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Stars, like the sun cannot run on chemical energy. If they did, we estimated that they would burn through all their fuel within a few thousand years.

Stars run on nuclear energy. In the sun, this is almost entirely the energy given off when hydrogen fuses together to form helium.

By measuring its gravity, we know that the mass of the sun is approximately,

The mass of a single hydrogen atom is
and every time two hydrogen atoms fuse to make one helium atom, they give off about  of energy.

In the prereading, we saw that the rate that the sun gives off energy, called the sun's power, is about 

Estimate the lifetime of the sun based on this information, assuming it will burn all its hydrogen into helium and then die, and assuming that it will continue shining at the same brightness that it does now for its entire life. You can assume that the sun began as pure hydrogen.

 

Write in Scientific notation.

 Apr 16, 2024
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Here's how to estimate the lifetime of the sun based on the given information:

 

Energy from Hydrogen Fusion:

 

Every time two hydrogen atoms fuse, they release 2.3 × 10^-13 J of energy.

 

Mass of Converted Hydrogen:

 

To find the total mass of hydrogen converted per second to generate the sun's power output, we can divide the power (energy per second) by the energy released per fusion:

 

Mass of converted hydrogen per second (m_h_converted) = Sun's power / Energy per fusion

 

m_h_converted = (3.6 × 10^26 J/s) / (2.3 × 10^-13 J/fusion)

 

m_h_converted ≈ 1.57 × 10^39 kg/s (This is the mass of hydrogen converted to helium every second)

 

Mass of Fused Hydrogen Over Time:

 

We are estimating the lifetime (t) of the sun. To find the total mass of hydrogen converted over that time, we multiply the mass converted per second by the lifetime:

 

Total mass of converted hydrogen (M_h_converted) = m_h_converted * t

 

Relating Converted Hydrogen to Initial Mass:

 

We know the initial mass of the sun (m_sun) is 2 × 10^30 kg. Since we assumed the sun started as pure hydrogen, this initial mass represents the total amount of hydrogen available for fusion.

 

Not all the mass of the hydrogen atom is converted to energy during fusion. A small amount is converted to helium, which has a slightly higher mass.

 

To account for this, we can introduce a factor (f) representing the fraction of the hydrogen mass that gets converted to energy. This factor is less than 1 (around 0.007).

 

M_h_converted = f * m_sun

 

Solving for Lifetime:

 

Now we can equate the total mass of converted hydrogen over time (from step 3) to the initial mass of the sun (adjusted for conversion efficiency) from step 4:

 

m_h_converted * t = f * m_sun

 

(1.57 × 10^39 kg/s) * t = f * (2 × 10^30 kg)

 

t = (f * 2 × 10^30 kg) / (1.57 × 10^39 kg/s)

 

Finding the Lifetime:

 

 

 

f = 0.01 (1% conversion):

 

 

 

Converting Seconds to Years:

 

There are 31,536,000 seconds in a year. Converting the estimated lifetimes:

 

For f = 0.01: Lifetime ≈ 1.27 × 10^11 seconds / (31,536,000 seconds/year) ≈ 4.03 × 10^3 years (around 4,000 years)

 Apr 17, 2024

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