Question

美国物理奥林匹克竞赛题

A star is born: According to one of the cosmological theories, the stars were formed from
interstellar dust by gravitational accretion. Estimate the time interval T it takes to form a star
according to this recipe.
The density of the dust is p = 2 × 10−17kg/m3, the gravitational constant is G = 6.67 ×
10−11Nm2/kg2. You may assume that the dust particles do not overtake one another during the
accretion process, and that initially the dusty region was a homogeneous spherical cloud.
Hint: use dimensional analysis to get an idea of the functional dependence of  on G and , then use Kepler’s
laws to get the correct proportionality coefficient.

恒星的形成：恒星是由星云做成的，看看下面这团星云要成星所用时间T。星云密度是p=2 × 10−17kg/m3， 重力常数G=6.67 × 10−11Nm2/kg2（六点六七乘以十的负十一次方牛米平方每千克平方）. 假设形成过程中，后面粒子不会撞上（overtake：赶上，突然来袭, 压倒）前面的灰尘粒子。星云开始时是均匀的（homogeneous）
提示：先用用空间解析来看T和G和p之间的函数因变量关系（functional dependence) 然后用开普勒定律得出正确的比值系数（proportionality coefficient)
a

Answer 1

r''=-4πρGr/3
t=(4πρG/3)^-0.5∫dr/（r0^2-r^2）^0.5 ,(0,r0)
=(3π^0.5/4) ρG^-0.5
~~~~~~~~~~~这是dimensional analysis（量纲分析）的部分，确定了数量级，前面是系数
数值结果是2.1e+13s ~= 6.7e+5 year

Answer 2

不行啊我，高人们快来吧。