# Differential Equations

## Second Order Equations # Newton’s Law of Universal Gravitation

## The Force of Gravity and Gravitational Potential

The law of universal gravitation was formulated by Isaac Newton $$\left(1643-1727\right)$$ and published in $$1687.$$

In accordance with this law, two point masses attract each other with a force that is directly proportional to the masses of these bodies $${m_1}$$ and $${m_2},$$ and inversely proportional to the square of the distance between them:

$F = G\frac{{{m_1}{m_2}}}{{{r^2}}}.$

Here, $$r$$ is the distance between the centers of mass of the bodies, $$G$$ is the gravitational constant, whose value found by experiment is $$G = 6,67 \times {10^{ - 11}}{\frac{{{\text{m}^3}}}{{\text{kg} \cdot {\text{s}^2}}}}.$$

The force of gravitational attraction is a central force, that is directed along a line passing through the centers of the interacting bodies.

In a system of two bodies (Figure $$2$$), the attraction force $${\mathbf{F}_{12}}$$ of the second body acts on the first body of mass $${m_1}.$$

Similarly, the attraction force $${\mathbf{F}_{21}}$$ of the first body acts on the second body of mass $${m_2}.$$ Both the forces $${\mathbf{F}_{12}}$$ and $${\mathbf{F}_{21}}$$ are equal and directed along $$\mathbf{r},$$ where

$\mathbf{r} = {\mathbf{r}_2} - {\mathbf{r}_1}.$

Using the Newton's second law we can write the following differential equations describing the motion of each body:

${m_1}\frac{{{d^2}{\mathbf{r}_1}}}{{d{t^2}}} = G\frac{{{m_1}{m_2}}}{{{r^3}}}\mathbf{r},\;\;\; {m_2}\frac{{{d^2}{\mathbf{r}_2}}}{{d{t^2}}} = - G\frac{{{m_1}{m_2}}}{{{r^3}}}\mathbf{r}$

or

$\frac{{{d^2}{\mathbf{r}_1}}}{{d{t^2}}} = G\frac{{{m_2}}}{{{r^3}}}\mathbf{r},\;\;\; \frac{{{d^2}{\mathbf{r}_2}}}{{d{t^2}}} = - G\frac{{{m_1}}}{{{r^3}}}\mathbf{r}.$

It follows from the last two equations that

$\frac{{{d^2}{\mathbf{r}_1}}}{{d{t^2}}} - \frac{{{d^2}{\mathbf{r}_2}}}{{d{t^2}}} = G\frac{{{m_2}}}{{{r^3}}}\mathbf{r} + G\frac{{{m_1}}}{{{r^3}}}\mathbf{r},\;\; \Rightarrow \frac{{{d^2}\mathbf{r}}}{{d{t^2}}} = -G\frac{{{m_1} + {m_2}}}{{{r^3}}}\mathbf{r}.$

This differential equation describes the change of the vector $$\mathbf{r}\left( t \right),$$ that is, the relative motion of two bodies under the force of gravitational attraction.

With a large difference in mass of the bodies, we can neglect the smaller body mass in the right side of this equation. For example, the mass of the Sun is $$333,000$$ times greater than the mass of the Earth. In this case, the differential equation can be written in a simpler form:

$\frac{{{d^2}\mathbf{r}}}{{d{t^2}}} = - G\frac{{{M_\text{S}}}}{{{r^3}}}\mathbf{r},$

where $${M_\text{S}}$$ is the mass of the Sun.

The gravitational interaction of bodies takes place through a gravitational field, which can be described by a scalar potential $$\varphi.$$ The force acting on a body of mass $$m,$$ placed in a field with potential $$\varphi,$$ is equal to

$\mathbf{F} = m\mathbf{a} = - m\,\mathbf{\text{grad}}\,\varphi .$

In the case of a point mass $$M,$$ the potential of the gravitational field is given by

$\varphi = - \frac{{GM}}{r}.$

The latter formula is also valid for distributed bodies with central symmetry, such as a planet or star.

## Kepler's Laws

The basic laws of planetary motion were established by Johannes Kepler $$\left(1571-1630\right)$$ based on the analysis of astronomical observations of Tycho Brahe $$\left(1546-1601\right)$$. In $$1609,$$ Kepler formulated the first two laws. The third law was discovered in $$1619.$$ Later, in the late $$17$$th century, Isaac Newton proved mathematically that all three laws of Kepler are a consequence of the law of universal gravitation.

## Kepler's First Law

The orbit of each planet in the solar system is an ellipse, one focus of which is the Sun (Figure $$3$$).

## Kepler's Second Law

The radius vector connecting the Sun and the planet describes equal areas in equal intervals of time. Figure $$4$$ shows the two sectors of the ellipse corresponding to the same time intervals.

According to Kepler's second law, the areas of these sectors are equal.

## Kepler's Third Law

The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit:

${T^2} \propto {a^3}.$

The proportionality coefficient is the same for all planets in the solar system. Therefore, for any two planets, one can write the relationship

$\frac{{T_2^2}}{{T_1^2}} = \frac{{a_2^3}}{{a_1^3}}.$

See solved problems on Page 2.