海をタップ → 水のかたまりを落とす ／ ドラッグでまわす・つまむ（ホイール）で寄るTap the sea → drop a lump of water · drag to turn · pinch (wheel) to zoom

ちいさな地球A Tiny Earth

地球を同じ材料のまま小さくしたら、海と森はどう動く？ 空気は透明な殻で閉じこめている（仮定）。

If Earth were shrunk but kept the same materials, how would its seas and forests behave? The air is held in by a transparent shell (an assumption).

幹の太さは 60 cm のまま、高さだけ変えます。自分の重さで曲がる高さをこえると、木は曲がって倒れます。風が吹くと木がゆれ、葉は風に運ばれます（ゆれの大きさは見た目）。Only the height changes; the trunk stays 60 cm thick. Above the height at which a tree bends under its own weight, the trees bend over. When the wind blows, trees sway and leaves are carried by the wind (the size of the sway is just for show).

    この星の物理ではWith this planet's physics

      さいきんのできごとRecent events

        光は見た目だけです（正直モードの温度は一日でならした日光で計算）。地形は見た目だけ段々にしています（陸は 400 m、海の底は 1000 m ごと。地球の大きさのときの高さで）。海と空気の計算は、なめらかな高さのまま。Light is only for show (honest-mode temperatures use sunlight averaged over a day). The terrain is drawn in steps just for looks (every 400 m on land and 1000 m on the sea floor, in Earth-size heights). The sea and air are calculated with smooth heights.

        この大きさの星ではOn a planet this size

        いまの海The sea now

        わかることWhat we learn

        このアプリについて

        地球と同じ材料（平均の密度 5513 kg/m³）で、半径だけを小さくした星を計算しています。重力は半径に比例して弱くなり（g = 4/3·π·G·ρ·R）、山の高さ・海の深さも同じ割合で小さくなります。木や人は小さくならないので、本当の大きさ（木は 30 m）で描いています。一日の長さは地球と同じ 24 時間です。

        海は、星の表面を約1万の「ます」に分けた浅水の式で計算しています（重力・コリオリ力・風の力・底の摩擦。陸にあふれる・引くも計算）。縮めた地球の海は、波の長さにくらべてとても浅いので、この式があてはまります。

