Astronomy & Astrophysics

Looking up at the night sky, we’re drawn into some of the most profound questions in science: How big is the universe? How far away are the stars? And how can we possibly measure such unimaginable distances? In this chapter, we leave the laboratory behind and step outside—sometimes far outside. Surprisingly, the tools in your pocket are enough to begin answering these ancient questions. With a smartphone’s sensors, camera, and a bit of creative thinking, you can estimate the size of the Earth, the distance to the Moon, the brightness of Venus, and even approximate the mass of the Sun.

These experiments span centuries of human inquiry, retracing the steps of Aristarchus, Eratosthenes, Hipparchus, and Galileo—now enhanced by digital precision and AI-powered analysis. From simple angular measurements using a sextant app to astrophotography through binoculars, your phone becomes a telescope, a calculator, and a celestial notebook all in one.

This chapter is a testament to how timeless curiosity meets twenty-first-century technology. Your smartphone may never leave Earth’s surface, but it can still carry you upward—giving you a foothold on the cosmic ladder, one calculation, one photo, and one thoughtful question at a time.

Earth-Based Astronomy

Measuring Earth’s Tilt: Polaris and the Smartphone Sextant (ASTR-01)

Sensors Used: Accelerometer, Gyroscope (via clinometer or sextant app), GPS (for comparison)
What’s Measured: Angle of Polaris above the horizon (≈ geographic latitude)

Description
On a clear night in the Northern Hemisphere, you can measure your latitude using nothing more than the stars and your smartphone. Locate Polaris—the North Star—positioned at the tip of the Little Dipper’s handle (Ursa Minor). This star sits nearly directly above Earth’s rotational axis, making it one of the most reliable fixed points in the night sky. Open a clinometer or sextant app and use your phone to measure the elevation angle from the horizon to Polaris. That angle corresponds closely to your latitude, thanks to the tilt of Earth’s axis relative to its orbital plane. To test your result, compare it to the latitude reported by your phone’s GPS. It’s an elegant demonstration of spherical geometry, astronomy, and ancient navigation brought into the digital age.

Measuring Angular Sizes with Digital Tools (ASTR-02)

Sensors Used: Camera, inclinometer or sextant app, optional polarizing filters
What’s Measured: Apparent angular size of the Moon and Sun

Description
With just a smartphone camera and a bit of geometry, students can estimate the angular size of celestial bodies such as the Moon and the Sun. The process begins by photographing the object — typically the Moon — and measuring its diameter in pixels. When compared to the known field of view of the smartphone’s camera, this pixel measurement translates into an angular size, usually close to 0.5 degrees. This simple technique provides a window into celestial geometry and allows students to calculate distances or physical sizes with surprising accuracy.

For a more direct method, a sextant or clinometer app can be used to measure the Moon’s angular diameter in real time. The Sun presents a greater challenge due to its intense brightness, but here, polarizing filters come to the rescue. By placing two polarizing filters over the camera lens and rotating them at 90 degrees to one another, much of the incoming light is blocked, making the Sun’s disk faintly visible. This makes it possible to measure its apparent diameter safely. Measurements typically yield values around 0.6 degrees — very close to the Moon’s, which explains the nearly perfect match seen during total solar eclipses.

These simple experiments reveal not only the elegance of observational astronomy but also the power of using digital tools to explore the sky quantitatively — bridging optics, physics, and cosmology with a handheld device.

Longitude and Latitude: Finding Your Place on Earth (ASTR-03)

Sensors Used: Inclinometer or sextant app, clock or timekeeping app, GPS (for comparison)
What’s Measured: Geographic latitude and (with effort) longitude via celestial navigation

Description
Determining your position on Earth used to be the pinnacle of navigational science — and with a smartphone, students can now retrace those steps. Latitude is the easier of the two coordinates to measure. At local noon, when the Sun is at its highest point in the sky, you can use your phone’s clinometer or sextant app to measure the elevation angle above the horizon. By referencing the date and the Sun’s declination — information easily found online or with a quick prompt to ChatGPT — you can calculate your latitude using basic trigonometry. This exercise gives students a real, hands-on feel for how early explorers and astronomers mapped the globe.

Longitude, on the other hand, is the trickier cousin. It requires precise timekeeping and knowledge of celestial timing. Historically, determining longitude meant comparing the local time (when the Sun is at its zenith) to a reference time at a known location — most famously Greenwich Mean Time. For modern learners, this becomes an opportunity to explore time zones, the rotation of the Earth, and the historical challenges that led to the invention of marine chronometers. While smartphones provide instant GPS data today, using them to rediscover old navigational techniques offers a powerful blend of astronomy, history, and mathematical reasoning.

Size of the Earth

Aristotle already made the assumption that the Earth is a sphere, based on the fact that Earth’s shadow cast on the moon during a solar eclipse is always circular. Starting with this premise, there are several different ways to measure the size of the Earth. All these methods have in common that you make a measurement at two different locations.

Using Polaris (ASTR-04)

Sensors Used: Sextant app, smartphone camera, GPS (for comparison)
What’s Measured: Latitude via Polaris elevation; change in latitude to estimate Earth’s circumference

Description
Polaris, the North Star, is not just a navigational beacon—it’s a natural measuring stick for Earth’s geometry. This experiment guides students through using a sextant app on their smartphone to determine latitude and, with some travel, even estimate the circumference of the Earth. Begin by locating Polaris in the night sky, ideally with the aid of a tripod-mounted phone, a 2× lens if available, and long exposure settings to make the stars more visible. Identify the Big Dipper to guide you to the Little Dipper, where Polaris sits at the end of the handle. Once Polaris is centered in your view, record the angle between it and the zenith using your sextant app—this value is the complement of the star’s elevation and correlates directly to your latitude.

