Введение
Стандартная поверхностная гравитация
The surface gravity, g, of an astronomical object is the gravitational acceleration experienced at its surface at the equator, including the effects of rotation. The surface gravity may be thought of as the acceleration due to gravity experienced by a hypothetical test particle which is very close to the object's surface and which, in order not to disturb the system, has negligible mass. For objects where the surface is deep in the atmosphere and the radius not known, the surface gravity is given at the 1 bar pressure level in the atmosphere. Surface gravity is measured in units of acceleration, which, in the SI system, are meters per second squared. It may also be expressed as a multiple of the Earth's standard surface gravity, which is equal to
In astrophysics, the surface gravity may be expressed as , which is obtained by first expressing the gravity in cgs units, where the unit of acceleration and surface gravity is centimeters per second squared (cm/s2), and then taking the base 10 logarithm of the cgs value of the surface gravity. Therefore, the surface gravity of Earth could be expressed in cgs units as 980.665, and then taking the base 10 logarithm ("log g") of 980.665, and we get 2.992 as "log g". The surface gravity of a white dwarf is very high, and of a neutron star even higher. A white dwarf's surface gravity is around 100,000 g whilst the neutron star's compactness gives it a surface gravity of up to 7 with typical values of order e=12 (that is more than 1011 times that of Earth). One measure of such immense gravity is that neutron stars have an escape velocity of around 100,000 km/s, about a third of the speed of light. For black holes, the surface gravity must be calculated relativistically.
Поверхностная гравитация, g, астрономического объекта — это гравитационное ускорение, испытываемое на его поверхности на экваторе, с учетом эффектов вращения. Поверхностную гравитацию можно рассматривать как ускорение, вызванное гравитацией, испытываемое гипотетической пробной частицей, находящейся очень близко к поверхности объекта и имеющей пренебрежимо малую массу, чтобы не возмущать систему. Для объектов, поверхность которых глубоко погружена в атмосферу, а радиус неизвестен, поверхностная гравитация определяется на уровне давления 1 бар в атмосфере. Поверхностная гравитация измеряется в единицах ускорения, которые в системе СИ равны метрам в секунду в квадрате. В астрофизике поверхностная гравитация может быть выражена как log g, которая получается путем первоначального выражения гравитации в единицах СГС, где единица ускорения и поверхностной гравитации составляет сантиметр в секунду в квадрате (см/с²), а затем вычисления десятичного логарифма значения поверхностной гравитации в СГС. Таким образом, поверхностная гравитация Земли может быть выражена в единицах СГС как 980,665, а затем, взяв десятичный логарифм ("log g") от 980,665, мы получаем 2,992 как значение "log g". Поверхностная гравитация белого карлика очень высока, а нейтронной звезды — еще выше. Поверхностная гравитация белого карлика составляет около 100 000 g, в то время как компактность нейтронной звезды обуславливает поверхностную гравитацию до 7, с типичными значениями порядка e = 12 (то есть более чем в 10¹¹ раз больше, чем у Земли). Одной из мер такой огромной гравитации является то, что у нейтронных звезд скорость убегания составляет около 100 000 км/с, примерно треть скорости света. Для черных дыр поверхностная гравитация должна быть рассчитана релятивистски.
The surface gravity, g, of an astronomical object is the gravitational acceleration experienced at its surface at the equator, including the effects of rotation. The surface gravity may be thought of as the acceleration due to gravity experienced by a hypothetical test particle which is very close to the object's surface and which, in order not to disturb the system, has negligible mass. For objects where the surface is deep in the atmosphere and the radius not known, the surface gravity is given at the 1 bar pressure level in the atmosphere. Surface gravity is measured in units of acceleration, which, in the SI system, are meters per second squared. It may also be expressed as a multiple of the Earth's standard surface gravity, which is equal to
In astrophysics, the surface gravity may be expressed as , which is obtained by first expressing the gravity in cgs units, where the unit of acceleration and surface gravity is centimeters per second squared (cm/s2), and then taking the base 10 logarithm of the cgs value of the surface gravity. Therefore, the surface gravity of Earth could be expressed in cgs units as 980.665, and then taking the base 10 logarithm ("log g") of 980.665, and we get 2.992 as "log g". The surface gravity of a white dwarf is very high, and of a neutron star even higher. A white dwarf's surface gravity is around 100,000 g whilst the neutron star's compactness gives it a surface gravity of up to 7 with typical values of order e=12 (that is more than 1011 times that of Earth). One measure of such immense gravity is that neutron stars have an escape velocity of around 100,000 km/s, about a third of the speed of light. For black holes, the surface gravity must be calculated relativistically.
