If an eclipsing binary system has a period of 32 days, an orbital velocity of 153km/s, and an orbit that is nearly edge-on, what is the circumference of the orbit? The radius of the orbit? The mass of the system?
I can't seem to find a starting point for this problem.
If the orbital velocity is a constant (in this case 153 km/s), the orbit of each star about the center of mass must be circular. You need to specify which of the stars has an orbital velocity of 153 km/s. Unless the stars have equal mass, they will move at different velocities about the center of mass, and at different distances.
Most eclipsing binaries are not in highly circular orbits. Just because the veolcity of one star in its orbit is 153 km/s at one particular time does not mean it will remain that value.
In a problem like this, for circular orbits, you can use the Newton form of Kepler's third law, if R is the distance between the two stars and M is the combined mass. That is how you should proceed.
If this confuses you, it is explained much better at
http://www.astro.cornell.edu/academics/courses/astro2201/kepler_binary.htm
Oh, yes, yes. This is explained in my book; I just didn't remember this formula or think to apply it to this problem. Thank you so much!
To find the circumference of the orbit, we can use the formula:
Circumference = 2πr
where "r" is the radius of the orbit.
To find the radius of the orbit, we can use the formula:
Radius = (Velocity * Period) / (2π)
where "Velocity" is the orbital velocity, and "Period" is the period of the orbit.
Lastly, to find the mass of the system, we need additional information. The mass can be determined using Kepler's Third Law, which relates the period of the orbit and the semi-major axis of the orbit:
M1 + M2 = (4π^2 * a^3) / (G * T^2)
where "M1" and "M2" are the masses of the two binary system objects, "a" is the semi-major axis of the orbit, "G" is the gravitational constant, and "T" is the period of the orbit.
However, since we do not have the semi-major axis or the second mass, we cannot determine the mass of the system accurately. The given information is insufficient for calculating the mass.
Let's calculate the circumference and the radius of the orbit using the provided data:
Orbital Velocity = 153 km/s
Period = 32 days
First, we need to convert the period and velocity into SI units for a consistent calculation:
Period = 32 days = 32 * 24 * 60 * 60 seconds = 2,764,800 seconds
Velocity = 153 km/s = 153,000 m/s
Now, we can calculate the radius of the orbit:
Radius = (Velocity * Period) / (2π)
= (153,000 m/s * 2,764,800 s) / (2π)
≈ 666,231,224.63 meters
To find the circumference of the orbit:
Circumference = 2π * Radius
= 2π * 666,231,224.63
≈ 4,186,153,306.11 meters
Therefore, the circumference of the orbit is approximately 4,186,153,306.11 meters and the radius of the orbit is approximately 666,231,224.63 meters.