In one example from Nuremberg, a recorded angle of 40.9° corresponded well with the actual latitude of 49.43°, since 90° − 49.43° = 40.6°. The agreement was within ±0.2°, showcasing the method’s surprising accuracy. The next challenge takes the activity even further: by traveling several hundred kilometers north or south and repeating the same measurement, students can determine how much the observed angle changes. Since one full trip around the Earth equals 360°, a small shift in Polaris’s angle provides a proportional segment of Earth’s total curvature. With just a smartphone and a bit of movement, learners can recreate Eratosthenes’ classic measurement of Earth’s circumference—one star at a time.

Observing the Zenith and the Meridian — Recreating Eratosthenes (ASTR-05)

Sensors Used: Camera, clock, clinometer or sextant app, GPS (for position and comparison)
What’s Measured: Solar elevation angle at local noon; used to estimate Earth’s curvature

Description
Eratosthenes’ experiment remains one of the most elegant scientific achievements in history—a demonstration that the shape and size of the Earth could be inferred from shadows and careful observation. This experiment invites students to follow in his footsteps using modern tools like a smartphone clinometer or sextant app.

At local noon on or around June 21st, find a tall, vertical stick or pole and observe its shadow. In places where the Sun is directly overhead—such as near the Tropic of Cancer on the summer solstice—there should be little to no shadow. But in locations further north or south, a shadow will appear. By measuring the angle of this shadow using your smartphone, and recording the exact time and location, you can calculate how far from the zenith the Sun appears.

Eratosthenes knew that Syene (modern Aswan) lay almost directly south of Alexandria, and he assumed the two were roughly on the same meridian. Using a measurement of the Sun’s angle in Alexandria at local noon, and knowing the approximate distance between the two cities, he used simple proportions to estimate Earth’s circumference. Students can replicate this process today by comparing solar elevation at two different latitudes and using GPS data to estimate the north-south distance traveled. With just a stick, a phone, and some ancient logic, learners can retrace one of the first true experiments in planetary science.

Solar Noon Imaging (ASTR-06)

Sensors Used: Camera, clock (or solar noon calculator), spirit level
What’s Measured: Shadow length at solar noon; meridian crossing of the Sun

Description
At the heart of ancient astronomy lies the concept of solar noon — the moment when the Sun reaches its highest point in the sky for a given location. This experiment allows students to detect solar noon by observing shadows using nothing more than a smartphone and a vertical reference.

To begin, use a spirit level or tripod to ensure your smartphone is perfectly horizontal. Position it so it can capture the shadow cast by a vertical object, such as a stick or pole. Wait until local solar noon — which may not coincide with 12:00 p.m. on the clock — and take a photograph of the shadow. Solar noon can be found using online calculators or astronomy apps that track the Sun’s position.

At that exact moment, you’ll notice the shadow is at its shortest. This is because the Sun is crossing your local meridian, appearing due south (or due north if you’re in the Southern Hemisphere). By repeating this observation over several days or comparing shadow lengths at different times, students can begin to understand solar motion, timekeeping, and how early civilizations tracked the passage of time using nothing but the sky and careful measurement.

Time of Meridian Crossing (ASTR-07)

Sensors Used: Camera, clock, compass or sextant app, solar position calculator
What’s Measured: Time of solar meridian crossing; longitude difference; estimate of Earth’s radius

Description
This experiment transforms your smartphone into a tool for measuring Earth’s rotation and estimating its size. Begin by aligning your smartphone with a true North-South line — ideally a wall, meridian marker, or string stretched along a compass-corrected bearing. Around the predicted time of solar noon, observe and record the exact moment when sunlight casts a shadow directly along this line. This is the moment of meridian crossing, when the Sun is due south (or due north, depending on hemisphere) and at its highest point in the sky.

To find the precise solar noon time for your location, consult the NOAA Solar Calculator or a similar app that accounts for your longitude and the date. Repeat this entire experiment at a second location approximately 300 to 400 kilometers east or west of the first. Carefully compare the time difference between the two meridian crossings. Since Earth rotates 360° in 24 hours, each hour corresponds to 15° of longitude. With a known distance and a measured time offset, you can estimate the size of Earth itself.

This is a modern take on ancient methods used to map the globe and determine time zones—one that blends celestial geometry, careful observation, and the remarkable precision of modern smartphone tools.

References:
[1] NOAA Solar Calculator, https://gml.noaa.gov/grad/solcalc/

Ship on the Horizon — Measuring Earth’s Curvature (ASTR-08)

Sensors Used: Camera (photo/video), GPS or map app, clock or timestamp, optional marine tracking app
What’s Measured: Disappearance distance of a ship due to Earth’s curvature; estimate of Earth’s radius

Description
One of the most elegant demonstrations of Earth’s curvature doesn’t require a telescope or a rocket—just a tall vantage point and a ship on the move. As a vessel sails away from shore, it gradually disappears from view—not all at once, but from the hull upward. This vanishing act isn’t due to haze or distance alone; it’s because the ship is dropping below the horizon as it follows the curve of the Earth.

To turn this visual phenomenon into an experiment, start from a high observation point—a cliff, tall building, or even a drone. Track a departing ship using your smartphone’s camera, taking either continuous video or a series of time-stamped photos. Try to estimate or verify the ship’s speed using marine tracking apps or known averages (e.g., 30 km/h for a large vessel). Note the moment the ship disappears completely, or when only the top remains visible.