Отношение гравитации поверхности к массе и радиусу
Поверхностная гравитация различных тел Солнечной системы (1 г = 9,80665 м/с², среднее ускорение свободного падения на Земле)
In the Newtonian theory of gravity, the gravitational force exerted by an object is proportional to its mass: an object with twice the mass produces twice as much force. Newtonian gravity also follows an inverse square law, so that moving an object twice as far away divides its gravitational force by four, and moving it ten times as far away divides it by 100. This is similar to the intensity of light, which also follows an inverse square law: with relation to distance, light becomes less visible. Generally speaking, this can be understood as geometric dilution corresponding to point source radiation into three dimensional space. A large object, such as a planet or star, will usually be approximately round, approaching hydrostatic equilibrium (where all points on the surface have the same amount of gravitational potential energy). On a small scale, higher parts of the terrain are eroded, with eroded material deposited in lower parts of the terrain. On a large scale, the planet or star itself deforms until equilibrium is reached. For most celestial objects, the result is that the planet or star in question can be treated as a near perfect sphere when the rotation rate is low. However, for young, massive stars, the equatorial azimuthal velocity can be quite high—up to 200 km/s or more—causing a significant amount of equatorial bulge. Examples of such rapidly rotating stars include Achernar, Altair, Regulus A and Vega. The fact that many large celestial objects are approximately spheres makes it easier to calculate their surface gravity. According to the shell theorem, the gravitational force outside a spherically symmetric body is the same as if its entire mass were concentrated in the center, as was established by Sir Isaac Newton. Therefore, the surface gravity of a planet or star with a given mass will be approximately inversely proportional to the square of its radius, and the surface gravity of a planet or star with a given average density will be approximately proportional to its radius. For example, the recently discovered planet, Gliese 581 c, has at least 5 times the mass of Earth, but is unlikely to have 5 times its surface gravity. If its mass is no more than 5 times that of the Earth, as is expected, and if it is a rocky planet with a large iron core, it should have a radius approximately 50% larger than that of Earth. Gravity on such a planet's surface would be approximately 2.2 times as strong as on Earth. If it is an icy or watery planet, its radius might be as large as twice the Earth's, in which case its surface gravity might be no more than 1.25 times as strong as the Earth's. For instance, Mars has a mass of 6.4185e23 = 0.107 Earth masses and a mean radius of 3,390 km = 0.532 Earth radii. The surface gravity of Mars is therefore approximately
times that of Earth. Without using the Earth as a reference body, the surface gravity may also be calculated directly from Newton's law of universal gravitation, which gives the formula
where M is the mass of the object, r is its radius, and G is the gravitational constant. If we let 1=ρ = M/V denote the mean density of the object, we can also write this as
so that, for fixed mean density, the surface gravity g is proportional to the radius r.
Since gravity is inversely proportional to the square of the distance, a space station 400 km above the Earth feels almost the same gravitational force as we do on the Earth's surface. A space station does not plummet to the ground because it is in an orbit around the Earth.
Название Поверхностная гравитация
Солнце 28,02 г
Меркурий 0,377 г
Венера 0,905 г
Земля 1 г (средние широты)
Луна 0,1657 г (среднее)
Марс 0,379 г (средние широты)
Фобос 0,000581 г
Деймос 0,000306 г
Паллас 0,022 г (экватор)
Веста 0,025 г (экватор)
Церера 0,029 г
Юпитер 2,528 г (средние широты)
Ио 0,183 г
Европа 0,134 г
Ганимед 0,146 г
Каллисто 0,126 г
Сатурн 1,065 г (средние широты)
Мимас 0,00648 г
Энцелад 0,0115 г
Тефия 0,0149 г
Диона 0,0237 г
Рея 0,0269 г
Титан 0,138 г
Япет 0,0228 г
Феба 0,0039–0,0051 г
Уран 0,886 г (экватор)
Миранда 0,0079 г
Ариэль 0,0254 г
Умбриэль 0,023 г
Титания 0,0372 г
Оберон 0,0361 г
Нептун 1,137 г (средние широты)
Протей 0,007 г
Тритон 0,0794 г
Плутон 0,063 г
Харон 0,0294 г
Эрида 0,084 г
Хаумеа 0,0247 г (экватор)
67P CG 0,000017 г
In the Newtonian theory of gravity, the gravitational force exerted by an object is proportional to its mass: an object with twice the mass produces twice as much force. Newtonian gravity also follows an inverse square law, so that moving an object twice as far away divides its gravitational force by four, and moving it ten times as far away divides it by 100. This is similar to the intensity of light, which also follows an inverse square law: with relation to distance, light becomes less visible. Generally speaking, this can be understood as geometric dilution corresponding to point source radiation into three dimensional space. A large object, such as a planet or star, will usually be approximately round, approaching hydrostatic equilibrium (where all points on the surface have the same amount of gravitational potential energy). On a small scale, higher parts of the terrain are eroded, with eroded material deposited in lower parts of the terrain. On a large scale, the planet or star itself deforms until equilibrium is reached. For most celestial objects, the result is that the planet or star in question can be treated as a near perfect sphere when the rotation rate is low. However, for