        簡単にしているところ

        • 透明な殻（仮定）: 小さな星は重力が弱く、空気は宇宙へ逃げてしまいます（半径 1 km なら脱出速度は約 1.8 m/s、空気の分子は約 500 m/s で飛んでいる）。ふだんは殻（半径の 1.25 倍の高さ）が 1 気圧・15℃ の空気を閉じこめているとしています。
        • 正直モード（殻を外す）: 空気の広がりは、半径の向きだけの1次元の流体の式（重力つき・熱のやりとりなし）で計算しています。空気が重力でとどまる大きな星（λ＞5）は流体の計算をせず、宇宙へ逃げる速さを式で見つもっています（上空まで 15℃ のままとした Parker の流れ・Jeans 逃散。上空が太陽で温まると、もっと速く逃げます）。
        • 海: 気圧が水の蒸気圧（15℃ で 1706 Pa）より下がると沸き、蒸発で冷えて凍ります。蒸発の速さは真空への蒸発の式（Hertz–Knudsen、蒸発係数 1 ＝ いちばん速い場合）。水の重さで圧力が蒸気圧より高い深い所は沸かず、氷の下で上から凍ります（氷の熱伝導）。氷の表面の温度は、一日と緯度でならした日光（太陽定数 1361 W/m²、氷のアルベド 0.5 は推定）と、熱の放射・昇華のつりあいで決めています。昼と夜・赤道と極の差、氷のふたで沸騰が弱まること、地熱は省いています。
        • 生きもの（シカ・ウサギ・小鳥・小魚・大きな魚）: 「地球と同じ動き方をするとしたら、この星の物理でどうなるか」を計算しています。大きさ・重さ・跳ぶ速さ・泳ぐ速さは代表の値（推定）。歩く速さは、歩きから走りに変わるフルード数 0.5（Kram ほか 1997）までとしています。跳んだあとは、重力（距離の2乗で弱まる）・空気の抵抗（抗力係数 1、推定）・風・コリオリ力で飛び方を計算し、殻にぶつかれば止まります。魚は体の高さの2倍より浅い所には入れないとしています（推定）。あふれた水が体の高さより深いと、おぼれるとしています（推定）。生まれる・食べる・増えるは入れていません。画面では見やすく 20 倍の大きさで描いています。
        • 正直モードの生きもの: 気圧が体温（37℃）の水の蒸気圧（約 6.3 kPa、アームストロング限界）を下回ると体液が沸きはじめます。犬を真空近くまで減圧した実験では 9〜12 秒で意識を失い、2 分より短ければ全頭が生きのびました（Bancroft & Dunn 1965）。ここでは 10 秒で意識を失い、2.5 分で死ぬとしています（推定）。魚は住む海が沸くか凍ると死ぬとしています。氷の下に水が残れば生きのびます（酸素は計算していません）。浮いていた・飛んでいた生きものは、空気が抜けるときの風が脱出速度より速ければ、空気といっしょに宇宙へ飛ばされます。
        • そばで見る（段階D）: 目の高さ 1.6 m に立ち、時間は本当の速さから。ジャンプは地球のその場とび（50 cm、とび出す速さ 3.1 m/s）と同じ力で、体重 70 kg・正面の面積 0.7 m²（推定）の人として、重力・空気の抵抗・風・コリオリ力で計算します。しぶき（直径 1〜6 mm）と雨つぶは、重力と空気の抵抗（抗力係数 0.5〜0.6、推定）で、風に運ばれます。雨つぶの大きさの上限は、地球でおよそ直径 6 mm（これより大きいとちぎれる。Villermaux & Bossa 2009 の議論）とし、表面張力と重力の比（毛管長）に合わせて √(地球の重力 ÷ この星の重力) 倍にしています（推定）。雨は降らせるとしたもので、雲は計算していません。
        • 森（段階B）: 自分の重さで曲がる高さは、一様な太さの柱が自重で曲がる長さ（0.788×(E D²/(ρg))^(1/3)、Greenhill の問題）で出しています（生木のヤング率 10 GPa・密度 800 kg/m³・幹の太さ 60 cm はおおよその値、推定）。本当の木は根元ほど太く、枝や葉の重さもあるので、これは目安です。風の力は入れていません。水を吸い上げる限界は、地球では重力のぶんと管の摩擦のぶんで決まります（Koch ほか 2004）。重力のぶんは計算できますが、摩擦のぶんは重力が弱い所で測られていないので、新しい限界の高さは出していません。葉の落ちる速さは、葉の面積あたりの重さ 0.1 kg/m²・抗力係数 1（推定）として、空気の抵抗と重さのつりあいから出しています。木のゆれの大きさは見た目です（風の力も木の硬さも重力と関係ないので、ゆれ方は地球とほとんど同じ）。
        • 森: 空気がなくなると葉の水が抜けて枯れます。枯れる速さは計算していません（色の変わり方は見た目だけ）。
        • 風はあなたが決めます（自然の風の計算はしていません）。風は、ある軸のまわりをまわる向きに吹きます。
        • 流れが勢いを運ぶ効果（移流）は省いています。波が砕ける・しぶきは計算していません。
        • 星の中身はかたいものとしています。地形は「本物の地球」（ETOPO1 を 1°ごと＝約 110 km ごとに間引いたもの。高い山の頂上は間引かれて低くなる）と「作りもの」（高さの分布だけ地球に近づけた。陸 29%、海の平均の深さ 3682 m）から選べます。森は陸の約 31%（FAO 2020）。本物の地球のときの森の場所は、緯度で決めた目安です（熱帯と緯度 45〜60° に多く、砂漠の多い 18〜35° に少なく。本物の森の地図ではありません）。1 ha に 400 本として数を出し、多すぎるときは間引いて描いています。
        • 底の摩擦係数 0.0025 は、潮や高潮の計算でよく使われる値（推定）です。
        • 半径 100 m より小さいと、表面張力と水の粘りけが主役になり、この式は合わなくなります。