Using the time and speed, calculate the distance DDD at which the ship vanished from view. Combine this with your observation height hhh, and apply the geometric relationship between line-of-sight distance and Earth’s curvature to estimate the planet’s radius. For example, at a height of 100 meters, the horizon distance is about 36 kilometers.

This experiment revives centuries-old sea navigation techniques and brings them into the digital age—using a smartphone to confirm, once again, that our world is round.

Mass of the Earth — From Size to Gravity (ASTR-09)

Sensors Used: None (calculation-based)
What’s Measured: Estimate of Earth’s mass based on radius and assumed density

Description
After determining the radius of Earth through observational experiments—such as tracking ships over the horizon or measuring solar angles—we can take the next bold step: estimating the planet’s mass. While we can’t place Earth on a scale, we can apply physics and a reasonable assumption about its average density to arrive at a surprisingly close figure.

Assume that Earth’s average density is roughly 5,000 kg/m³—a simplification, but one that aligns fairly well with reality. Using the known formula for the volume of a sphere, V=43πR3V = \frac{4}{3} \pi R^3V=34πR3, and combining it with density ρ=mV\rho = \frac{m}{V}ρ=Vm, we can rearrange to calculate the mass: m=ρ⋅Vm = \rho \cdot Vm=ρ⋅V. If you’ve previously measured the Earth’s radius in a separate experiment (say, around 6.37 × 10⁶ meters), you can plug that into the formula and compute an estimated mass.

The currently accepted value of Earth’s mass is about 5.97 × 10²⁴ kilograms. With just a smartphone, a few observations, and some help from ChatGPT or a calculator, students can derive a result remarkably close to this astronomical number—an excellent example of how far reasoning and data can take us. In the next chapter we will use this to calculate the gravitational constant.

Distance of the Moon

Once we know the size of the Earth, estimating the distance to the Moon becomes an achievable next step. Over time, astronomers have developed a number of clever methods — some simple enough for amateur observers, others requiring sophisticated tools. Here are some of the key approaches:

Angular Size During a Lunar Eclipse: By photographing the Earth’s shadow on the Moon during a lunar eclipse, and comparing it to the Moon’s known angular diameter, we can estimate how far away the Moon must be.
Timing the Duration of a Lunar Eclipse: Since we know Earth’s size and orbital speed, measuring how long the Moon takes to pass through Earth’s shadow provides a way to estimate the Moon’s distance.
Frequency of Solar and Lunar Eclipses: The regularity and geometric constraints of eclipses help astronomers refine Moon-Earth-Sun distances. While not easily usable in a single school experiment, it’s an excellent concept to explore with AI.
Parallax from Two Earth Locations: By observing the Moon from two distant points on Earth at the same time, and measuring the apparent shift in position (parallax), we can triangulate the Moon’s distance — using basic trigonometry.
Geometry of a Solar Eclipse: During a total solar eclipse, the Moon and Sun appear the same size in the sky. This cosmic coincidence lets us calculate the Moon’s distance, using the known size and distance of the Sun.
Laser Ranging: By bouncing lasers off retroreflectors placed on the Moon by Apollo astronauts, scientists can measure the Moon’s distance to within centimeters — the most precise method available.

Measuring the Distance to the Moon via Angular Size During a Lunar Eclipse (ASTR-10)

Sensors Used: Camera
What’s Measured: Angular diameter of Earth’s shadow during a lunar eclipse

Description
A lunar eclipse offers more than just a celestial spectacle—it provides a rare opportunity to measure the distance to the Moon using simple observations and basic geometry. During the eclipse, Earth casts a shadow that appears as a large, dark circle moving across the Moon’s face. Careful analysis of eclipse photographs reveals that Earth’s shadow is about three times wider than the Moon itself.

Given that the Moon’s angular diameter is approximately 0.5°, we can estimate the angular diameter of Earth’s shadow to be about 1.5°. Using this angular measurement and the known radius of Earth (roughly 6,000 kilometers), we apply simple trigonometry to estimate the Earth-Moon distance. The result comes out to around 460,000 kilometers—a surprisingly accurate approximation when compared to the accepted average of 384,400 kilometers.

This method, first used by ancient astronomers, shows just how much we can learn from a few angles and a well-timed photo. It’s a beautiful example of classical astronomy reborn through modern tools like smartphone photography and a bit of clever geometry.

Lunar Velocity from Eclipse Duration (ASTR-11)

Sensors Used: Clock or stopwatch (smartphone), camera (optional)
What’s Measured: Duration of lunar eclipse, used to estimate lunar velocity and distance

Description
Long before spacecraft, radar, or lasers, ancient astronomers like Aristarchus deduced the Moon’s distance using nothing more than the timing of a lunar eclipse. This experiment revisits that method using a modern timekeeper—your smartphone. A total lunar eclipse, from the Moon’s entry into Earth’s shadow to its exit, typically lasts about three hours. Assuming that the Sun is distant enough to cast a nearly cylindrical shadow, the Moon must traverse a path roughly twice Earth’s radius in length—about 12,000 km.

By dividing this shadow width by the eclipse duration, we get the Moon’s orbital speed. Then, knowing that the Moon completes one full orbit around Earth in approximately 27.3 days (a sidereal month), we can calculate the circumference of its orbit. From that, we derive the Moon’s average distance by working backward through basic circular motion formulas.

