Mass-radius relation of a white dwarf - calculating radius

In summary, to calculate the radius of a 1.3 Msun white dwarf, you can use the mass-radius relation for white dwarfs, which states that $$R \propto M^{-\frac{1}{3}}$$. By plugging in the mass of 1.3 Msun and a known example of a white dwarf, such as Sirius B, you can solve for the constant of proportionality and use it to find the radius in solar radius units.
  • #1
Ryaners
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Homework Statement


Calculate the radius of a 1.3 Msun white dwarf using the mass-radius relation for white dwarfs. Give the answer in solar radius.

Homework Equations


Mass-radius relation: $$R \propto M^{-\frac{1}{3}}$$

The Attempt at a Solution


So I've tried the following:
$$R_{D} \propto M_{D}^{-\frac{1}{3}} \Rightarrow \frac {R_{D}} {R_{sun}} = \frac{M_{D}^{-\frac{1}{3}}} {M_{sun}^{-\frac{1}{3}}} \Rightarrow R_{D} = \frac{M_{D}^{-\frac{1}{3}}} {M_{sun}^{-\frac{1}{3}}} R_{sun}$$
$$ \Rightarrow R_{D} = \left( {\frac {1.3 M_{sun}} {M_{sun}}} \right) ^{-\frac{1}{3}} R_{sun} = \left( 1.3 \right) ^{-\frac{1}{3}} R_{sun} $$

This gives me an answer of about ##0.916~{R_{sun}}## , which is incorrect. Where am I going wrong here?

Thanks in advance for any help.
 
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  • #2
Start with ##M_{Sun}^{1/3}R_{Sun}=M_{D}^{1/3}R_{D}## and replace ##M_D=1.3M_{Sun}##. The algebra is less confusing when you eliminate the proportionality constant.
 
  • #3
kuruman said:
Start with ##M_{Sun}^{1/3}R_{Sun}=M_{D}^{1/3}R_{D}## and replace ##M_D=1.3M_{Sun}##. The algebra is less confusing when you eliminate the proportionality constant.

That's a fair point - though I get the same result:

$$ \left( M_{sun} \right) ^\frac {1}{3} R_{sun} = \left( M_{D} \right) ^{\frac {1}{3}} R_D $$
$$ \Rightarrow R_D = \left( \frac {M_{sun}} {M_D} \right) ^{\frac{1}{3}} R_{sun} $$
$$ \Rightarrow R_D = \left( \frac {1}{1.3} \right) ^{\frac {1}{3}} R_{sun} = \left( 1.3 \right) ^{-\frac {1}{3}} R_{sun} $$
 
  • #4
At this point you need to question why you think that the answer is incorrect. What you think is the correct answer may be a misprint or a miscalculated answer by whoever gave it to you. The only other thing I can think of is the starting equation which is approximate and may have to be refined.
 
  • #5
kuruman said:
At this point you need to question why you think that the answer is incorrect. What you think is the correct answer may be a misprint or a miscalculated answer by whoever gave it to you. The only other thing I can think of is the starting equation which is approximate and may have to be refined.

Perhaps the problem is that the sun is not a white dwarf?
 
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  • #6
Dick said:
Perhaps the problem is that the sun is not a white dwarf?
Perhaps, but the problem clearly states that you should use "the mass-radius relation for white dwarfs."

I am sorry, but my resources regarding this question have been exhausted. I took a single astrophysics course several decades ago and I have reached the point where I can no longer help you. Perhaps someone else may be able to step in.
 
  • #7
kuruman said:
Perhaps, but the problem clearly states that you should use "the mass-radius relation for white dwarfs."

I am sorry, but my resources regarding this question have been exhausted. I took a single astrophysics course several decades ago and I have reached the point where I can no longer help you. Perhaps someone else may be able to step in.

It's not really an serious astrophysics point. The mass-radius relation gives you a proportionality. To get the constant of proportionality you need an example mass and radius of a white dwarf. The sun isn't one.
 
  • #8
Hint: Sirius B is a pretty well known example of a white dwarf :wink:
 

Related to Mass-radius relation of a white dwarf - calculating radius

1. What is the mass-radius relation for a white dwarf?

The mass-radius relation for a white dwarf is a mathematical equation that describes the relationship between the mass and radius of a white dwarf, a type of star that has exhausted its nuclear fuel and collapsed under its own gravity.

2. How is the radius of a white dwarf calculated?

The radius of a white dwarf can be calculated using the Chandrasekhar limit, which states that the maximum mass of a white dwarf is approximately 1.4 times the mass of the sun. Using this limit and the known mass of a white dwarf, the radius can be calculated using the equation R = (2.18 x 10^-5)(M^-1/3), where R is the radius in kilometers and M is the mass in solar masses.

3. What factors affect the mass-radius relation of a white dwarf?

The mass-radius relation of a white dwarf is affected by the mass of the star, the composition of its core, and the equation of state that governs the relationship between pressure and density within the star.

4. Why is the mass-radius relation important in the study of white dwarfs?

The mass-radius relation is important in the study of white dwarfs because it allows us to understand the physical properties and evolution of these stars. It also helps us to predict the behavior of white dwarfs in different conditions, such as in binary star systems or during a supernova explosion.

5. Can the mass-radius relation be applied to other types of stars?

While the mass-radius relation is specifically derived for white dwarfs, similar relationships can be found for other types of stars, such as neutron stars and main sequence stars. However, the specific equations and factors may vary depending on the type of star being studied.

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