young, massive stars, the equatorial azimuthal velocity can be quite high—up to 200 km/s or more—causing a significant amount of equatorial bulge. Examples of such rapidly rotating stars include Achernar, Altair, Regulus A and Vega. The fact that many large celestial objects are approximately spheres makes it easier to calculate their surface gravity. According to the shell theorem, the gravitational force outside a spherically symmetric body is the same as if its entire mass were concentrated in the center, as was established by Sir Isaac Newton. Therefore, the surface gravity of a planet or star with a given mass will be approximately inversely proportional to the square of its radius, and the surface gravity of a planet or star with a given average density will be approximately proportional to its radius. For example, the recently discovered planet, Gliese 581 c, has at least 5 times the mass of Earth, but is unlikely to have 5 times its surface gravity. If its mass is no more than 5 times that of the Earth, as is expected, and if it is a rocky planet with a large iron core, it should have a radius approximately 50% larger than that of Earth. Gravity on such a planet's surface would be approximately 2.2 times as strong as on Earth. If it is an icy or watery planet, its radius might be as large as twice the Earth's, in which case its surface gravity might be no more than 1.25 times as strong as the Earth's. For instance, Mars has a mass of 6.4185e23 = 0.107 Earth masses and a mean radius of 3,390 km = 0.532 Earth radii. The surface gravity of Mars is therefore approximately
times that of Earth. Without using the Earth as a reference body, the surface gravity may also be calculated directly from Newton's law of universal gravitation, which gives the formula
where M is the mass of the object, r is its radius, and G is the gravitational constant. If we let 1=ρ = M/V denote the mean density of the object, we can also write this as
so that, for fixed mean density, the surface gravity g is proportional to the radius r.
Since gravity is inversely proportional to the square of the distance, a space station 400 km above the Earth feels almost the same gravitational force as we do on the Earth's surface. A space station does not plummet to the ground because it is in an orbit around the Earth.
В ньютоновской теории гравитации гравитационная сила, действующая на объект, пропорциональна его массе: объект с вдвое большей массой производит вдвое большую силу. Ньютоновская гравитация также следует закону обратного квадрата, так что перемещение объекта в два раза дальше уменьшает его гравитационную силу в четыре раза, а перемещение его в десять раз дальше уменьшает ее в 100 раз. Это аналогично интенсивности света, которая также следует закону обратного квадрата: по мере увеличения расстояния свет становится менее заметным. В общем, это можно понимать как геометрическое разбавление, соответствующее излучению точечного источника в трехмерном пространстве. Большой объект, такой как планета или звезда, обычно будет приблизительно круглым, приближаясь к гидростатическому равновесию (где все точки на поверхности имеют одинаковое количество гравитационной потенциальной энергии). В небольших масштабах более высокие части рельефа подвергаются эрозии, а эродированный материал откладывается в более низких частях рельефа. В больших масштабах планета или звезда деформируется до достижения равновесия. Для большинства небесных объектов результат таков, что рассматриваемую планету или звезду можно рассматривать как почти идеальную сферу при низкой скорости вращения. Однако для молодых массивных звезд экваториальная азимутальная скорость может быть довольно высокой — до 200 км/с или более — вызывая значительную экваториальную выпуклость. Примерами таких быстро вращающихся звезд являются Ахернар, Альтаир, Регулус А и Вега. Тот факт, что многие крупные небесные объекты приблизительно сферические, облегчает расчет их поверхностной гравитации. Согласно теореме о оболочке, гравитационная сила вне сферически симметричного тела такая же, как если бы вся его масса была сосредоточена в центре, что было установлено сэром Исааком Ньютоном. Следовательно, поверхностная гравитация планеты или звезды с заданной массой будет приблизительно обратно пропорциональна квадрату ее радиуса, а поверхностная гравитация планеты или звезды с заданной средней плотностью будет приблизительно пропорциональна ее радиусу. Например, недавно открытая планета Gliese 581 c имеет массу, по крайней мере, в 5 раз больше массы Земли, но вряд ли имеет в 5 раз большую поверхностную гравитацию. Если ее масса не более чем в 5 раз больше земной, как ожидается, и если это скалистая планета с большим железным ядром, ее радиус должен быть примерно на 50% больше, чем у Земли. Гравитация на поверхности такой планеты будет примерно в 2,2 раза сильнее, чем на Земле. Если это ледяная или водяная планета, ее радиус может быть в два раза больше земного, в этом случае ее поверхностная гравитация может быть не более чем в 1,25 раза сильнее земной. Например, масса Марса составляет 6,4185e23 = 0,107 земных масс, а средний радиус — 3390 км = 0,532 земных радиусов. Поэтому поверхностная гравитация Марса примерно в два раза меньше, чем у Земли. Без использования Земли в качестве опорного тела поверхностную гравитацию также можно рассчитать непосредственно из закона всемирного тяготения Ньютона, который дает формулу, где M — масса объекта, r — его радиус, а G — гравитационная постоянная. Если мы обозначим 1=ρ = M/V средней плотностью объекта, мы также можем записать это так, чтобы при фиксированной средней плотности поверхностная гравитация g была пропорциональна радиусу r.
