Observations show that a celestial body traveling at 1.2...

Observations show that a celestial body traveling at $1.2 \times 10^{6} \mathrm{mi} / \mathrm{h}$ appears to be describing about point $B$ a circle of radius equal to 60 light years. Point $B$ is suspected of being a very dense concentration of mass called a black hole. Determine the ratio $M_{i} / M_{S}$ of the mass at $B$ to the mass of the sun. (The mass of the sun is 330,000 times the mass of the earth, and a light year is the distance traveled by light in 1 year at $186,300 \mathrm{mi} / \mathrm{s}$.)

Key Concepts

Unit Conversions in Astrophysics

Astrophysical problems often require the integration of various units such as distances in light years, speeds in miles per hour or meters per second, and masses in multiples of known bodies like the Earth or the sun. Accurate unit conversion is crucial to ensure consistency in calculations, particularly when applying formulas derived from physical laws.

Mass Scaling in Astronomy

Comparing masses in astronomy typically involves expressing an unknown mass relative to a well-known standard, such as the sun. This scaling is useful for putting large astronomical masses into perspective, and it often requires converting between different mass units, such as from Earth masses to solar masses, to facilitate meaningful comparisons.

Centripetal Force in Circular Motion

An object moving in a circular path requires an inward force, known as centripetal force, to keep it in its curved trajectory. In orbital dynamics, this force is provided by gravity and is directly linked to the orbital speed and radius, making it fundamental for determining the mass responsible for the observed motion.

Newton's Law of Universal Gravitation

This principle establishes that every mass attracts every other mass with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. It forms the basis for calculating the gravitational pull exerted by a massive object such as a black hole on nearby bodies in orbit.

Orbital Dynamics

Orbital dynamics connects the motion of celestial bodies with the gravitational forces at play. By equating the gravitational force (which depends on the mass of the central object) to the required centripetal force for circular motion, one can derive expressions that allow for the estimation of the mass of objects like black holes from observational data.

Transcript

00:01 In this problem, we're showing that we have a celestial body at traveling at 1 .2 times 10 to the 6th miles per hour.

00:14 At circling something as a rate as equal to 60 light years, so 60 times c.

00:24 Actually, i could, i should need to 60 times c times one year.

00:36 So we have the, we're asked to find is the density, the term of the ratio, the mass of this point b, what the thing the body is orbiting about, to the mass of the sun.

00:53 And we're told that the mass of the sun is 3 ,30 ,000 times the mass of the earth, and the light year we're given the speed of light there.

01:09 We need to again make sure we calculate this in consistent units so that everything we have miles per hour here.

01:18 We can use do this in miles.

01:19 We can convert this to feet per second and we can do this in feet.

01:23 So we just, it's kind of a lot of unit conversions.

01:25 Just we got to make sure everything, all the units are correctly identified.

01:30 And so what we have here is then we have the universal gravitational constant times the mass of that.

01:40 Body at that point b times the mass of the orbiting body divided by the radius squared.

01:50 So this is newton's law here for the body at point b.

01:55 And so we see that again we get that the mass of this body is v squared r over g.

02:09 So we know that that g times m .e...

Please give Ace some feedback