20 Nov 12
I just read about a newly discovered planet around a star Kappa Andromedae, about 170 light years out where the planet is about 60 AU away from the star and masses 7 Jupiters. How can you go from that to a radius? Assuming the planets have the same density of course, just the simplest case for a rough comparison. Given equal density I would assume that would factor out of the equation so you could compare say, Mercury with another planet with IT's density but say 10 times more massive.
- Joined
- 10 Dec 06
- Moves
- 8528
20 Nov 12
Originally posted by sonhouseWell,
I just read about a newly discovered planet around a star Kappa Andromedae, about 170 light years out where the planet is about 60 AU away from the star and masses 7 Jupiters. How can you go from that to a radius? Assuming the planets have the same density of course, just the simplest case for a rough comparison. Given equal density I would assume that woul ...[text shortened]... could compare say, Mercury with another planet with IT's density but say 10 times more massive.
Assume a sphere
m= mass_ Jupiter
p = density_ Jupiter
r = radius of planet
n = number of times more massive than Jupiter
n*m = 4/3*p*PI*r^3
solve for "r"
r = ((3*n*m)/(4*PI*p))^(1/3)
r= (n^(1/3))*(3*m/(4*PI*p))^(1/3)
r = (n^(1/3))*(r_ Jupiter)
so if it is 7 times more massive than Jupiter than the radius is
r = (7^(1/3))*(r_Jupiter)
r= 1.913*(r_ Jupiter)
20 Nov 12
Originally posted by joe shmoAh, so it's the cube root of the difference in size. Thanks! So it would be roughly twice the size of Jupiter assuming the same density. That's getting up there since Jove is about 80,000 (about 130,000 km) miles across so this new planet would be something like 150,000 (240,000 Km)miles across. That's getting within reach of the size of the sun which clocks in at about 880,000 miles (1,400,000 Km) across!
Well,
Assume a sphere
m= mass_ Jupiter
p = density_ Jupiter
r = radius of planet
n = number of times more massive than Jupiter
n*m = 4/3*p*PI*r^3
solve for "r"
r = ((3*n*m)/(4*PI*p))^(1/3)
r= (n^(1/3))*(3*m/(4*PI*p))^(1/3)
r = (n^(1/3))*(r_ Jupiter)
so if it is 7 times more massive than Jupiter than the radius is
r = (7^(1/3))*(r_Jupiter)
r= 1.913*(r_ Jupiter)
- Joined
- 10 Dec 06
- Moves
- 8528
20 Nov 12
Originally posted by sonhouseYour welcome!
Ah, so it's the cube root of the difference in size. Thanks! So it would be roughly twice the size of Jupiter assuming the same density. That's getting up there since Jove is about 80,000 (about 130,000 km) miles across so this new planet would be something like 150,000 (240,000 Km)miles across. That's getting within reach of the size of the sun which clocks in at about 880,000 miles (1,400,000 Km) across!
21 Nov 12
1 edit
Originally posted by joe shmoIt occurred to me by just doing the cube root of the difference and comparing the real sizes, for instance if you did that for the sun Vs Jupiter, you could get a rough idea of the density difference. So the sun, 1.4 E6 Km diameter Vs 1.28E5 Km works out to be about 11 times bigger than Jupiter but if you take the cube root of that, the sun would be only about 2.2 times bigger than Jupiter so the density of the sun must be a LOT less than the density of Jupiter.
Your welcome!
So it looks like the density of Jupiter is about 5 times that of the sun. That is solely based on the numbers. I have no idea what the real numbers are, but I bet it's not far off what I said, about 5 times more dense than the sun.
- Joined
- 10 Dec 06
- Moves
- 8528
21 Nov 12
4 edits
Originally posted by sonhouseI don't think that works, the formula doesn't compare densities, because it was derived assuming the densities of the two compared planets were equal...If you want to compare densities based on size its going to be a whole new formula. I'll work it out for you.
