# Binary Stars energy loss calculation

• Pseudopro
In summary, the energy loss of binary stars is calculated using the formula for gravitational potential energy and is affected by factors such as the masses of the stars, distance between them, and their orbital period. This energy loss can impact the stars' evolution by causing them to spiral towards each other and can be observed or measured using various methods. However, there are challenges and uncertainties in calculating this energy loss due to the complexity and dynamic nature of the process.

#### Pseudopro

Two stars of equal mass M orbit a common centre. The radius of the orbit of each star is R. Assume that each of the stars has a mass equal to 1.5 solar masses (solar mass=2x10^30kg) and that the initial separation of the stars is 2.0x10^9m.

Note: the figure shows two stars on opposite sides of a circular orbit and a distance 2R separating them.

Force on each star=GM^2/4R^2
Period of revolution of each star T^2=16pi^2 R^3/GM
Period=7.8
Total energy of two stars E=-GM^2/4R
The two-star system loses energy as a result of emitting gravitational radiation so that the stars will move closer to each other.

1)Explain why the fractional loss of energy per unit time may be calculated from the expression delta E/E/delta t=3/2 delta T/T/delta t where delta T/T/delta t is the fractional decrease in period per unit time.
2)The orbital period decreases at a rate of delta T/delta t=72 micro s yr^-1. Estimate the fractional energy loss per year. (Ans: 1.7x10^-19 J yr^-1)
3)The two stars will collapse into each other when delta E~E. Estimate the lifetime, in years, of this binary star system. (Ans: 6x10^8 yr)

Basically, I attempted by saying E is proportionate to R^-1 so delta E/delta t is proportionate to R^-2, T proportion to R^3/2 so delta T... proportion to 3/2T^1/2... I replaced these 'proportionates' into the equation and it worked out. But when I place the numbers for 2), the answer is totally different. I don't understand how 3) would work.

Long question I know. Thank you so much for your patience. Thank you very much if you can help me!

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I suspect that the proportionality constant should be -2/3, not 3/2.

If you solve the expression for the period for R as a function of T, then substitute that into the energy expression, you end up with E being proportional to -(1/T)2/3.

So write E = -k (1/T)2/3. Find dE/dT, and carry on.

Pseudopro said:
1)Explain why the fractional loss of energy per unit time may be calculated from the expression delta E/E/delta t=3/2 delta T/T/delta t where delta T/T/delta t is the fractional decrease in period per unit time.
2)The orbital period decreases at a rate of delta T/delta t=72 micro s yr^-1. Estimate the fractional energy loss per year. (Ans: 1.7x10^-19 J yr^-1)
3)The two stars will collapse into each other when delta E~E. Estimate the lifetime, in years, of this binary star system. (Ans: 6x10^8 yr)

Basically, I attempted by saying E is proportionate to R^-1 so delta E/delta t is proportionate to R^-2, T proportion to R^3/2 so delta T... proportion to 3/2T^1/2... I replaced these 'proportionates' into the equation and it worked out. But when I place the numbers for 2), the answer is totally different. I don't understand how 3) would work.

Long question I know. Thank you so much for your patience. Thank you very much if you can help me!
For 1:

$$\frac{d}{dt}E=-\frac{d}{dt}\left(\frac{G\,M^2}{4\,R}\right)\quad\to\quad\frac{d}{dt}E=\left(\frac{G\,M^2}{4\,R^2}\right)\,\frac{dR}{dt}$$

$$\frac{(\Delta E)/E}{\Delta t}=\frac{\left(\frac{G\,M^2}{4\,R^2}\right)}{\left(\frac{G\,M^2}{4\,R}\right)}\,\frac{\Delta R}{\Delta t}\quad\to\quad\frac{(\Delta E)/E}{\Delta t}=\frac{(\Delta R)/R}{\Delta t}$$

Note: You can also get this by taking d/dt of   $$\ln(-E)=\ln\left(\frac{G\,M^2}{4\,R}\right)\,.$$

$$T^2=\left(\frac{16\pi^2}{GM}\right)\,R^3$$

$$2T\,\frac{dT}{dt}=3\left(\frac{16\pi^2}{GM}\right)\,R^2\frac{dR}{dt}$$

Divide the left side by 2T2 and the right side by 2×16π2 R3/GM .

$$\frac{(\Delta T)/T}{\Delta t}=\frac{3}{2}\,\frac{(\Delta R)/R}{\Delta t}$$

So, it looks as if the proportionality constant should be 2/3, i.e. (ΔE/E)/Δt = (2/3) (ΔT/T)/Δt, which is what you get if you follow gneil's suggestion.
gneil said:
...
So write E = -k (1/T)2/3. Find dE/dT, and carry on.

Pseudopro said:
...

Period=7.8 hours
...

1)Explain why the fractional loss of energy per unit time may be calculated from the expression delta E/E/delta t=2/3 delta T/T/delta t where delta T/T/delta t is the fractional decrease in period per unit time.
2)The orbital period decreases at a rate of delta T/delta t=72 micro s yr -1. Estimate the fractional energy loss per year. (Ans: 1.7x10^-19 J yr^-1)
...
After "playing around" with the numbers, I made some additions/corrections in red.

delta T/delta t=72 μs/yr, and T = (7.8 hours)×(3600 seconds/hour)

So, the fractional decrease in the period is: (72x10-6)/(7.8)/(3600)yr-1.

Multiply this by 2/3 to get the fractional energy loss per year.

SammyS said:
After "playing around" with the numbers, I made some additions/corrections in red.

delta T/delta t=72 μs/yr, and T = (7.8 hours)×(3600 seconds/hour)

So, the fractional decrease in the period is: (72x10-6)/(7.8)/(3600)yr-1.

Multiply this by 2/3 to get the fractional energy loss per year.

Thank you for your help (SammyS and gneill), the expression derivation makes sense. SammyS, would the answer be 1.7x10-9 as opposed to 1.7x10-19?

Yes, I got 1.7x10-9yr-1.

That seems consistent with:
Estimate the lifetime, in years, of this binary star system. (Ans: 6x10^8 yr)

## 1. How is the energy loss of binary stars calculated?

The energy loss of binary stars is calculated using the formula for gravitational potential energy, which is given by U = -Gm1m2/r, where G is the gravitational constant, m1 and m2 are the masses of the two stars, and r is the distance between them.

## 2. What factors affect the energy loss of binary stars?

The energy loss of binary stars is affected by the masses of the two stars, the distance between them, and their orbital period. Other factors that can influence the energy loss include the eccentricity of the orbit, the rotation of the stars, and any interactions with other objects in the system.

## 3. How does the energy loss of binary stars impact their evolution?

The energy loss of binary stars can impact their evolution by causing them to spiral towards each other, potentially leading to a merger or the formation of a new binary system. The amount and rate of energy loss can also affect the stars' lifetimes, as it can alter their internal structure and nuclear reactions.

## 4. Can the energy loss of binary stars be observed or measured?

Yes, the energy loss of binary stars can be observed and measured using various methods, such as analyzing the stars' orbital motion, measuring their luminosity, or studying their spectral lines. These observations can provide valuable insights into the physical processes occurring in binary star systems.

## 5. Are there any challenges or uncertainties in calculating the energy loss of binary stars?

Yes, there can be challenges and uncertainties in calculating the energy loss of binary stars, as it is a complex and dynamic process. The accuracy of the calculations may be affected by factors such as the assumptions made in the models used, the precision of the measurements, and the interactions with other objects in the system. Additionally, the energy loss may vary over time, making it difficult to accurately predict the future evolution of the stars.