm = 4/3 p (Rm3 – Rc3)
Crust Volume Vcrust = 4/3 p (Rp3 – Rm3)
So the total Mass = (9000 Vcore + 4500Vm + 3300Vcrust)/Mp
Rc ,Rm and Rp are the core, mantle and planet radii in meters, and the total mass of the model is given in multiples of the exoplanet’s mass Mp.
Let’s do a test case that we work by hand to make sure we understand what we are doing.
Choose Rc = 30% of Rp and Rm = 80% of Rp. What is the predicted total mass of the exoplanet?
Rc = 0.3 x 9.6×106 meters = 2.9×106 meters.
Rm = 0.8x 9.6×106 meters = 7.7×106 meters.
Then
Vcore = 4/3p (2.9×106)3 = 1.0×1020 m3
Vm = 4/3p ( (7.7×106)3 – (2.9×106)3) = 1.8×1021 m3
Then Mass = (9000 Vcore + 4500 Vm + 3300Vcrust)/Mp
Mass = (9×1023 kg + 8.1×1024 kg + 5.9×1024 kg)/1.5×1025 kg = 1.0 Mp
Now lets use ChatGPT to generate some models from which we can make a choice.
Enter the following query into ChatGPT to check your answer to the above test problem.
ChatGPT Query: A spherical planet with a radius of Rp consists of three interior zones; a core with a radius of Rcore, a mantle with an inner radius of Rc and an outer radius of Rm, and a crust with an inner radius of Rm and an outer radius of Rp=9.6×10^6 meters. If the density of the core is 9000 kg/m^3, the mantle is 4500 kg/m^3 and the crust is 3300 kg/m^3, What is the total mass of the planet if Rc = 30% of Rp and Rm = 80% of Rp? Give your answer for the planet’s total mass in multiples of the planet’s known mass of 1.5×10^25 kg, and to two significant figures.
Re-run this ChatGPT query but change the values for the mantle radius Rm and core radius Rc each time. Plot your models on a graph of Rc versus the calculated mass Mp on curves for which Rm is constant. An example of this plot is shown in the excel spreadsheet plot below.
For example, along the black curve we are using Rm=0.8. At Rc = 0.5 we have a model where the core extends to 50% of the radius of the exoplanet .The mantle extends to 80% of the radius, and so the crust occupies the last 20% of the radius to the surface. With densities of 9000, 4500 and 3300 kg/m3 respectively, the Y-axis predicts a total mass of about 1.1 times the observed mass of the exoplanet (1.00 in these units). With a bit of fine-tuning we can get to the desired 1.00 of the mass. But what about the solution at (0.3, 1.00) ? In fact, all of the solutions along the horizontal line along y = 1.00 are mathematically valid.
Question 1: The exoplanet is located close to its star where iron and nickel can remain in solid phase but the lower density silicates remain in a gaseous phase. Which of the models favors this location at formation?
Answer: The exoplanet should have a large iron/nickel core and not much of a mantle or crust. This favors solutions on the y=1.00 line to the right of x=0.5.
Question 2: The exoplanet is located far from its star where it is cool enough that silicates can condense out of their gas phase as the exoplanet forms. Which of the models favor this location?
Answer: The exoplanet will have a small iron/nickel core and a large mantle and crust. This favors models to the left of x= 0.5.
So here you have some examples for how ChatGPT can be used as an intelligent calculator once the students understands how to use the equations and is able to explain why they are being used for a given modeling scenario.
I would be delighted to get your responses and suggestions to this approach . Just include your comment in the Linkedin page where I have posted this idea.
One year ago, I posted a fun problem of predicting when we will have the very last total solar eclipse viewable from Earth. It was a fun calculation to do, and the answer seemed to be 700 million years from now, but I have decided to revisit it with an important new feature added: The slow but steady evolution of the sun’s diameter. For educators, you can visit the Desmos module that Luke Henke and I put together for his students.
