Why can textbook pictures not show sizes and distances at the same time?

The Sun is 1.39 million km across but Neptune is 4.5 billion km away — a ratio of about 3,200. If the Sun is drawn 110 px wide, Neptune must sit about 355,000 px away, and Earth would be a single pixel. Any picture that fits on a page has to exaggerate the sizes, compress the distances, or both.

How big is the Sun compared with Earth?

The Sun's diameter (1,391,400 km) is about 109 Earth diameters (12,756 km), and roughly 1.3 million Earths would fit inside its volume. Jupiter is about 11.2 Earth diameters across.

What does "light travel time" mean and how is it computed?

It is the time light needs to cross the orbit radius: t = a / c. For Earth, 149.6 million km ÷ 299,792 km/s ≈ 499 s ≈ 8.3 min; for Neptune (30.07 AU) it is about 4.2 h. When you look at a planet you see it as it was that long ago.

How do the football numbers come out?

Everything is multiplied by the same factor k = 0.22 m ÷ 1,391,400 km ≈ 1.6 × 10⁻⁷ m per km. Earth's diameter becomes 12,756 km × k ≈ 2.0 mm (a peppercorn) at 149.6 million km × k ≈ 23.7 m; Neptune is a 7.8 mm pea about 711 m away, and the nearest star would be another football roughly 6,300 km from the first.

Solar System to Scale: Sizes and Distances

Real radii and orbit sizes in one honest scale: sizes-only, distances-only or everything to scale; Sun-as-football preset.

Zum Modell

This simulator takes the real equatorial radii of the Sun and planets and the semi-major axes of their orbits (NASA planetary fact sheets) and draws them along one horizontal line using a single, consistent scale factor at a time. In "Sizes to scale" the diameters are honest — the Sun is 109 Earth diameters across, Jupiter 11.2 — but the planets are spread out evenly, so distances mean nothing. In "Distances to scale" the orbit radii are honest (Neptune at 30 AU fits the width) while planet diameters are multiplied by an adjustable factor N; the Sun is exaggerated by at most ×58 so that it still fits inside Mercury's orbit. "Everything to scale" cheats nowhere: with the Sun 110 px wide, Earth is a one-pixel dot about 15 screen widths to the right and Neptune 450 screens away — the mini-map and the "screens away" readout show how far you must pan, and wheel zoom lets you travel between scales. Light travel time is a/c, and a light pulse can be sent from the Sun at a chosen number of light-minutes per real second. The football preset shrinks the Sun to a 22 cm ball and lists every body's size and distance in millimetres and metres. Orbits are treated as circles of radius a (eccentricity, inclination and the planets' actual positions on a given date are ignored), oblateness is ignored, the Kuiper belt is a decorative 30–50 AU band, and the horizontal layout is a cross-section, not a sky view.

Für wen: Grades 7–11 astronomy and general-science classes; anyone who has only ever seen the usual compressed textbook picture of the solar system.

Wichtige Begriffe

astronomical unit
scale model
equatorial radius
semi-major axis
light-minute
light travel time
Kuiper belt
scale factor

So funktioniert es

Real equatorial radii and semi-major axes (NASA fact sheets) are rendered with one consistent scale factor at a time. "Sizes to scale" keeps the diameters honest but compresses the distances; "Distances to scale" keeps the orbit radii honest but multiplies the planet diameters by N (the Sun by at most ×58 so it still fits inside Mercury's orbit); "Everything to scale" cheats nowhere — the Sun is a 110-px disc and Earth is a one-pixel dot about 15 screens to the right. Orbits are treated as circles of radius a. The light pulse moves at c in scaled time: 1 real second = the chosen number of light-minutes.

Wichtige Gleichungen

size on screen = real size × k, k = px per km (one k for everything in "true" mode)

D_Sun / D_Earth = 1 391 400 km / 12 756 km ≈ 109; D_Jupiter / D_Earth ≈ 11.2

t_light = a / c; 1 AU / c = 499 s ≈ 8.3 min; Neptune: 30.07 AU → 4.2 h

football: k = 0.22 m / 1 391 400 km → Earth ⌀ 2.0 mm at 23.7 m, Neptune ⌀ 7.8 mm at 711 m

Häufige Fragen

Why can textbook pictures not show sizes and distances at the same time?: The Sun is 1.39 million km across but Neptune is 4.5 billion km away — a ratio of about 3,200. If the Sun is drawn 110 px wide, Neptune must sit about 355,000 px away, and Earth would be a single pixel. Any picture that fits on a page has to exaggerate the sizes, compress the distances, or both.
How big is the Sun compared with Earth?: The Sun's diameter (1,391,400 km) is about 109 Earth diameters (12,756 km), and roughly 1.3 million Earths would fit inside its volume. Jupiter is about 11.2 Earth diameters across.
What does "light travel time" mean and how is it computed?: It is the time light needs to cross the orbit radius: t = a / c. For Earth, 149.6 million km ÷ 299,792 km/s ≈ 499 s ≈ 8.3 min; for Neptune (30.07 AU) it is about 4.2 h. When you look at a planet you see it as it was that long ago.
How do the football numbers come out?: Everything is multiplied by the same factor k = 0.22 m ÷ 1,391,400 km ≈ 1.6 × 10⁻⁷ m per km. Earth's diameter becomes 12,756 km × k ≈ 2.0 mm (a peppercorn) at 149.6 million km × k ≈ 23.7 m; Neptune is a 7.8 mm pea about 711 m away, and the nearest star would be another football roughly 6,300 km from the first.

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Einflusssphäre

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Radius der Einflusssphäre r_H ≈ a (m/3M)^{1/3}: Gebiet, in dem die Gravitation des Planeten die Gezeiten des Sterns überwiegt. Schema von Bahn und r_H nach Massen und großer Halbachse.

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