        おもな出どころ

        • 地球の半径・平均の密度: NASA NSSDC, Earth Fact Sheet
        • 海の平均の深さ 3682 m: Charette, M. A. & Smith, W. H. F. (2010) The volume of Earth’s ocean. Oceanography 23(2): 112–114.
        • 海面の風の力（抗力係数 約 1.2×10⁻³）: Large, W. G. & Pond, S. (1981) Open ocean momentum flux measurements in moderate to strong winds. J. Phys. Oceanogr. 11: 324–336.
        • 浅水の式: Vallis, G. K. (2017) Atmospheric and Oceanic Fluid Dynamics, 2nd ed. Cambridge University Press.
        • 大気が宇宙へ逃げるしくみ: Catling, D. C. & Kasting, J. F. (2017) Atmospheric Evolution on Inhabited and Lifeless Worlds, 5章. Cambridge University Press.
        • 氷と水の蒸気圧: Murphy, D. M. & Koop, T. (2005) Review of the vapour pressures of ice and supercooled water for atmospheric applications. Q. J. R. Meteorol. Soc. 131: 1539–1565.
        • 水の蒸発係数: Eames, I. W., Marr, N. J. & Sabir, H. (1997) The evaporation coefficient of water: a review. Int. J. Heat Mass Transfer 40: 2963–2973.
        • 太陽定数: Kopp, G. & Lean, J. L. (2011) A new, lower value of total solar irradiance. Geophys. Res. Lett. 38: L01706.
        • 歩きから走りに変わる速さ（フルード数 0.5、重力が弱いとき）: Kram, R., Domingo, A. & Ferris, D. P. (1997) Effect of reduced gravity on the preferred walk-run transition speed. J. Exp. Biol. 200: 821–826.
        • 真空近くでの意識と生存（犬）: Bancroft, R. W. & Dunn, J. E. (1965) Times of consciousness, collapse, and survival and pathology examinations of dogs rapidly decompressed to near vacuum environment. USAF School of Aerospace Medicine SAM-TR-65-48.
        • 自分の重さで曲がる柱の高さ: Greenhill, A. G. (1881) Determination of the greatest height consistent with stability that a vertical pole or mast can be made, and of the greatest height to which a tree of given proportions can grow. Proc. Camb. Phil. Soc. 4: 65–73。木への当てはめ: McMahon, T. (1973) Size and shape in biology. Science 179: 1201–1204.
        • 木の高さの限界（水）: Koch, G. W., Sillett, S. C., Jennings, G. M. & Davis, S. D. (2004) The limits to tree height. Nature 428: 851–854.
        • 雨つぶがちぎれる大きさ: Villermaux, E. & Bossa, B. (2009) Single-drop fragmentation determines size distribution of raindrops. Nature Physics 5: 697–702.
        • 本物の地球の地形: NOAA National Geophysical Data Center (2009) ETOPO1 1 Arc-Minute Global Relief Model. NOAA National Centers for Environmental Information（doi:10.7289/V5C8276M、2026-10-12 に取得、1°ごとに間引いて使用）。Amante, C. & Eakins, B. W. (2009) ETOPO1 1 Arc-Minute Global Relief Model: Procedures, Data Sources and Analysis. NOAA Technical Memorandum NESDIS NGDC-24.
        • 森の割合: FAO (2020) Global Forest Resources Assessment 2020.

        使っているもの

        three.js r128（MIT License）、フォント: Google Fonts（SIL Open Font License）

        About this app

        This app calculates a planet made of the same material as Earth (mean density 5513 kg/m³) with only its radius made smaller. Gravity gets weaker in proportion to the radius (g = 4/3·π·G·ρ·R), and mountain heights and sea depths shrink by the same ratio. Trees and people do not shrink, so they are drawn at their real size (trees are 30 m). A day lasts 24 hours, as on Earth.

        The sea is calculated with the shallow-water equations on about 10,000 cells covering the planet (gravity, the Coriolis force, wind, and friction at the bottom, including water spilling onto land and draining back). The sea of a shrunken Earth is very shallow compared with the length of its waves, so these equations apply.