The apparent lunar diameter during a total solar eclipse depends on whether the moon is at perigee or apogee, or at some intermediate distance from Earth. This is represented by the two red curved lines and the red area in between them. The upper red line is the angular diameter viewed from Earth when the moon is at perigee (closest to Earth) and will have the largest possible diameter. The lower red curve is the moon’s angular diameter at apogee (farthest from Earth) when its apparent diameter will be the smallest possible. As I mentioned in the previous posting, these two curves will slowly drift to smaller values because the Moon is moving away from Earth at about 3cm per year. Using the best current models for lunar orbit evolution, these curves will have the shapes shown in the above graph and can be approxmately modeled by the quadratic equations:
Perigee: Diameter = T2 – 27T +2010 arcseconds
Apogee: Diameter = T2 -23T +1765 arcseconds.
where T is the time since the present in multiples of 100 million years, so a time 300 million years ago is T=-3, and a time 500 million years in the future is T=+5.
The blue region in the graph shows the change in the diameter of the Sun and is bounded above by its apparent diameter at perihelion (Earth closest to Sun) and below by its farthest distance called aphelion. This is a rather narrow band of possible angular sizes, and the one of interest will depend on where Earth is in its orbit around the Sun AND the fact that the elliptical orbit of Earth is slowly rotating within the plane of its orbit so that at the equinoxes when eclipses can occur, the Sun will vary in distance between its perihelion and aphelion distances over the course of 100,000 years or so. We can’t really predict exactly where the Earth will be between these limits so our prediction will be uncertain by at least 100,000 years. With any luck, however, we can estimate the ‘date’ to within a few million years.
Now in previous calculations it was assumed that the physical diameter of the Sun remained constant and only the Earth-Sun distance affected the angular diameter of the Sun. In fact, our Sun is an evolving star whose physical diameter is slowly increasing due to its evolution ‘off the Main Sequence’. Stellar evolution models can determine how the Sun’s radius changes. The figure below comes from the Yonsei-Yale theoretical models by Kim et al. 2002; (Astrophysical Journal Supplement, v.143, p.499) and Yi et al. 2003 (Astrophysical Journal Supplement, v.144, p.259).
The blue line shows that between 1 billion years ago and today, the solar radius has increased by about 5%. We can approximate this angular diameter change using the two linear equations:
Perihelion: Diameter = 18T + 1973 arcseconds.
Aphelion: Diameter = 17T + 1908 arrcseconds.
where T is the time since the present in multiples of 100 million years, so a time 300 million years ago is T=-3, and a time 500 million years in the future is T=+5. When we plot these four equations we get
There are four intersection points of interest. They can be found by setting the lunar and solar equations equal to each other and using the Quadratic Formula to solve for T in each of the four possible cases.:
Case A : T= 456 million years ago. The angular diameter of the Sun and Moon are 1890 arcseconds. At apogee, this is the smallest angular diameter the Moon can have at the time when the Sun has its largest diameter at perihelion. Before this time, you could have total solar eclipses when the Moon is at apogee. After this time the Moon’s diameter is too small for it to block out the large perihelion Sun disk and from this time forward you only have annular eclipses at apogee.
Case B : T = 330 million years ago and the angular diameters are 1852 arcseconds. At this time, the apogee disk of the Moon when the Sun disk is smallest at aphelion just covers the solar disk. Before this time, you could have total solar eclipses even when the Moon was at apogee and the Sun was between its aphelion and perihelion distance. After this time, the lunar disk at apogee was too small to cover even the small aphelion solar disk and you only get annular eclipses from this time forward.
Case C : T = 86 million years from now and the angular diameters are both 1988 arcseconds. At this time the large disk of the perigee Moon covers the large disk of the perihelion Sun and we get a total solar eclipse. However before this time, the perigee lunar disk is much larger than the Sun and although this allows a total solar eclipese to occur, more and more of the corona is covered by the lunar disk until the brightest portions can no longer be seen. After this time, the lunar disk at perigee is smaller than the solar disk between perihelion and aphelion and we get a mixture of total solar eclipses and annular eclipses.
Case D : T = 246 million years from now and the angular diameters are 1950 arcseconds. The largest lunar disk size at perigee is now as big as the solar disk at aphelion, but after this time, the maximum perigee lunar disk becomes smaller than the solar disk and we only get annular eclipses. This is approximately the last epoc when we can get total solar eclipses regardless of whether the Sun is at aphelion or perihelion, or the Moon is at apogee or perigee. The sun has evolved so that its disk is always too large for the moon to ever cover it again even when the Sun is at its farthest distance from Earth.
