Question

Какова плотность белого карлика, масса которого равна массе Солнца, а диаметр порядка диаметра Земли? Во сколько раз она меньше плотности нейтронной звезды, масса которой 2М, а радиус 10 км?

Answer 1

Ответ:   Плотность белого карлика ≈ 1,83575*10^9 кг/м³

Плотность нейтронной звезды в ≈ 517193205 раз больше плотности белого карлика.

Объяснение: Дано:

Масса белого карлика Мк = 1,9885*10^30 кг

Радиус белого карлика Rк = 6371 км = 6,371*10^6 м

Масса нейтронной звезды Мн = 2Мк

Радиус нейтронной звезды Rн = 10 км

Найти:

Плотность белого карлика ρк - ?

Плотность нейтронной звезды ρн - ?

Во сколько раз плотности нейтронной звезды больше плотности белого карлика ρн/ρк - ?

В общем случае плотность определяется, как частное от деления массы тела на его объем,  т.е.  ρ = М/V.

Белый карлик и нейтронная звезда имеют шарообразную форму. Таким образом, объем белого карлика будет равен:

Vк = 4πRк³/3, а объем нейтронной звезды Vн = 4πRн³/3.

Тогда плотность белого карлика ρк = 3Мк/4πRк³ =

= 3*1,9885*10^30/4π*(6,371*10^6)³ ≈ 1,83575*10^9 кг/м³

Плотность нейтронной звезды ρн = 3Мн/4πRн³.

С учетом принятых обозначений ρн = 3*2Мк/4π*Rн³.

Найдем отношение ρн/ρк = (3*2Мк/4π*Rн³)/(3Мк/4πRк³) =

= 3*2Мк*4πRк³/3Мк*4π*Rн³ = 2(Rк/Rн)³. Подставив числовые значения параметров, имеем: ρн/ρк = 2(6371/10)³ ≈ 517193205

Плотность нейтронной звезды в ≈ 517193205 раз больше плотности белого карлика

Answer 2

White Dwarf Density Calculation

To calculate the density of a white dwarf with a mass equal to that of the Sun and a diameter approximately the size of the Earth, we can use the formula for density:

Density = Mass / Volume

First, let's calculate the volume of the white dwarf. The volume of a sphere can be calculated using the formula:

Volume = (4/3) * π * (radius^3)

Given that the diameter of the white dwarf is approximately the diameter of the Earth, we can calculate the radius:

Radius = Diameter / 2 = Earth's Diameter / 2

The diameter of the Earth is approximately 12,742 kilometers Therefore, the radius of the white dwarf is:

Radius = 12,742 km / 2 = 6,371 km

Now, let's calculate the volume of the white dwarf:

Volume = (4/3) * π * (6,371 km)^3

Next, we need to convert the volume from cubic kilometers to cubic centimeters, as the mass of the white dwarf is given in solar masses and the density is typically expressed in grams per cubic centimeter.

1 cubic kilometer (km^3) is equal to 1 x 10^15 cubic centimeters (cm^3).

Volume (cm^3) = Volume (km^3) * (1 x 10^15 cm^3 / 1 km^3)

Now, we can calculate the density of the white dwarf:

Density = Mass (in grams) / Volume (in cm^3)

Given that the mass of the white dwarf is equal to the mass of the Sun, which is approximately 1.989 x 10^33 grams and the volume we calculated earlier, we can substitute these values into the formula to find the density.

Neutron Star Density Calculation

To calculate the density of a neutron star with a mass of 2 times the mass of the Sun and a radius of 10 kilometers, we can use the same formula:

Density = Mass / Volume

First, let's calculate the volume of the neutron star. The volume of a sphere can be calculated using the formula:

Volume = (4/3) * π * (radius^3)

Given that the radius of the neutron star is 10 kilometers, we can calculate the volume:

Volume = (4/3) * π * (10 km)^3

Next, we need to convert the volume from cubic kilometers to cubic centimeters, as the mass of the neutron star is given in solar masses and the density is typically expressed in grams per cubic centimeter.

1 cubic kilometer (km^3) is equal to 1 x 10^15 cubic centimeters (cm^3).

Volume (cm^3) = Volume (km^3) * (1 x 10^15 cm^3 / 1 km^3)

Now, we can calculate the density of the neutron star:

Density = Mass (in grams) / Volume (in cm^3)

Given that the mass of the neutron star is 2 times the mass of the Sun, which is approximately 1.989 x 10^33 grams and the volume we calculated earlier, we can substitute these values into the formula to find the density.

Comparison of Densities

To compare the densities of the white dwarf and the neutron star, we can calculate the ratio of their densities:

Density Ratio = Density of White Dwarf / Density of Neutron Star

By substituting the calculated densities of the white dwarf and the neutron star into the formula, we can find the density ratio.

Please note that the actual calculations may involve more precise values and additional factors, but this simplified approach should provide a reasonable estimate of the densities.

Let's perform the calculations and find the density ratio.