        What is simplified (this is a learning tool)

        • Transparent shell (an assumption): on a small planet gravity is weak and the air would escape into space (on a 1 km planet the escape speed is about 1.8 m/s, while air molecules fly at about 500 m/s). Normally a shell (at 1.25 times the radius) is assumed to hold air at 1 atm and 15 °C.
        • Honest mode (remove the shell): the spreading of the air is calculated with one-dimensional fluid equations along the radius (with gravity, no heat exchange). For large planets where gravity holds the air (λ > 5), the fluid is not calculated; the escape rate is estimated with formulas (a Parker flow kept at 15 °C high up, and Jeans escape). If the Sun heats the upper air, it escapes faster.
        • Sea: when the air pressure drops below the vapor pressure of water (1706 Pa at 15 °C), the water boils, cools by evaporation, and freezes. The evaporation rate uses the formula for evaporation into a vacuum (Hertz–Knudsen, evaporation coefficient 1 = the fastest case). Deep water, where the weight of the water keeps the pressure above the vapor pressure, does not boil and freezes from the top under the ice (heat conduction through ice). The ice surface temperature balances sunlight averaged over a day and over latitude (solar constant 1361 W/m²; ice albedo 0.5 is an estimate) against heat radiation and sublimation. Day and night, the equator and the poles, boiling weakened by an ice lid, and heat from inside the planet are left out.
        • Animals (deer, rabbits, small birds, small fish, large fish): the app calculates “what would happen with this planet’s physics if they moved the way they do on Earth.” Sizes, weights, jump and swimming speeds are typical values (estimates). Walking speed is limited to a Froude number of 0.5, where walking turns into running (Kram et al. 1997). After a jump, the flight is calculated with gravity (weakening with the square of the distance), air drag (drag coefficient 1, estimate), wind and the Coriolis force; it stops if it hits the shell. Fish cannot enter water shallower than twice their body height (estimate). Animals drown if spilled water is deeper than their body height (estimate). Birth, eating and growth are not included. Animals are drawn 20× larger so you can see them.
        • Animals in honest mode: when the pressure falls below the vapor pressure of water at body temperature (37 °C; about 6.3 kPa, the Armstrong limit), body fluids begin to boil. In experiments where dogs were decompressed to near vacuum, they lost consciousness in 9–12 seconds, and all dogs exposed for less than 2 minutes survived (Bancroft & Dunn 1965). Here animals lose consciousness after 10 s and die after 2.5 min (estimate). Fish die when their sea boils or freezes; they survive if liquid water remains under the ice (oxygen is not calculated). Animals that were floating or flying are carried into space with the escaping air if that wind is faster than the escape speed.
        • Look up close: you stand with your eyes 1.6 m high, and time starts at real speed. A jump uses the same push as a standing jump on Earth (50 cm, take-off speed 3.1 m/s) for a 70 kg person with a front area of 0.7 m² (estimate), with gravity, air drag, wind and the Coriolis force. Splash drops (1–6 mm across) and raindrops move with gravity and air drag (drag coefficient 0.5–0.6, estimate) and are carried by the wind. The largest raindrop is taken as about 6 mm across on Earth (larger drops break up; discussed by Villermaux & Bossa 2009) and scaled by √(Earth’s gravity ÷ this planet’s gravity), following the balance of surface tension and gravity (the capillary length) — an estimate. The rain is something you make fall; clouds are not calculated.
        • Forest: the height at which a tree bends under its own weight uses the length at which a uniform column buckles under its own weight (0.788×(E D²/(ρg))^(1/3), Greenhill’s problem), with green-wood stiffness 10 GPa, density 800 kg/m³ and a 60 cm trunk (rough values, estimates). Real trees are thicker at the base and carry branches and leaves, so this is a rough guide. Wind loads are not included. On Earth, the limit to how high trees can lift water is set by gravity plus friction in the water pipes (Koch et al. 2004). The gravity part can be calculated, but the friction part has not been measured in weak gravity, so no new height limit is given. The falling speed of leaves assumes 0.1 kg of leaf per m² and a drag coefficient of 1 (estimates). The size of the sway is for show (wind force and wood stiffness do not depend on gravity, so trees sway almost as on Earth).
        • Forest: without air, water leaves the leaves and the trees die. How fast they die is not calculated (the color change is only for show).
        • You choose the wind (natural winds are not calculated). The wind blows around an axis.
        • The flow carrying its own momentum (advection) is left out. Breaking waves are not calculated.
        • The planet’s interior is treated as rigid. The terrain is either “Real Earth” (ETOPO1 sampled every 1° ≈ 110 km; high peaks are lowered by the sampling) or “Made-up” (only the spread of heights is made Earth-like: 29% land, mean sea depth 3682 m). Forests cover about 31% of the land (FAO 2020). On the real Earth, where forests grow is a rough guide based on latitude (more in the tropics and at 45–60°, fewer in the dry belt at 18–35°; this is not a real forest map). Tree numbers assume 400 trees per hectare; when there are too many, fewer are drawn.
        • The bottom friction coefficient 0.0025 is a value often used for tides and storm surges (estimate).
        • Below a radius of 100 m, surface tension and the stickiness of water take over and these equations no longer fit.