The answer to our initial question is that the last total solar eclipse is likely to occur about 246 million years from now when we include the slow increase in the solar diameter due to its evolution as a star.
Once again, if you want to use the Desmos interactive math module to exolore this problem, just visit the Solar Eclipses – The Last Total Eclipse? The graphical answers in Desmos will differ from the four above cases due to rounding errors in the Desmos lab, but the results are in close accord with the above analysis solved using quadratic roots.
During a solar eclipse, the lansdcape will slowly dim until it is nearly complete darkness along the path of totality. other observers wil see te landscape dim a bit but then brighten to normal intensity. If you didn;t know that an eclipse was going on you might not even notice the dimming, mistaking it for a cloud passing across the sun. The geometric condition for this dimming have to do with the area of exposed solar surface and how this changes as the disk of the moon passes across it. Below is a simple mathematical model for ambient light dimming that you can put to the test the next time a solar eclipse passes over your geographic location.
I have reanalyzed the geometry and defined it in terms of the center-to-center distance, L, between the sun and moon, and their respective radii Rs and Rm as the figure of the upper half-plane of the intersection shows, with the yellow area on the left representing the disk of the sun and the white area on the right the disk of the moon. This problem was previously considered in 2000 by British astronomer David Hughes who used the distance defined by the segment FE, which he called alpha, but L = 1+M-a. The figure shows the moon overlapping the disk of the sun in a lens-shaped zone whose upper half is represented by the area AFDE.
The basic idea is that we want to compute the area of the lunar arc cap AFD by computing the area of the sector BAF and subtracting the triangle BAE from the sector area. That leaves the area of the cap as the left-over area. We perform the same calculation for the solar sector CAE and subtract the triangle CAD from this. The resulting area of the full lens-shaped overlap region is then
Occulting Area = 2x(AreaAFD + AreaAED).
Because of the geometry, the resulting area should only depend on the center-to-center separation and the radii of the sun and moon. You should not have to specify any angles as part of the final calculation. In the following we will use degree measure for all angles.
The area of the sector of a circle is just A = (Theta/360)piR2 so that gives us the first two relationships:
To simplify the problem, we are only interested in the fraction of the full sun disk that is illuminated. The full sun has an area of pi Rs2, so we divide Am and As by pi Rs2 , and if we define Rs=1.0 and M = Rm/Rs we get:
Although M is fixed by the solar-lunar ratio, we seem to have two angular variables alpha and theta that we also have to specify. We can reduce the number of variables because the geometry gives a relationship between these two angles because they share a common segment length given by h.
so that the EQ-1 for A can be written entirely in terms of the center-to-center distance, L, and moon-to-sun disk ratio M = Rm/Rs. This is different than the equation used by Hughes, which uses the width of the lens (the distance between the lunar and solar limbs) segment FDE=a as the parameter, which is defined as L = 1+M-a.
During a typical total solar eclipse lasting 4 minutes, we can define L as
L = 1900 – 900*(T/240) arcseconds where T is the elapsed time from First Contact in seconds. Since L is in units of the current solar diameter (1900 asec) we have
EQ 3) L = 1 – T/480.
If we program EQ 1, 2, and 3 into an excel spreadsheet we get the following plot for the April 8, 2024 eclipse.
First Contact occurs at 16:40 UT and Fourth Contact occurs at 19:57 UT so the full duration is 197 minutes. During this time L varies from -(1+M) to +(1+M). For the April 8, 2024 eclipse we have the magnitude M = 1.0566, so L varies from -2.0566 (t=0) to +2.0566 (t=197m). As the moon approaches the full 4-minute overlap of the solar disk between L=-0.05 and L=+0.05 (t =97m to t=102m), we reach full eclipse.
We can re-express this in terms of the landscape lighting. The human eye is sensitive to a logarithmic variation in brightness, which astronomers have developed into a ‘scale of magnitudes’. Each magnitude represents the minimum change in brightness that the human eye can discern and is equivalent to a factor change by 2.51-times. The full-disk solar brightness is equal to -26.5m, full moon illumination is -18.0m on this scale. The disk brightness, S, is proportional to the exposed solar disk area, where E is the solar surface emission in watts/m2 due to the Planck distribution for the solar temperature of T=5770 k. This results in the formula:
m = -26.5 – 2.5log10(F)
where F is the fraction of the full disk exposed and is equal to Equation 1. For a sun disk where 90% has been eclipsed, f=0.10 and the dimming is only 2.5log(1/10) = 2.5m. How this translates into how humans perceive ambient lighting is complicated.