It occurred to me by just doing the cube root of the difference and comparing the real sizes, for instance if you did that for the sun Vs Jupiter, you could get a rough idea of the density difference. So the sun, 1.4 E6 Km diameter Vs 1.28E5 Km works out to be about 11 times bigger than Jupiter but if you take the cube root of that, the sun would be only ab ...[text shortened]... eal numbers are, but I bet it's not far off what I said, about 5 times more dense than the sun.
let
p_A = density of planet A
p_B = density of planet B
the ratio, density of planet B: density of planet A = n'
n' = p_B/p_A (n' will be the number of times more dense or less dense "planet B" is than "planet A" )
now each of the respective densities of the planet are as follows
p_A = m_A/V_A ( that is the mass of planet A divided by the volume of planet A, and the same goes for planet B)
p_B = m_B/V_B
now we are still trying to find the ratio n', so we back substitute as follows.
n' = (m_B/V_B)/(m_A/V_A)
once again assuming a sphere
V_A = 4/3*PI*r_A^3
V_B = 4/3*PI*r_B^3
so we back substitute these equations into the most recent equation for n' and after simplification
n' = (m_B/m_A)*(r_A/r_B)^3
now we assume that planet B is n times more massive than planet A, which means
m_B = n*m_B
back substitute and simplify and we arrive at the final equation
n' = n*(r_A/r_B)^3
!AND A FURTHER NOTE!:
all the variables on the right you should know...that is to say, don't solve for r_A from the first equation we derived earlier when we assumed the planets A and B had equal densites. For If you do, n' will just be a bogus result.
- Joined
- 31 May 06
- Moves
- 1795
22 Nov 12
Originally posted by joe shmoSmall typo...
I don't think that works, the formula doesn't compare densities, because it was derived assuming the densities of the two compared planets were equal...If you want to compare densities based on size its going to be a whole new formula. I'll work it out for you.
let
p_A = density of planet A
p_B = density of planet B
the ratio, density of planet B: ...[text shortened]... nets A and B had equal densites. For If you do, n' will just be a bogus result.
now we assume that planet B is n times more massive than planet A, which means
m_B = n*m_B
I think you mean m_B = n*m_A
Otherwise good work.
- Joined
- 10 Dec 06
- Moves
- 8528
22 Nov 12
1 edit
!AND A FURTHER NOTE!:
all the variables on the right you should know...that is to say, don't solve for r_A from the first equation we derived earlier when we assumed the planets A and B had equal densites. For If you do, n' will just be a bogus result.
To further explain this note, I would like to add that the first equation derived in this thread is in fact just a particular solution to the latest equation (n' = 1). So if you were to substitute the the first into the last, regardless of what any of the variables (n, r_A,r_B) you will always arrive at the solution n' =1 (indicating that the densities of planets A & B are equal)
22 Nov 12
Originally posted by joe shmoThanks again for the work! What do you do for a living? Are you a math professor or something?!AND A FURTHER NOTE!:
all the variables on the right you should know...that is to say, don't solve for r_A from the first equation we derived earlier when we assumed the planets A and B had equal densites. For If you do, n' will just be a bogus result.
To further explain this note, I would like to add that the first equation derived in t ...[text shortened]... ways arrive at the solution n' =1 (indicating that the densities of planets A & B are equal)
- Joined
- 10 Dec 06
- Moves
- 8528
22 Nov 12
Originally posted by sonhouseNo, I'm not a Math professor. Currently I'm unemployed, hopefully not for much longer. I just have a Bachelors in Mechanical Engineering, and a desire to understand things well enough so that if the opportunity should arise, I can help explain them to others.
Thanks again for the work! What do you do for a living? Are you a math professor or something?
I'm lucky I found this site a few years back when I had (what would probably have been) a fleeting interest intellectual activities. Before I did, I virtually new nothing of math, or physics. I would say in no small part, RHP's problem solvers facilitated the math/physics bug ( that I think I originally obtained from some plug and chug equations I had to use at my job ), which inevitably lead me to college, where I could really dig in and change my brain. Thanks people. (In fact you were probably one them, but most of the others don't seem to be around anymore). So I guess I'm just doing what I can to keep the good problem solving karma going!
22 Nov 12
Originally posted by joe shmoWorks for me🙂
No, I'm not a Math professor. Currently I'm unemployed, hopefully not for much longer. I just have a Bachelors in Mechanical Engineering, and a desire to understand things well enough so that if the opportunity should arise, I can help explain them to others.
I'm lucky I found this site a few years back when I had (what would probably have been) a fl ...[text shortened]... more). So I guess I'm just doing what I can to keep the good problem solving karma going!