Last week, based on my rudimentary knowledge of physics, I published the article "A Superficial Analysis of Solar Sail Technology" and shared it on the Mufu Astronomy Forum, hoping to invite more expert opinions. Fortunately, I received corrections from experts on the forum. They pointed out that the condition a = a_{ray} - a_G > 0 mentioned in my article was too harsh. This is because, in addition to solar radiation pressure, there is another force that can overcome solar gravity—inertial centrifugal force.
Let us restate a result from the previous article: a = a_{ray} - a_G = \left( \frac{L}{2\pi c (\rho h + m'/S)} - GM_{sun} \right) \frac{1}{r^2}
In the context of celestial mechanics, we are more accustomed to writing this as: \begin{aligned} \ddot{\vec{r}} &= -\mu \frac{\vec{r}}{|\vec{r}|^3} \\ \mu &= GM_{sun} - \frac{L}{2\pi c (\rho h + m'/S)} \end{aligned}
We should be pleased because solar radiation pressure, like gravity, is inversely proportional to the square of the distance. This allows us to directly apply existing research from celestial mechanics to this problem, which saves us a significant amount of effort. For example, knowledge of the two-body problem can be applied as a first-order approximation; all we need to do is “modify the mass of the Sun.” Since \mu = G \bigl( M_{sun} - \frac{L}{2G\pi c (\rho h + m'/S)} \bigr), it is evident that the effect of solar radiation pressure is equivalent to reducing the mass of the Sun.
To leave the solar system, we need to accelerate the probe to the third cosmic velocity. Earth itself already provides us with an orbital velocity of approximately 30 km/s, while the solar escape velocity at Earth’s distance is 42 km/s. Therefore, we only need to add about 12 km/s of velocity. However, this velocity only accounts for escaping the Sun’s gravity and does not yet account for escaping Earth’s gravity. Earth’s escape velocity is 11.2 km/s. According to the principle of conservation of energy, the third cosmic velocity required to launch a probe from the ground is \sqrt{11.2^2 + 12^2} = 16 \text{ km/s}. That is to say, the total velocity relative to the Sun is 46 km/s.
To save fuel, we hope that Earth’s orbital velocity itself becomes the escape velocity for the probe. We can envision a scheme: launch the solar sail into a certain orbit and then deploy it. Once deployed, it automatically reaches escape velocity. This is possible because, as we have stated, the solar sail acts to reduce the effective mass of the Sun. Since the effective mass decreases, the escape velocity naturally decreases as well. According to the results of the two-body problem, at Earth’s orbit, the solar mass for which the escape velocity is 30 km/s is half of the original mass. Thus, we require: GM_{sun} - \frac{L}{2\pi c (\rho h + m'/S)} \leq 1/2 GM_{sun}
This slightly expands the permissible range: \rho h + m'/S < \frac{L}{\pi G M_{sun} c} \approx 3 \text{ g/m}^2
There is still room to improve this figure; for instance, an “ion-solar sail” hybrid propulsion system could be used to lower the requirements. Regardless, the prospects for solar sail technology remain very promising.
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