Amazingly, this method yields a result that closely matches the true average Earth-Moon distance of 384,400 kilometers. It’s a powerful reminder of how centuries-old reasoning, paired with simple timing tools, can still illuminate the architecture of the cosmos.

Parallax — Measuring the Distance to the Moon with Two Photos (ASTR-12)

Sensors Used: Camera, GPS (for location), Sextant app (optional)
What’s Measured: Apparent angular displacement of the Moon due to parallax

Description
The Moon doesn’t appear in quite the same place when viewed from different spots on Earth — it shifts slightly against the background stars. This subtle effect, known as parallax, allows us to measure the Moon’s distance by comparing observations from two different geographic locations. There are two main ways to capture this shift. The first uses a sextant or clinometer app to measure the Moon’s elevation above the horizon from each location at the same time. The second — more visual and accessible — involves taking photos of the Moon from both locations and comparing its position relative to nearby stars.

A clever variation of this method can be done while traveling. Before you leave, take a photo of the Moon from your starting point — say, Germany — including a nearby background star for reference. Once you arrive at your destination — perhaps Spain — take another photo of the Moon under similar sky conditions. Do the same on your return trip. The Moon’s apparent position will have shifted due to two influences: Earth’s orbital motion around the Sun and your change in terrestrial location. By comparing image pairs from the outbound and return trips, you can subtract the orbital component and isolate the parallax due to your movement on Earth’s surface.

Using pixel measurements, angular overlays, or trigonometric formulas, this displacement reveals the Moon’s distance — a beautiful example of how motion, light, and simple geometry allow us to reach across space from our own backyard.

References:
[1] Measuring the Moon’s Diameter, http://practicalphysics.org/diameter-moon.html
[2] Determining the Moon’s Distance, http://practicalphysics.org/moon-distance-earth.html

Using Eclipse Frequency to Estimate Lunar Size (ASTR-13)

Sensors Used: None (historical analysis)
What’s Measured: Ratio of eclipse types to infer relative radii of Earth and Moon

Description
How might someone estimate the size of the Moon without using a telescope, spacecraft, or even a smartphone? The answer lies in a surprisingly elegant approach rooted in historical astronomy: analyzing the frequency of eclipses. By simply recording how often lunar and solar eclipses occur from a fixed location over many years, it’s possible to deduce something remarkable about the size of the Moon relative to Earth.

Imagine a careful observer living in Nuremberg between 1901 and 2000. According to NASA’s eclipse catalog, that person would have witnessed 38 solar eclipses and 163 lunar eclipses — a ratio of approximately 0.23 solar to lunar. This ratio isn’t arbitrary. Because a larger object casts a bigger shadow and is visible from a wider area, it is statistically more likely to produce an observable eclipse. The ratio of eclipse frequency is therefore tied to the ratio of the cross-sectional areas of Earth and Moon, which in turn is related to the square of their radii.

Taking this 0.23 ratio and assuming Earth’s radius is about 6,400 kilometers, one can estimate the Moon’s radius as roughly 0.23 × 6,400, or about 1,472 kilometers. That’s remarkably close to the Moon’s actual radius of approximately 1,737 kilometers. Though far from perfect, this simple and clever method provides powerful evidence of how consistent celestial patterns, paired with basic reasoning, can reveal the dimensions of the cosmos — long before modern tools existed.

References:
[1] NASA Eclipse Catalog, https://eclipse.gsfc.nasa.gov/eclipse.html

Calculating Lunar Size via Angular Diameter (ASTR-14)

Sensors Used: Camera, Sextant app, or clinometer
What’s Measured: Angular diameter of the Moon, used with known distance to estimate size

Description
The size of the Moon can be estimated in more than one way—each rooted in simple geometry and careful observation. Earlier, we explored how eclipse frequency can offer a statistical estimate of the Moon’s size, yielding a radius about 23% that of Earth. If Earth’s radius is approximately 6,400 kilometers, this method suggests a lunar radius of around 1,472 kilometers—a decent approximation, though a bit low.

A more precise technique involves measuring the Moon’s angular diameter, which can be done with a sextant app, clinometer, or calibrated photograph. The Moon’s angular size as seen from Earth is roughly 0.5°, a figure surprisingly consistent due to the regularity of its orbit. When paired with the known average Earth-Moon distance of about 400,000 kilometers, this angle allows us to calculate the Moon’s actual diameter using basic trigonometry. The result comes out to around 3,474 kilometers—remarkably close to the true value confirmed by modern astronomy.

This experiment bridges two methods—one observational and historical, the other geometric and real-time—to illuminate the scale of our nearest celestial neighbor using tools no more advanced than a smartphone and a clear night sky.

Mass of the Moon from Volume and Density (ASTR-15)

Sensors Used: None directly; based on calculated diameter
What’s Measured: Estimated lunar mass using known diameter and assumed density

Description
Once the size of the Moon has been estimated—whether through eclipse frequency, angular measurements, or photographs—it’s possible to go a step further and calculate its mass. Assuming a roughly spherical shape, we can compute the volume of the Moon using the standard formula for the volume of a sphere, V=43πr3V = \frac{4}{3}\pi r^3V=34πr3, where rrr is the Moon’s radius.

With an estimated radius of about 1,737 kilometers, we convert this to meters and calculate the total volume. To find the mass, we multiply that volume by an assumed average density—typically around 5,000 kg/m³ for rocky bodies like the Moon. This approach yields a mass estimate that comes very close to the accepted value of approximately 7.35×10227.35 \times 10^{22}7.35×1022 kilograms.