In the Newtonian theory of gravity, the gravitational force exerted by an object is proportional to its mass: an object with twice the mass produces twice as much force. Newtonian gravity also follows an inverse square law, so that moving an object twice as far away divides its gravitational force by four, and moving it ten times as far away divides it by 100. This is similar to the intensity of light, which also follows an inverse square law: with relation to distance, light becomes less visible. Generally speaking, this can be understood as geometric dilution corresponding to point source radiation into three dimensional space. A large object, such as a planet or star, will usually be approximately round, approaching hydrostatic equilibrium (where all points on the surface have the same amount of gravitational potential energy). On a small scale, higher parts of the terrain are eroded, with eroded material deposited in lower parts of the terrain. On a large scale, the planet or star itself deforms until equilibrium is reached. For most celestial objects, the result is that the planet or star in question can be treated as a near perfect sphere when the rotation rate is low. However, for young, massive stars, the equatorial azimuthal velocity can be quite high—up to 200 km/s or more—causing a significant amount of equatorial bulge. Examples of such rapidly rotating stars include Achernar, Altair, Regulus A and Vega. The fact that many large celestial objects are approximately spheres makes it easier to calculate their surface gravity. According to the shell theorem, the gravitational force outside a spherically symmetric body is the same as if its entire mass were concentrated in the center, as was established by Sir Isaac Newton. Therefore, the surface gravity of a planet or star with a given mass will be approximately inversely proportional to the square of its radius, and the surface gravity of a planet or star with a given average density will be approximately proportional to its radius. For example, the recently discovered planet, Gliese 581 c, has at least 5 times the mass of Earth, but is unlikely to have 5 times its surface gravity. If its mass is no more than 5 times that of the Earth, as is expected, and if it is a rocky planet with a large iron core, it should have a radius approximately 50% larger than that of Earth. Gravity on such a planet's surface would be approximately 2.2 times as strong as on Earth. If it is an icy or watery planet, its radius might be as large as twice the Earth's, in which case its surface gravity might be no more than 1.25 times as strong as the Earth's. For instance, Mars has a mass of 6.4185e23 = 0.107 Earth masses and a mean radius of 3,390 km = 0.532 Earth radii. The surface gravity of Mars is therefore approximately
times that of Earth. Without using the Earth as a reference body, the surface gravity may also be calculated directly from Newton's law of universal gravitation, which gives the formula
where M is the mass of the object, r is its radius, and G is the gravitational constant. If we let 1=ρ = M/V denote the mean density of the object, we can also write this as
so that, for fixed mean density, the surface gravity g is proportional to the radius r.
Since gravity is inversely proportional to the square of the distance, a space station 400 km above the Earth feels almost the same gravitational force as we do on the Earth's surface. A space station does not plummet to the ground because it is in an orbit around the Earth.
Поскольку гравитация обратно пропорциональна квадрату расстояния, космическая станция, находящаяся на высоте 400 км над Землей, испытывает почти ту же гравитационную силу, что и мы на поверхности Земли. Космическая станция не падает на землю, потому что она находится на орбите вокруг Земли.
