# Is Newton’s gravitation law able to explain the circular motion of the earth around the sun?

I have read in Forbes website that the orbit of the earth around the sun is expanding (by 1.5 cm in 2019).

I have also heard that big bang theory suggests that the universe would collapse into itself under gravitational force, unless it was expanding.

Is Newton gravitation model able to explain why the orbit of the earth around the sun does not follow an inward spiral path which eventually falls into the sun?

Does it mean that we require the big bang theory (an expanding universe) to explain why the earth orbit does not follow an inward spiral path?

• Earth moves in elliptical orbit around the sun – VK_fan Nov 14 at 6:46
• Newton’s gravitational law combined with his second law of motion, $\mathbf F=m\mathbf a$, explains the motion of the planets to high accuracy. No Big Bang is necessary to explain this. – G. Smith Nov 14 at 6:50
• The expansion of the universe is not the same scenario as a planetary orbit. In the expansion of the universe, galaxies are moving directly away from each other. Planets move around, not away from, their star. – G. Smith Nov 14 at 7:06
• Do you know the equations of motion for when an object is accelerating at a constant rate and direction? – Anders Gustafson Nov 14 at 8:24
• @Hooman Bahreini please see the edited version of my answer which may help – VK_fan Nov 14 at 8:49

Newton's theory does imply that the earth will fall into the sun - if it never had any tangential velocity with regards to the sun. As it obviously does have such velocity, it follows an elliptical path, as described by Newton's laws.

As to why the earth does not follow a spiral path into the sun, that would indeed happen on condition that a) there is something to slow it down, e.g. interplanetary dust and b) there are no other planets or moons. The effect of Jupiter and other planets is by far the greater of the two effects.

Once you add other planets, a planet's orbit is chaotic: it's (very) long term motion is impossible to predict unless you know the starting position and velocity with impossible precision. Over the long term (many billions of years) the effect of Jupiter may be to fling us out of the solar system, to crash us into the sun, or any of many other possible scenarios.

While Newton's laws (as corrected by Einstein) may be exactly correct - the future will tell - we still cannot compute the earth's long term path. In fact, calculating the orbits of 3 or more bodies is not possible except in special cases. In general cases we can only use computers to approximate the result, as our current maths do not give us an exact solution.

As for the Big Bang, it only affects the overall expansion of the universe. It has no effect on the movement of stars and planets. It does not make the Milky Way expand and in fact, the Andromeda galaxy is travelling towards the Milky Way instead of away from it.

In Newtonian Mechanics the Acceleration of an object is equal to the rate of change vector in the objects velocity, or to put it another way is the second derivative of an objects position with respect to time. Also $$\vec{a}=\frac{\vec{F}}{m}$$ with $$\vec{a}$$ being the acceleration, $$\vec{F}$$ being the force exerted on the object, and $$m$$ being the mass of the object.

In the case of an object accelerating at a constant rate you can find the final position of the object using the equation $$\vec{x_f}=\frac{1}{2}\vec{a}t^2+\vec{v_0}t+\vec{x_0}$$, and final velocity of the object using the equation $$\vec{v_f}=\vec{a}t+\vec{v_0}$$ with $$\vec{x_0}$$ being the initial position vector, $$\vec{v_0}$$ the initial velocity vector, $$\vec{a}$$ being the acceleration vector, $$t$$ being the time passed since the initial conditions, $$\vec{x_f}$$ being the final position of the object, and $$\vec{v_0}$$ being the final velocity vector. The Gravitational acceleration vectors of the Earth and Sun are not constant, but there is a way to use the equations I mentioned to approximate the positions and velocities of the Earth and Sun at a time $$t$$.

Given that you mentioned Newtons Law of Gravity, you probably saw the equation $$F_g=-\frac{GM_1M_2}{r^2}$$ This is the equation describing the magnitude of the Gravitational force between both objects, however in order to approximate the motion of the Earth and Sun we need the equations for the force vectors, which will have the same magnitude as the force in the above equation but also have directions. If you multiply $$Fg$$ by the displacement of the Earth relative to the Sun, and the Sun relative to the Earth, then you will get vector quantities that are in the correct direction, but we need to then divide the displacement by the magnitude of the displacement, which is the distance, in order to maintain the correct magnitude. So the equations for the Gravitational Force vectors between two objects are $$\vec{F_{g1}}=-\frac{\vec{r_1}}{r}\frac{GM_1M_2}{r^2}$$ and $$\vec{F_{g2}}=-\frac{\vec{r_2}}{r}\frac{GM_1M_2}{r^2}$$ with $$r$$ being the distance between the two masses, $$\vec{r_1}$$ being the displacement of body 1 with respect to body 2, $$\vec{r_2}$$ being the displacement of body 2 with respect to body 1 $$G$$ being the Gravitational Constant,$$M_1$$ being the Mass of body 1, $$M_2$$ being the Mass of body 2, $$\vec{F_{g1}}$$ being the gravitational force on body 1 from body 2, and $$\vec{F_{g2}}$$ being the gravitational force on body 2 from body 1.

A 3d vector $$\vec{x}$$ can be expressed using coordinates $$(x_1,x_2,x_3)$$. Also using cartesian coordinates the way to add two vectors is to add their like components so for example $$\vec{x}+\vec{y}=(x_1+y1,x_2+y_2,x_3+y_3)$$ if $$\vec{x}$$ and $$\vec{y}$$ are both 3d vectors.

In order to figure out the exact equation describing the positions of two masses as a function of time requires knowing how to solve differential equations, however even if you don't know differential equations there is a way to approximate the positions of two masses as a function of time using the equations for the position and velocity of a single object as a function of time when the object is accelerating at a constant rate. In order to do this, for a vary small increment of time you can calculate as if the acceleration of each body was constant.

Step 1: Set the initial positions, as well as their masses, and set the initial time as $$0$$.

Step 2: Use the initial positions and masses of both bodies to calculate the accelerations of both masses.

Step 3: Set up the initial velocities to be nearly perpendicular to the displacements of both bodies relative to each other, and set up the magnitudes of the velocities of the two bodies to be less than $$\sqrt{a_1x_1}$$ and $$\sqrt{a_2x_2}$$ with $$a_1$$ being the acceleration of body 1, $$a_2$$ being the initial acceleration of body 2, $$x_1$$ being the distance between body 1 and the center of mass of the system when the center of mass of the system is at the origin, and $$x_2$$ being the distance between body 2 and the center of mass of the system when the center of mass of the system is at the origin.

Step 4: Set up the time step increments to be several tens of thousand times smaller than the initial position multiplied by $$2\pi$$ divided by the initial velocity.

Step 5: Plug the initial positions, velocities, accelerations, and size of the time step increments, with the size of the time step increments replacing the time passed since the initial conditions, into the equations of motion I gave for when an object accelerates at a constant rate to approximate the positions and velocities of the two masses at the end of the first time step.

Step 6: Find the time passed at the end of the first time step increment by adding the time at the beginning of the first time step increment to the size of the time step increments.

Step 7: Repeat steps 2, 5, and 6.

There are computer programs, that you can program to do the same task repeatedly. This can help you get an idea of how orbital mechanics is derived from Newtons law of Gravity.