The concept of a Just Noticeable Difference is an active research area in psychophysics. In assessing heaviness, for example, the difference between two stimuli of 10 and 11 grams could be detected, but we would not be able to detect the difference between 100 and 101 grams. As the magnitude of the stimuli grow, we need a larger actual difference for detection. The percentage of change remains constant in general. To detect the difference in heaviness, one stimulus would have to be approximately 2 percent heavier than the other; otherwise, we will not be able to spot the difference. Psychologists refer to the percentages that describe the JND as Weber fractions, named after Ernst Weber (1795-1878), a German physiologist whose pioneering research on sensation had a great impact on psychological studies. For example, humans require a 4.8% change in loudness to detect a change; a 7.9% change in brightness is necessary. These values will differ from one person to the next, and from one occasion to the next. However, they do represent generally accurate values.
The minimum perceivable light intensity change is sometimes stated to be 1%, corresponding to +5.0m, but for the Weber Fraction a 7.8% change is required in brightness corresponding to only -2.5log(0.078) = +2.7m. This is compounded by whether the observer is told beforehand that a change is about to happen. If they are not informed, this threshold magnitude dimming could be several magnitudes higher and perhaps closer to the +5.0m value.
This is my new book for the general public about our sun and its many influences across the solar system. I have already written several books about space weather but not that specifically deal with the sun itself, so this book fills that gap.
We start at the mysterious core of the sun, follow its energies to the surface, then explore how its magnetism creates the beautiful corona, the solar wind and of course all the details of space weather and their nasty effects on humans and our technology.
I have sections that highlight the biggest storms that have upset our technology, and a discussion of the formation and evolution of our sun based on Hubble and Webb images of stars as they are forming. I go into detail about the interior of our sun and how it creates its magnetic fields on the surface. This is the year of the April 2024 total solar eclipse so I cover the shape and origin of the beautful solar corona, too. You will be an expert among your friends when the 2024 eclipse happens.
Unlike all other books, I also have a chapter about how teachers can use this information as part of their standards-based curriculum using the NASA Framework for Heliospheric Education. I even have a section about why our textbooks are typically 10 – 50 years out of date when discussinbg the sun.
For the amateur scientists and hobbyists among you, there is an entire chapter on how to build your own magnetometers for under $50 that will let you monitor how our planet is responsing to solar storms, which will become very common during the next few years.
There hasn’t been a book like this in over a decade, so it is crammed with many new discoveries about our sun during the 21st century. Most books for the general public about the sun have actually been written in a style appropriate to college or even graduate students.
My book is designed to be understandable by my grandmother!
Generally, books on science do not sell very well, so this book is definitely written without much expectation for financial return on the effort. Most authors of popular science books make less than $500 in royalties. For those of you that do decide to get a copy, I think it will be a pleasurable experience in learning some remarkable things about our very own star! Please do remember to give a review of the book on the Amazon page. That would be a big help.
Yep…I want to get the e-book version ($5): Link to Amazon.
Yep…I want to get the paperback version ($15): Link to Amazon.
Oh…by the way…. I am a professional astronomer who has been working at NASA doing research, but also education and public outreach for over 20 years. Although I have published a number of books through brick-and-morter publishing houses, I love the immediacy of self-publishing on topics I am excited about, and seeing the result presented to the public within a month or two from the time I get the topic idea. I don’t have to go through the lengthy (month-year) tedium of pitching an idea to several publishers who are generally looking for self-help and murder mysteries. Popular science is NOT a category that publishers want to support, so that leaves me with the self-publishing option.
Other books you might like:
Exploring Space Weather with DIY Magnetometers. ($7). Link to Amazon.
History of Space Weather: From Babylon to the 21st Century. (paperback, $30) (ebook, $5). Link to Amazon.
Solar Storms and their Human Impacts (e-book; $2) Link to Amazon.
The 23rd CycleL Learning to live with a stormy star. – Out of print.
An astronomer's point-of-view on matters of space, space travel, general science and consciousness
