by Eleftheria Safarika , IB 1-Pierce ACG.
Most of us remember Newton’s laws, as we were taught in school, and consider them as one of the relatively easy chapters of Physics. Most of us also usually consider calculating the mass of the sun as one arduous and perplexing undertaking. But what we do not know is that simple concepts such as Newton’s laws, could help us calculate quickly and easily not only the mass our bright star, but also the mass of our earth, or the satellites of the other planets in our solar system.
First of all, however, before making any calculations, we need to understand how the Newtonian Laws apply in the movements of the planets around the stars, such as in our own solar system. Newton suggested that every object in the universe pulls on to every other. This is what we call the law of universal gravitation. So what keeps a planet in orbit around the sun, is the gravitational pull of the sun to the planet. When an object is moving in a circular trajectory around another one, like the planets orbiting the sun, there is a force acting on them, which we call the centripetal force. Let’s note here that in this case, the planet in orbit has acceleration, even though its speed remains constant. In the case of the planet, the gravitational force from the sun on the planets is also the centripetal force, which keeps the planets in orbit. Newton’s law for the gravitational force states that: the gravitational force between two bodies is proportional to the product of their masses, and inversely proportional to the square of the distance between them. Therefore, we can write where F is the gravitational force, which is equal for both objects, is the sun’s mass and is the planet’s mass, and r the distance between the bodies. G is what we call the gravitational constant, it’s the same everywhere in the universe, and has a value of about .
Now that we have the general information required to make our calculations, we need to make some assumptions to simplify the process. These assumptions will, however, slightly divert from reality, but will still allow us to make our calculations with relative accuracy and ease. These will be: firstly, that the planet we choose moves around the sun in a perfectly circular orbit, and secondly, that the sun is stationary, so the radius of the orbit is always the same. What we need to know to complete our calculations will be the distance between our chosen planet and the sun, and the time needed for the planet to complete one full orbit around the sun.
So let’s start. According to Newton’s second law, for the planet it will be true that , where is the planet’s mass. The only force acting on the planet is the gravitational force, caused by the sun. So we write: = . Now, the centripetal acceleration is equal to , where is the speed of the planet , and r is the radius between the planet and the sun. So our equation becomes: = . We can see now, that is equal to the centripetal force, so we understand that , where is the gravitational force, and is the centripetal force. Also, the linear speed of the planet is equal to , meaning it is equal to the circumference of its orbit over the period needed to complete one full revolution. So, by replacing v in the formula, we get that = . But , therefore: = ó ó ó . This is the final formula we will use to find the mass of the sun. Notice that the mass of the planet is absent from it. We know that , , and if we know the distance of the planet from the sun and the time to complete one full orbit around it, we can very easily calculate approximately the mass of the sun.
Let’s take an example using the earth: Firstly, we’ll need to know that the distance between the earth and the sun is about m. Also, we know that earth needs 365 days to complete one full revolution: T= 365 days/rev = 31.536.000 sec/rev . So now the only thing that remains is to substitute the values we have in the formula: ó ó
. This is the mass of the sun, within approximation.
We can also try an example with a planet and satellites: Earth and moon: If we know the distance between the Earth and the moon and the duration of the moon’s orbit around the Earth, we can find the Earth’s mass. First of all, distance between the earth and the moon: about , and the duration of one rotation of the moon around the earth is about 27,32 days = . Substituting these numbers in the equation , we get that the Earth’s mass is around.
The true mass of the sun is 1.9891 × 1030 kg, a number close to the we found, and as for the earth’s actual mass, it’s 5,974 x kg. The difference in the numbers in both these situations is caused by the fact that we supposed that both the earth’s orbit around the sun, and the moon’s orbit around the earth are perfectly circular, while they are elliptic. Also, in reality, r is not constant (the distance between the sun and the earth, or earth and the moon), and, the sun and the earth are actually orbiting around the center of mass of the system consisting of the sun and the earth, and the same is true for the system earth-moon.
Nevertheless, there are other simple ways to find the mass of our planet, using again Newton’s laws. For example, let’s imagine a body on free fall. Let’s consider there is no air resistance. While it is falling, we understand that, since it the body gains acceleration, Newton’s second law must be in act. Therefore: ΣF = ma. The net force acting on the body is the gravitational force, and the acceleration it is experiencing is the gravitational acceleration, g . So we get: where is the mass of the earth, and is the mass of the object. So, to find the mass of the earth: ó ó . Knowing that , that /, and that , we can find that the mass of the Earth is kg, a more accurate number than the previous one.
Newton’s Laws are also valid outside our solar system. They can explain the motion of the stars, the double stars, the galaxies. Thus, we can understand the importance of such simple, but elegant and surprisingly precise laws, proving that they are essential and are encountered in every aspect of physics.
[AK LECTURES]. (2013, Aug 13) Calculation of Mass of the Sun [Video File]. Retrieved from: https://www.youtube.com/watch?v=7KZuahtnwy8
Lewin, W., & Goldstein, W. (2011) For the Love of Physics – From the End of the Rainbow to the Edge of Time: a Journey through the Wonders of Physics. New York, NY: Free Press.