PRINTABLE ACTIVITIES / BLACK-HOLE HORIZONS
More Mass.
A Larger Horizon.
Calculate Schwarzschild radii and diameters, compare adopted masses and distinguish a horizon from an observed bright ring.
Download printable worksheet & educator notes · PDF ↓Open the Black-hole horizons interactive tool →
Print learner pages 1–2; answers and educator notes are on pages 3–4. The complete activity is also readable below. Optional data and code downloads are inside the offline-verification section.
BLACK-HOLE HORIZONS / CHECK THIS RESULT
Check a horizon-size result.
Use the Black-hole horizons interactive tool. Enter Black hole A = 10 and Black hole B = 20 in nominal solar mass units. Predict the change before reading the result: under this Schwarzschild model, doubling mass doubles horizon radius and diameter.
Calculated comparisons, not measured horizons: cases 1 and 2 are hypothetical. Sagittarius A* and M87* use rounded mass inputs from the literature. We apply the same nonrotating, uncharged model to compare scales; we do not fit their actual spin, observe their horizons or derive masses. Extra digits describe the arithmetic.
| Step | Reference result | What it means |
|---|---|---|
| Nominal solar mass parameter μ | 1.3271244 × 1020 m³/s² | Exact IAU conversion convention; μ already includes G. |
| Schwarzschild areal radius | rs = 2μm/c² ≈ 29.532501 km | Use c = 299,792,458 m/s; convert meters to kilometers. |
| Horizon diameter convention | D = 2rs ≈ 59.065002 km | Diameter is twice radius, not the same quantity. |
| Equivalent diameter in miles | 36.70129 miles | Divide kilometers by exactly 1.609344. |
| Double the adopted mass | m = 20 → D ≈ 118.130003 km | DB/DA = mB/mA = 2. |
Let m be the numerical mass in nominal solar units. The formula rs = 2μm/c² gives meters when μ and c use SI units. It gives about 2.953250076 km of radius, or 5.906500152 km of diameter, per nominal solar unit. Do not insert another factor of G: μ is already a gravitational mass parameter.
Repeat the calculation in a spreadsheet
For the 10-unit case, calculate radius in kilometers with =2*1.3271244E20*10/(299792458^2)/1000. Multiply that result by 2 for diameter; divide the diameter by 1.609344 for international miles. Replace 10 with 20 to check the doubling relation.
To recover an assumed mass from a model diameter in kilometers, divide by the full diameter coefficient: =118.130/(4*1.3271244E20/(299792458^2)/1000) gives approximately 20 nominal solar units. This is a model inverse question, not a measured mass of an unknown object.
Why is the bright ring not the horizon diameter?
The event horizon is a causal boundary, not a glowing solid surface. Telescope images contain emission from surrounding plasma altered by light bending, with a central shadow associated with photon capture. The shadow and bright ring are not identical to the horizon boundary. Their interpretation requires a radiative and gravitational model.
This activity calculates a Schwarzschild horizon size only. It does not turn the guide illustration into a calibrated diagram or the lab’s marker dots into a telescope image. Read the EHT M87* paper and EHT Sagittarius A* explanation for the separate observations.
Read the logarithmic markers and the model boundary
The existing plot uses marker position on a logarithmic axis. Equal factors in diameter give equal horizontal steps; marker dot sizes are not drawings of the horizon diameters. The worksheet’s two circles instead compare cases 1 and 2 on a common linear size scale.
The Schwarzschild radius is an areal/circumferential coordinate: the horizon has area 4πrs². D = 2rs is the size convention used here, not a ruler distance through the black-hole interior. Real black holes may spin, changing the geometry. These calculations do not estimate tidal survival, an accretion disk size or a measured shadow.
Verify the generated cases offline
Extract the full kit and run python3 verify_analysis.py --data-dir . (Windows: py -3 verify_analysis.py --data-dir .). Python 3.9+ and its standard library are sufficient. Compare emitted JSON with worked-result.json.
The independent implementation checks four pinned conversion rows, four generated cases, radius versus diameter, kilometers/miles/AU, mass-size ratios and inverse mass. It changes no inputs and makes no network request. Matching the original model is an arithmetic check, not observational validation.
FOR EDUCATORS / INDEPENDENT LEARNERS
A radius is half
a diameter.
Suggested use: ages 14+ through introductory college; 30–40 minutes with a calculator. Learners need multiplication, division and ratios. A supplied numerical coefficient supports readers unfamiliar with scientific notation. Use the live tool or paper cases; spreadsheet and Python checks are optional. No account, submitted answers or student information is required.
