# How to use ChatGPT to model exoplanet interiors as a lab for high school students.

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?

Vc = 4/3p Rc3  =   4/3p (3000000m)3 = 1.1×1020 m3

Vmantle = 4/3 p Rp3 – Vc  =  1.1×1021 m3 – 1.1×1020 m3 = 9.8×1020 m3.

Solve equation for Dm:

Dm =  ( M – Dc x Vc ) /Vm

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

Vcrust =  4/3p ((9.6×106)3  – (7.7×106)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.

# Smartphone Photography of the April 8, 2024 Eclipse

Many people, including me, will try to capture some images of the eclipse on Monday, April 8. This blog is aimed at people in the Greater Washington DC area who will experience a partial eclipse. If you travel 6 hours due-west of Washington DC you will be on the Path of Totality and your experiences will be dramatically different.

In the Washington DC area, the eclipse will start at 2:04 PM with the dark lunar disk taking its first little bite out of the solar disk, and end at 4:33 PM as the moon leaves the disk. The maximum partial eclipse will occur at 3:20 PM when the moon will block about 89% of the solar disk. Here’s what that looks like:

You will notice a rapid darkening of the daylight sunshine so that instead of the normal mid-afernoon sun it will look more like early twilight for about 5-10 minutes before the daylight finally starts to return.

Safety:

This is a partial eclipse. Only use approved ‘eclipse glasses’ and not sunglasses. You will not be able to see the fabulous corona unless you are on the path of Totality.

I know it is tempting, but good photography practice is NOT to point your camera at the sun with no filter…including your smartphone. Smartphones have faint light meters for twilight photography and you run the risk of damaging this meter so that you may not be able to take low-light-level photos anymore.

Manage your expectations. You will not see the corona that everyone talks about. With your Eclipse Glasses you will see a sequence of partial stages that look something like this. This was taken by NASA/Noah Moran at the Johnson Space Center during the August 21, 2017 eclipse which was only a partial eclipse over Houston, TX. Also, instead of seeing a super-huge image with the naked eye, you will only see a disk as large as the full moon in the night sky.

Smartphone Photography Tips.

1. On sunday at 3:30 pm go outside and check that your viewing location will give you a good view of the sun. Put on your Eclipse Glasses and check that your view of the solar disk is unobscurred. The higher the sun is above the clutter at your chosen location the better your experience will be.
2. On Friday, Saturday or Sunday before the eclipse, place one of the lenses of the Eclipse Glasses over the selfi-camera lens located at the top edge of your smartphone just below the center of the top edge.
3. Start-up your camera and place it in selfi mode.
4. With the sun’s disk over your left or right sholder, check that your camera display shows a bright orange disk of the sun and adjust your camera angle so that the disk is centered and unobstructed by your head.
5. Your camera should automatically be able to focus on the edge of the sun disk and set the camera’s exposure. This photo on a cloudy morning on Friday April 4 without editing was automatically taken by my iPhone 13 Pro camera at 1/15 of a second at an ISO of 1600.

With no clouds, the sun disk should be crisp with a good clean edge. You may need to experiment with the manual focus if your camera allows you to do this.

6. You might want to experiment with manipulating this test image of the filtered sun to get the best clarity and background. With stray clouds this is a challenge as the image below, adjusted with Photoshop, shows. I adjusted the brightness and contrast.

The crispness of the solar disk was compromized by the diffusing of sunlight in the foreground clouds. You will have a better experience if there are no clouds in the way. To get an idea of what this optimal picture would look like in selfie-mode, here is an image of the moon taken by my iPhone 13 Pro in selfie mode. You will see a similar-sized solar disk with the filter covering the selfi camera lens but you will only see the sun disk and not the foreground trees etc. This was the best focus my camera in this mode was able to provide with its smaller lens.

A second mode of solar photography is to take your camera out of ‘selfie’ mode and use the normal forward camera. It has better lenses and resolution than the selfie camera. Here is an example of a photo of the moon taken in this mode. Notice that the lunar disk is much clearer. Your eclipse picture in this mode will have the same clarity but of course the sky and foreground will be completely black through the filter.

The set-up for a higher-resolution direct image requires some preparation. You might want to create a sun shield out of foamboard that covers a 1-foot x 1-foot area.

Cut out a square hole in the center that your camera lens can peak through as shown in the left-hand image.

Cover the camera lens opening with the Eclipse Glasses filter and secure all pieces in place with tape as shown in the middle picture.

When you want to photograph the sun, start up your camera in its normal ‘forward’ mode and place it over the filter opening as shown in the right-hand image. Keep your eye close to the camera display so that your head is shadowed by the shield. If you want, you can secure your smartpone to the foamboard with tape, but be sure that you place a 3×5 index card over the display so that you don’t get glue on it. Otherwise, you can hold the camera to the filter opening manually.

As before, your camera should be able to automatically focus on the eclipsed solar disk to give you the best clarity your particular camera is able to provide.

Good luck…but make sure you take the time to enjoy the eclipse and not worry about getting a perfect photo with your smartphone!!!