What’s powerful about this experiment is not just the accuracy it achieves with basic math and assumptions, but the insight it offers into how scientists originally pieced together the properties of distant celestial bodies long before the space age.

The Sun: Distance, Size, and Mass

The ancient Greek astronomer Aristarchus was the first to attempt a measurement of the Earth-Sun distance using pure geometry. His method relied on this key observation:When the Moon is exactly half illuminated, it forms a right triangle with Earth and the Sun. If you know the distance to the Moon, and can measure the angle between the Moon and the Sun at that moment, you can calculate the distance to the Sun using trigonometry.

Aristarchus and the Quarter Moon Triangle (ASTR-16)

Sensors Used: Camera or observational timing (optional), geometry-based
What’s Measured: Relative distance to the Sun from Moon-Sun-Earth angle during quarter phase

Description
Long before telescopes or space probes, the ancient Greek astronomer Aristarchus of Samos attempted one of the most ambitious measurements in astronomy: estimating the distance to the Sun using only the geometry of the Moon’s phases. His idea was simple but ingenious. When the Moon appears exactly half-lit—known as the first or last quarter—the Earth, Moon, and Sun form a right triangle. If one could measure the angle between the Moon and the Sun in the sky at that moment, then basic trigonometry would reveal how far away the Sun must be relative to the Moon.

Aristarchus estimated that this angle was about 87°, which led him to conclude that the Sun was roughly 19 times farther away than the Moon. This was a substantial underestimate: the actual angle is closer to 89.83°, which means the Sun is about 400 times farther than the Moon. But here’s the fascinating twist—detecting the exact geometric half-phase of the Moon is extraordinarily difficult. The visual transition occurs only about 12 minutes off from the ideal angle, making precise timing nearly impossible with the naked eye.

Still, we can use a simplified version of Aristarchus’s method to put a lower bound on the Sun’s distance. For example, if on the sixth day after new moon the Moon is not yet at first quarter, we know the Sun must be quite far away. By estimating that the angular gap from a right triangle is at least 13°, we can say the Sun must be at least rmoon/sin⁡(13∘)r_{moon} / \sin(13^\circ)rmoon/sin(13∘), or about 1.2 million kilometers away. While this is still far from the actual 149.6 million kilometers, it dramatically improves upon earlier estimates—and more importantly, it shows how geometry and observation alone can give deep insights into the structure of the cosmos.

DIY: Photographing the Quarter Moon (ASTR-17)

Sensors Used: Camera, astronomy app
What’s Measured: Visual accuracy of lunar phase determination; inferred lower bound on Sun’s distance

Description
The quarter moon—the moment when exactly half the Moon’s surface appears illuminated—is a subtle but powerful astronomical marker. In this hands-on project, students can try to capture this fleeting moment themselves, and use it to explore both the geometry of the Earth-Moon-Sun system and the challenges of observational astronomy.

Begin by photographing the Moon on the days surrounding the first and third quarters—typically one to two days before and after each event. Using image processing tools, such as brightness filters or edge detection, students can compare these photos and attempt to identify when the Moon appears truly half-lit. Overlaying images from consecutive nights can reveal subtle curvature in the shadow line, helping to pinpoint the most geometrically accurate moment of quarter phase.

To validate their observations, students can consult astronomy apps like Stellarium to find the predicted times of the quarter moon. Even small discrepancies—on the order of hours or minutes—highlight the precision required for ancient methods like those of Aristarchus. If the Moon appears to be half-lit slightly earlier or later than expected, that difference in angle can be translated into a lower bound for the Sun’s distance using basic trigonometry.

This experiment not only builds observational and analytical skills, but also provides a direct and humbling connection to one of the earliest scientific attempts to measure the scale of the solar system.

References:
[1] Distance to the Sun, http://practicalphysics.org/distance-sun.html

Size and Mass of the Sun from Angular Diameter and Distance (ASTR-18)

Sensors Used: Camera (for angular diameter, optional), astronomical constants
What’s Measured: Solar diameter, solar mass, and average density (via indirect calculation)

Description
Though the Sun is unimaginably far away, its size and mass can be estimated using just a few well-known values and the basic laws of physics. Begin with the Sun’s angular diameter, which is approximately 0.53° as seen from Earth. Knowing the average distance to the Sun—about 1.496 × 10¹¹ meters—we can apply basic trigonometry to estimate the Sun’s true diameter. Using the formula for arc length, we find the Sun’s diameter to be approximately 1.39 million kilometers. This aligns closely with the accepted value, which is a powerful demonstration of geometry applied to the sky.

Next, we can estimate the mass of the Sun using Newton’s law of universal gravitation in combination with the centripetal force required to keep Earth in its nearly circular orbit. Setting the gravitational attraction equal to the centripetal force and solving for the Sun’s mass yields a value of roughly 1.99 × 10³⁰ kilograms—again matching the accepted figure with impressive accuracy.

To take the analysis one step further, we can compute the Sun’s average density by dividing this mass by the volume of a sphere with the previously calculated radius. The result is around 1,400 kg/m³, only slightly denser than water. This may seem surprisingly low given the Sun’s enormous mass, but it reflects the fact that the Sun is composed primarily of hydrogen and helium—light elements under extreme pressure.

This experiment elegantly combines geometry, astronomy, and physics, and reveals how a few measurements and simple equations can unlock profound insights into our nearest star.

