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Mass-radius relation of a white dwarf - calculating radius

  1. Apr 1, 2017 #1
    1. The problem statement, all variables and given/known data
    Calculate the radius of a 1.3 Msun white dwarf using the mass-radius relation for white dwarfs. Give the answer in solar radius.

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

    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.
     
  2. jcsd
  3. Apr 1, 2017 #2

    kuruman

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    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.
     
  4. Apr 2, 2017 #3
    That's a fair point - though I get the same result:

    $$ \left( M_{sun} \right) ^\frac {1}{3} R_{sun} = \left( M_{D} \right) ^{\frac {1}{3}} R_D $$
    $$ \Rightarrow R_D = \left( \frac {M_{sun}} {M_D} \right) ^{\frac{1}{3}} R_{sun} $$
    $$ \Rightarrow R_D = \left( \frac {1}{1.3} \right) ^{\frac {1}{3}} R_{sun} = \left( 1.3 \right) ^{-\frac {1}{3}} R_{sun} $$
     
  5. Apr 2, 2017 #4

    kuruman

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    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.
     
  6. Apr 2, 2017 #5

    Dick

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    Perhaps the problem is that the sun is not a white dwarf???
     
  7. Apr 2, 2017 #6

    kuruman

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    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.
     
  8. Apr 2, 2017 #7

    Dick

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    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.
     
  9. Apr 2, 2017 #8

    gneill

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    Staff: Mentor

    Hint: Sirius B is a pretty well known example of a white dwarf :wink:
     
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