The four-page packet contains two learner pages, two educator pages and an original labeled size-comparison diagram. The kit adds reference inputs, generated values, worked results, an independent Python checker and both unchanged model dependencies. The tasks and answers below are a text-first alternative; diagrams use labels as well as color.
Learning goals
- Distinguish Schwarzschild radius from the diameter convention D = 2rs.
- Calculate model sizes with consistent units and a nominal mass parameter.
- Use proportional reasoning to compare masses and sizes.
- Distinguish a modeled horizon from observed emission and shadow structure.
Investigation record
- Predict: decide whether doubling mass doubles, quadruples or leaves diameter unchanged, then explain your choice.
- Calculate: find rs and D for the four adopted masses below. Use rs ≈ 2.953250076 × m km, then D = 2rs.
- Compare: test A = 10 and B = 20 in the tool, then B = 4,000,000. Check that diameter ratios match adopted mass ratios. Explain the logarithmic axis and fixed marker dot sizes.
- Convert and invert: convert the Sagittarius A* comparison’s diameter from km to AU. Recover the hypothetical mass for D ≈ 118.130 km. Explain why neither operation supplies an observation.
- Explain: state a claim with two numerical clues, one assumption and one limitation. Distinguish radius, diameter, horizon boundary and observed bright ring. Optional: use A = 4πrs² to find the area factor when mass doubles.
| Case | Adopted mass m | Input status |
|---|---|---|
| 1 / 10-unit example | 10 | Hypothetical stellar-mass case. |
| 2 / doubled example | 20 | Hypothetical comparison. |
| 3 / Sagittarius A* | 4,000,000 | Rounded literature comparison; no spin fit. |
| 4 / M87* | 6,500,000,000 | Rounded literature comparison; no spin fit. |
Educator notes and suggested answers for black-hole horizons
Timing: prediction 5 min; cases 10; ratios/units 8; inverse 5; discussion 7; optional area 5. For 30 minutes use cases 1–3 and provide the coefficient. Accept sensible rounding rather than implying precise measured masses.
Answers, radius then diameter in km: case 1: 29.532501 and 59.065002; case 2: 59.065002 and 118.130003; case 3: 11,813,000.304 and 23,626,000.609; case 4: 19,196,125,494.652 and 38,392,250,989.303. Case 1’s diameter is about 36.701290 miles. The nominal parameter μ already contains G.
Ratios and units: case 2 / case 1 = 2; Sagittarius A* / case 1 = 400,000; M87* / Sagittarius A* = 1,625. These apply to radius, diameter and adopted mass under the stated model. Case 3’s diameter is about 0.157930059 AU, or 14,680,516.166 miles; changing units preserves size.
Inverse and extension: the rounded 118.130-km model diameter gives m ≈ 19.9999995, hence about 20 nominal solar units. This is an assumed model size, not a mass measurement. Doubling mass doubles radius and diameter but quadruples area. Do not infer an ordinary interior volume or a survival time.
Assessment: require the radius/diameter factor of two, consistent units, linear scaling and recognition that an observed ring involves emission and light bending. A logarithmic marker dot is not a linear size drawing. Exact conversion conventions do not remove uncertainty from rounded astronomical inputs. No formal curriculum alignment or independent scientific review is claimed.
Keep the geometry
and source trail.
Suggested citation: The Last Transmission. “Scale Lab: More Mass. A Larger Horizon.” DarkScienceDocs.com. Teaching method 1.0, resource release 269; updated . https://darksciencedocs.com/exploration-guides/black-hole-horizons/#horizon-teaching. Add your access date.
Formula and conversions: Andrew Hamilton / JILA, Schwarzschild geometry; IAU 2015 Resolution B3, nominal conversion constants; BIPM, exact SI c; Observatoire de Paris, IAU AU definition; NIST, international mile. Nominal constants are fixed conversion factors, not exact measured solar properties.
Named comparisons and image interpretation: EHT, Sagittarius A* image announcement and EHT Collaboration, First M87 Results / Paper I. We adopt rounded 4-million and 6.5-billion nominal-solar comparisons. The arithmetic here is separate from those observations and their inferred masses.
Sources checked October 25, 2026. Original activity, comparison diagram, generated cases and verifier by The Last Transmission. Not an EHT, IAU, BIPM, NIST or JILA product or endorsement. No independent scientific review is claimed. Continue with the Black Holes Field Guide. Corrections: select Scientific correction and include input, result and evidence.