Energy of the Sun — From Lux to Fusion (ASTR-19)

Sensors Used: Light sensor (lux meter or smartphone)
What’s Measured: Solar illuminance; calculated solar luminosity and power output

Description
We can estimate the Sun’s immense power output using something as simple as the light falling on Earth. By measuring the Sun’s illuminance in lux using a smartphone or handheld light meter, and combining that with the known distance to the Sun, we can work backward to calculate the Sun’s total luminosity.

First, it’s important to understand the difference between lumens and lux. Lumens measure the total amount of visible light emitted by a source, while lux measures how much of that light falls on a given surface area (lumens per square meter). Because of the inverse square law, lux values decrease rapidly with distance. For instance, a 60-watt incandescent bulb may emit around 850 lumens, but just one meter away, it produces about 68 lux.

Now turn to the Sun. Under clear skies at noon, the Sun provides an illuminance of approximately 100,000 lux at the surface of Earth. Knowing the distance from Earth to the Sun is about 1.5 × 10¹¹ meters, and applying the inverse square law, we can calculate the total luminous flux reaching Earth’s orbit. This value is roughly 3.75 × 10²⁸ lumens.

To convert this to power, we use the luminous efficacy of sunlight, which averages around 93 lumens per watt. Dividing the total lumens by this value gives a staggering total power output: approximately 4 × 10²⁶ watts. This is the total energy the Sun emits into space every second, in all directions.

Such an enormous energy output can’t be sustained by chemical burning or combustion of any known fuel. If the Sun were powered by wood, oil, or any chemical process, it would burn out in just a few thousand years. The only mechanism capable of producing this energy over billions of years is nuclear fusion in the Sun’s core. Through this simple chain of measurements and reasoning, students connect a reading from their phone to the heart of stellar physics — and grasp why fusion is not just a theory, but a necessity.

References:
[1] What’s in a Lux?, https://www.allaboutcircuits.com/technical-articles/understanding-illuminance-whats-in-a-lux/
[2] Wikipedia: Sun, https://en.wikipedia.org/wiki/Sun
[3] The Sun’s luminosity (Practical Physics), practicalphysics.org/sun’s-luminosity.html

Stars, Brightness, and Distances

Estimating Distance to Sirius via Apparent Magnitude (ASTR-20)

Sensors Used: None required (calculation-based)
What’s Measured: Apparent magnitude difference; estimated stellar distance using brightness comparison

Description
Even without a telescope, we can estimate how far Sirius is from Earth using its brightness in the night sky compared to that of our Sun. This method relies on two ideas: the apparent magnitude scale, which compares how bright stars appear from Earth, and the inverse square law, which relates brightness to distance.

Sirius is the brightest star visible in the night sky and has an apparent magnitude of about −1.47. The Sun, overwhelmingly brighter because of its proximity, has an apparent magnitude of −26.74. The difference in magnitude between the two is Δm ≈ 25.3. Using the magnitude-brightness formula, we can calculate how much dimmer Sirius appears than the Sun: each magnitude step represents a brightness factor of about 2.512, so Sirius appears roughly 13 billion times fainter.

Now, if we assume Sirius has the same intrinsic luminosity as the Sun, then the inverse square law tells us that for an object to appear 13 billion times dimmer, it must be √13 billion times farther away. Since the Sun is 1 astronomical unit (AU) from Earth, this places Sirius at about 3.6 million AU, or around 1.8 light-years. This would be Sirius’s distance if it were a Sun-like star.

However, Sirius is not identical to the Sun — it’s approximately 25 times more luminous. Taking this into account and adjusting the brightness ratio accordingly, the actual distance estimate comes much closer to the accepted value of about 8.6 light-years. This thought experiment shows how apparent magnitude, combined with basic physics and a few assumptions, lets us reach across space to estimate stellar distances — even with the naked eye and a little math.

Apparent Brightness Using Smartphone Astrophotography (ASTR-21)

Sensors Used: Camera (with manual exposure or RAW mode), optional light sensor
What’s Measured: Relative apparent brightness of celestial bodies via pixel intensity

Description
Can you measure how bright a star is using just your smartphone? Surprisingly, the answer is yes — at least in a relative sense. By capturing celestial objects like planets, stars, and the Moon using carefully controlled exposure settings, students can analyze the brightness of each and explore the apparent magnitude scale firsthand.

The key is consistency: take photographs of two different celestial bodies (for instance, Venus and Sirius) using identical exposure settings. Venus shines at a magnitude of around −4.7, while Sirius, the brightest star, is about −1.5. This magnitude difference of roughly 3.2 corresponds to a brightness ratio of about 25. If you take RAW images or images with manual settings (especially fixed ISO and shutter speed), you can compare the summed pixel values in the core or surrounding region of each object. Alternatively, adjust the exposure until both objects appear equally bright on camera, and then infer their relative brightness from the difference in exposure settings.

You can also use the Moon as a reference point, especially during its crescent or half phases (a full Moon is often overexposed). Measure its brightness in photos, or use a smartphone light sensor to directly measure the lux from moonlight. From there, it’s even possible to estimate its luminous flux and use that to scale brightness comparisons to other objects like planets.

To get meaningful results, students must avoid overexposing the images — blown-out stars can’t be analyzed properly — and aim to capture their subjects at similar altitudes in the sky to reduce atmospheric extinction effects. Using RAW image capture or apps that minimize post-processing helps reduce distortion and ensures a cleaner dataset. While it’s not perfect, this experiment allows for direct, hands-on engagement with the concept of apparent magnitude, bringing astrophotometry into reach for any student with a modern phone and a clear night sky.