In the Newtonian theory of gravity, the gravitational force exerted by an object is proportional to its mass: an object with twice the mass produces twice as much force. Newtonian gravity also follows an inverse square law, so that moving an object twice as far away divides its gravitational force by four, and moving it ten times as far away divides it by 100. This is similar to the intensity of light, which also follows an inverse square law: with relation to distance, light becomes less visible. Generally speaking, this can be understood as geometric dilution corresponding to point source radiation into three dimensional space. A large object, such as a planet or star, will usually be approximately round, approaching hydrostatic equilibrium (where all points on the surface have the same amount of gravitational potential energy). On a small scale, higher parts of the terrain are eroded, with eroded material deposited in lower parts of the terrain. On a large scale, the planet or star itself deforms until equilibrium is reached. For most celestial objects, the result is that the planet or star in question can be treated as a near perfect sphere when the rotation rate is low. However, for young, massive stars, the equatorial azimuthal velocity can be quite high—up to 200 km/s or more—causing a significant amount of equatorial bulge. Examples of such rapidly rotating stars include Achernar, Altair, Regulus A and Vega. The fact that many large celestial objects are approximately spheres makes it easier to calculate their surface gravity. According to the shell theorem, the gravitational force outside a spherically symmetric body is the same as if its entire mass were concentrated in the center, as was established by Sir Isaac Newton. Therefore, the surface gravity of a planet or star with a given mass will be approximately inversely proportional to the square of its radius, and the surface gravity of a planet or star with a given average density will be approximately proportional to its radius. For example, the recently discovered planet, Gliese 581 c, has at least 5 times the mass of Earth, but is unlikely to have 5 times its surface gravity. If its mass is no more than 5 times that of the Earth, as is expected, and if it is a rocky planet with a large iron core, it should have a radius approximately 50% larger than that of Earth. Gravity on such a planet's surface would be approximately 2.2 times as strong as on Earth. If it is an icy or watery planet, its radius might be as large as twice the Earth's, in which case its surface gravity might be no more than 1.25 times as strong as the Earth's. For instance, Mars has a mass of 6.4185e23 = 0.107 Earth masses and a mean radius of 3,390 km = 0.532 Earth radii. The surface gravity of Mars is therefore approximately
times that of Earth. Without using the Earth as a reference body, the surface gravity may also be calculated directly from Newton's law of universal gravitation, which gives the formula
where M is the mass of the object, r is its radius, and G is the gravitational constant. If we let 1=ρ = M/V denote the mean density of the object, we can also write this as
so that, for fixed mean density, the surface gravity g is proportional to the radius r.
Since gravity is inversely proportional to the square of the distance, a space station 400 km above the Earth feels almost the same gravitational force as we do on the Earth's surface. A space station does not plummet to the ground because it is in an orbit around the Earth.
Газовые гиганты
Для газовых гигантов, таких как Юпитер, Сатурн, Уран и Нептун, значение поверхностной гравитации приводится на уровне давления в 1 бар в атмосфере.
Несферически симметричные объекты
Большинство реальных астрономических объектов не обладают полной сферической симметрией. Одна из причин этого заключается в том, что они часто вращаются, испытывая на себе одновременное воздействие гравитационной и центробежной сил. Это приводит к тому, что звезды и планеты приобретают форму сплюснутого сфероида, то есть поверхностная гравитация на экваторе у них меньше, чем на полюсах. Этот эффект был использован Халом Клементом в его научно-фантастическом романе «Миссия гравитации», где описывается массивная, быстро вращающаяся планета с гораздо большей гравитацией на полюсах, чем на экваторе. Насколько внутреннее распределение массы объекта отклоняется от симметричной модели, настолько мы можем использовать измеренную поверхностную гравитацию для определения характеристик его внутренней структуры. Этот принцип был практически применен начиная с 1915–1916 годов, когда торсионные весы Роланда Эотвёша использовались для поиска нефти в районе города Эгбелл (ныне Гбели, Словакия). В 1924 году торсионные весы были использованы для обнаружения нефтяного месторождения Нэш-Дом в Техасе.
Динамические черные дыры
Гравитация поверхности для стационарных чёрных дыр хорошо определена. Это связано с тем, что у всех стационарных чёрных дыр горизонт является горизонтом Киллинга. В последнее время наблюдается тенденция к определению гравитации поверхности динамических чёрных дыр, пространство-время которых не содержит поля Киллинга, касательного времени. За прошедшие годы различные авторы предложили несколько определений, таких как гравитация поверхности отслаивания и гравитация поверхности Кодамы. На данный момент нет единого мнения или согласия относительно того, какое определение, если таковое имеется, является верным. Полуклассические результаты указывают на то, что гравитация поверхности отслаивания не определена для переходных объектов, формирующихся за конечное время с точки зрения удалённого наблюдателя.