        Main sources

        • Earth’s radius and mean density: NASA NSSDC, Earth Fact Sheet
        • Mean sea depth 3682 m: Charette, M. A. & Smith, W. H. F. (2010) The volume of Earth’s ocean. Oceanography 23(2): 112–114.
        • Wind force on the sea (drag coefficient about 1.2×10⁻³): Large, W. G. & Pond, S. (1981) Open ocean momentum flux measurements in moderate to strong winds. J. Phys. Oceanogr. 11: 324–336.
        • Shallow-water equations: Vallis, G. K. (2017) Atmospheric and Oceanic Fluid Dynamics, 2nd ed. Cambridge University Press.
        • How atmospheres escape to space: Catling, D. C. & Kasting, J. F. (2017) Atmospheric Evolution on Inhabited and Lifeless Worlds, chapter 5. Cambridge University Press.
        • Vapor pressure of ice and water: Murphy, D. M. & Koop, T. (2005) Review of the vapour pressures of ice and supercooled water for atmospheric applications. Q. J. R. Meteorol. Soc. 131: 1539–1565.
        • Evaporation coefficient of water: Eames, I. W., Marr, N. J. & Sabir, H. (1997) The evaporation coefficient of water: a review. Int. J. Heat Mass Transfer 40: 2963–2973.
        • Solar constant: Kopp, G. & Lean, J. L. (2011) A new, lower value of total solar irradiance. Geophys. Res. Lett. 38: L01706.
        • Walk–run transition (Froude number 0.5 in weak gravity): Kram, R., Domingo, A. & Ferris, D. P. (1997) Effect of reduced gravity on the preferred walk-run transition speed. J. Exp. Biol. 200: 821–826.
        • Consciousness and survival near vacuum (dogs): Bancroft, R. W. & Dunn, J. E. (1965) Times of consciousness, collapse, and survival and pathology examinations of dogs rapidly decompressed to near vacuum environment. USAF School of Aerospace Medicine SAM-TR-65-48.
        • Height of a column bending under its own weight: Greenhill, A. G. (1881) Determination of the greatest height consistent with stability that a vertical pole or mast can be made, and of the greatest height to which a tree of given proportions can grow. Proc. Camb. Phil. Soc. 4: 65–73. Applied to trees: McMahon, T. (1973) Size and shape in biology. Science 179: 1201–1204.
        • Limits to tree height (water): Koch, G. W., Sillett, S. C., Jennings, G. M. & Davis, S. D. (2004) The limits to tree height. Nature 428: 851–854.
        • Size at which raindrops break up: Villermaux, E. & Bossa, B. (2009) Single-drop fragmentation determines size distribution of raindrops. Nature Physics 5: 697–702.
        • Real Earth terrain: NOAA National Geophysical Data Center (2009) ETOPO1 1 Arc-Minute Global Relief Model. NOAA National Centers for Environmental Information (doi:10.7289/V5C8276M, accessed 2026-10-12, sampled every 1°). Amante, C. & Eakins, B. W. (2009) ETOPO1 1 Arc-Minute Global Relief Model: Procedures, Data Sources and Analysis. NOAA Technical Memorandum NESDIS NGDC-24.
        • Share of forest: FAO (2020) Global Forest Resources Assessment 2020.

        Software used

        three.js r128 (MIT License); fonts: Google Fonts (SIL Open Font License)