References:
[1] Smartphone Astrophotography: How I Capture the Moon and Planets with My Phone, https://petapixel.com/2015/02/20/smartphone-astrophotography-how-i-capture-the-moon-and-planets-with-my-phone/

Estimating Planetary Size from Brightness: Venus vs. Mars (ASTR-22)

Sensors Used: Camera (optional, for brightness comparison), data lookup tools
What’s Measured: Apparent brightness of planets, used to infer relative planetary radii

Description
How large is Venus compared to Mars? One clever way to approach this question is by analyzing how bright each planet appears in the sky. The brightness we observe from Earth depends on a few key factors: how reflective the planet is (its albedo), how far it is from both the Sun (which illuminates it) and Earth (from which we view it), and its actual size — specifically, the square of its radius. With some known astronomical parameters and a bit of algebra, we can use apparent brightness as a tool to estimate relative planetary sizes.

The brightness of a planet seen from Earth scales proportionally with its albedo and the square of its radius, and inversely with the square of its distances to both the Sun and Earth. Mathematically, this is: Brightness ∝ (Albedo × Radius²) / (Distance to Sun² × Distance to Earth²)

By comparing Venus and Mars, and using known values for each — apparent magnitudes, albedos, and distances — we can solve for the ratio of their radii. Using values for when Venus is near its full phase (which is critical, since Venus shows phases like the Moon), we find:

Apparent magnitudes: −4.4 for Venus and −2.0 for Mars
Brightness ratio from these magnitudes: approximately 2.29
Distance from Earth: 1.72 AU for Venus, 0.37 AU for Mars
Distance from the Sun: 0.72 AU for Venus, 1.52 AU for Mars
Albedos: 0.75 for Venus, 0.15 for Mars

Plugging these into the formula and solving, we find that the estimated radius ratio is about 1.49 — meaning Venus is approximately 1.5 times larger in radius than Mars. The actual ratio is about 1.78, which means this method gets us within roughly 20% of the correct answer. That’s remarkably accurate for such a simple approach.

What’s most striking is the role of albedo. Without factoring in the reflectivity of each planet, the calculated radius ratio jumps to about 3.3 — a dramatic overestimate that shows just how crucial surface reflectivity is in interpreting astronomical brightness. This experiment doesn’t require complex instruments; just a few known values, some simple math, and the observational data available to anyone with a smartphone and a clear view of the night sky.

Practical Stargazing: Imaging, Mapping, and Measuring the Sky (ASTR-23)

Sensors Used: Camera, gyroscope, compass, orientation sensors
What’s Measured: Lunar phases, star motion, celestial orientation

Description
The night sky is not only beautiful — it’s full of measurable phenomena, and your smartphone can turn stargazing into hands-on science. This experiment combines several practical astronomy techniques that deepen observation skills while illustrating key physical principles.

Start by attaching your smartphone to a pair of binoculars or a telescope using a simple mount or even rubber bands. This setup lets you capture surprisingly detailed images of the Moon’s surface, as well as brighter planets like Venus, Jupiter, and Saturn. Over a month, photograph the Moon each night at the same zoom level and framing to create a time-lapse of its changing phases. This personal lunar calendar helps students visualize the Moon’s cycle, understand orbital motion, and observe how its position and orientation shift over time.

Next, use a manual camera app to take long-exposure photos of the night sky from a stable surface or tripod. As the exposure builds, stars trace graceful arcs across the image — visible evidence of Earth’s rotation. The curvature and length of these trails can even be used to estimate the planet’s rotational speed, revealing how celestial photography can double as a measurement tool.

To map the sky in real time, open a planetarium app like Stellarium or Sky Map. Turn off location services and cover the camera, forcing the app to rely solely on the phone’s compass and gyroscope. Now challenge yourself or your students to identify planets and stars manually, using just orientation and a printed sky map. This approach turns passive screen-watching into active sky-reading, encouraging students to internalize the layout of constellations and celestial markers.

Together, these methods blend technology, observation, and imagination — transforming casual stargazing into a powerful, multi-sensory exploration of astronomy.

References & Tools:
[1] Does the Earth Move? Photographing the Night Sky, http://practicalphysics.org/does-earth-move-photographing-night-sky.html
[2] Smartphone Astrophotography, https://petapixel.com/2015/02/20/smartphone-astrophotography-how-i-capture-the-moon-and-planets-with-my-phone/
[3] Sky Map Generator, https://in-the-sky.org/skymap2.php
[4] Astronomie.de Sky Viewer, https://www.astronomie.de/der-himmel-aktuell/ansicht-des-sternenhimmels
[5] Open Camera App (Manual Control), https://play.google.com/store/apps/details?id=net.sourceforge.opencamera

[6] PIPP - Planetary Imaging PreProcessor, https://sites.google.com/site/astropipp/
[7] http://www.autostakkert.com/
[8] https://stargazerslounge.com/topic/184821-beginners-guide-to-stacking-planetary-images-with-autostakkert2/

Capstone / Research Extension

Building an All-Sky Camera with Raspberry Pi (ASTR-24)

Sensors Used: Digital camera module (via Raspberry Pi), optional light sensor, gyroscope (in phone for alignment)
What’s Measured: Night sky brightness, meteor trails, cloud cover, celestial event timing

Description
This capstone project invites students to build their own all-sky camera — a wide-angle sky-monitoring station that opens the door to long-term celestial observation. Using a Raspberry Pi connected to a high-quality camera module and equipped with a fisheye or wide-angle lens, you can capture an entire hemisphere of the night sky in one frame. Once assembled and weatherproofed, the system can be mounted outdoors to continuously photograph the sky, either as a live feed or with time-lapse intervals.

This DIY observatory provides a low-cost, scalable way to monitor meteor showers, star trails, auroras, passing satellites, and other transient sky events. Students can use their smartphones to align the camera (leveraging compass and gyroscope apps), and later analyze the images or video sequences on a computer. Combined with open-source software, the system can log timestamps and even correlate events with atmospheric data or astronomical predictions.

Beyond its educational value in astronomy and physics, the project offers practical experience in engineering, programming, and digital image analysis. It’s an ideal long-term initiative that brings together curiosity, code, and cosmic wonder — transforming a simple Raspberry Pi into a personal window on the universe.

References:
[1] “Roll Your Own All-Sky, Raspberry Pi Camera”, https://spectrum.ieee.org/all-sky-camera

Radio Ears: Build Your Own Radio Telescope for the Invisible Sky (ASTR-25)

Sensors Used: Satellite dish with LNB (low-noise block), radio frequency amplifier, SDR receiver (Software-Defined Radio)
What’s Measured: 21-cm hydrogen line emissions from neutral hydrogen in the Milky Way

Description
What if, instead of watching the night sky, you could listen to it? This capstone experiment challenges students to construct a functioning radio telescope that can detect the invisible whisper of hydrogen gas permeating the galaxy. Using a satellite dish, a low-noise block downconverter (LNB), and a software-defined radio (SDR) dongle or similar amplifier setup, the telescope tunes in to the 21-centimeter emission line — a signal produced by neutral hydrogen atoms, the most common substance in the universe.

These radio waves are not blocked by dust, clouds, or daylight, making them a powerful tool for observing deep into our galaxy, even during the day or in poor weather. Once configured, the device can track the Milky Way’s rotation by recording changes in signal intensity and frequency as different regions of the galaxy move in and out of view.

The experiment is not just an engineering challenge — it introduces students to radio astronomy, Doppler shift, spectral analysis, and the large-scale structure of the universe. With guidance from open-source software and online communities, students can process the incoming data, map hydrogen density in the galactic plane, and even identify shifts in frequency due to relative motion. It’s a deeply immersive project that transforms a backyard into a window onto the cosmos.

References:
[1] Track the Milky Way with a DIY Radio Telescope (IEEE Spectrum), https://spectrum.ieee.org/geek-life/hands-on/track-the-movement-of-the-milky-way-with-this-diy-radio-telescope

Historical Inquiry

Tracking Planetary Motion: Repeating Kepler with Your Smartphone (ASTR-26)

Sensors Used: Smartphone camera, orientation sensors (optional)
What’s Measured: Apparent motion of planets against the background stars over time

Description
In the early 17th century, Johannes Kepler used the meticulous naked-eye observations of Tycho Brahe to uncover the elliptical nature of planetary orbits — one of the great turning points in astronomy. With nothing more than a smartphone, a steady hand, and a few weeks of clear skies, students today can follow in his footsteps. By capturing nightly images of a bright planet like Mars, Jupiter, or Venus and comparing its changing position relative to the fixed stars, you can trace the same celestial patterns that led Kepler to his revolutionary laws of motion.

The process is surprisingly accessible. Using a tripod or smartphone mount, students take pictures at roughly the same time each night from the same location. The star background remains mostly unchanged, while the planet slowly moves — sometimes even appearing to loop backward in its retrograde path. With basic image alignment or stacking tools (available in apps or free software like GIMP or online astro-imaging tools), the motion becomes clear, even dramatic. Over a span of two to three weeks, the planet’s shifting position begins to outline the curve of its orbit.

While full orbital fitting requires more advanced data and tools, simply watching the arc unfold across your own sky connects students directly to the history of science. It’s a living experiment that echoes one of the most profound discoveries in astronomy: that the heavens move not in perfect circles, but in elegant ellipses.

Optional Extensions: Compare your photos to simulated sky charts, look for retrograde loops, or use overlays to estimate orbital eccentricity.

References:
[1] Terence Tao on how we measure the cosmos, https://www.youtube.com/watch?v=YdOXS_9_P4U

Hipparchus, Parallax, and the Distance to the Moon (ASTR-27)

Sensors Used: None required (historical reasoning and geometry)
What’s Measured: Lunar distance inferred from solar eclipse geometry

Description
Long before telescopes or smartphones, the Greek astronomer Hipparchus made one of the earliest known attempts to measure the distance to the Moon — using little more than logic, geography, and a lucky eclipse. In 129 BCE, a total solar eclipse was seen from the Hellespont, near modern-day Dardanelles, while observers in Alexandria, hundreds of kilometers south, saw only a partial eclipse — with about four-fifths of the Sun obscured.

Hipparchus understood that during a solar eclipse, the Moon must lie directly between Earth and the Sun. Because the Sun and Moon appear nearly the same size in the sky, even a small shift in viewing position on Earth changes the degree to which the Sun is covered — a subtle but measurable parallax effect. By comparing the observed difference in eclipse coverage between the two cities, and knowing the distance between them, Hipparchus calculated the Moon’s angular shift. Using this parallax angle, he could then apply basic trigonometry to estimate the Moon’s distance.

Though his methods lacked the precision of modern instruments, his reasoning was sound — a brilliant demonstration of how geometry and astronomical observation could reveal the scale of the cosmos. For students today, retracing Hipparchus’s logic offers not just a lesson in ancient science, but a deeper appreciation for the timeless power of human curiosity and mathematical thinking.