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Valter правка от 3 сент. 2026 г., 18:36

правка #20521 позже →

en/2.2.7.md +25 −0

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		+	### Statement
		+
		+	$2.2.7.$ [Insert the problem statement]
		+
		+	### Solution
		+
		+	### Statement
		+
		+	$2.2.7.$ During observations from Earth, it is possible to determine only the radial velocity of partner stars that are part of a binary star (i.e., the projection of the velocity onto the Earth-star line). Upon measurements, the values of the radial velocity $v_1$ and $u_1$ of the partner stars of the binary star were obtained. During repeated measurements taken a year later, the values of this velocity turned out to be $v_2$ and $u_2$. Find the ratio of the masses of the partner stars comprising this binary star. Why is it necessary to change the calculations if the repeated measurement is taken after a month or half a year?
		+
		+	### Solution
		+
		+	From the law of conservation of momentum $\vec{p}_1 = \vec{p}_2$:
		+	$$m_1 v_1 + m_2 u_1 = m_1 v_2 + m_2 u_2$$
		+	$$m_1/m_2 = (u_2 - u_1)/(v_1 - v_2)$$
		+
		+	If measuring not after a year, it is necessary to introduce a correction for the change in the position and velocity of the Earth, since the radial velocities are relative to the velocity of the Earth.
		+
		+	#### Answer
		+	$$\frac{m_1}{m_2} = \frac{u_2 - u_1}{v_1 - v_2}$$
		+	It is necessary to take into account the change in the Earth's velocity.
		+
		+	#### Answer
		+
		+	[Insert a concise answer or boxed result]

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			### Statement

			$2.2.7.$ [Insert the problem statement]

			### Solution

			### Statement

			$2.2.7.$ During observations from Earth, it is possible to determine only the radial velocity of partner stars that are part of a binary star (i.e., the projection of the velocity onto the Earth-star line). Upon measurements, the values of the radial velocity $v_1$ and $u_1$ of the partner stars of the binary star were obtained. During repeated measurements taken a year later, the values of this velocity turned out to be $v_2$ and $u_2$. Find the ratio of the masses of the partner stars comprising this binary star. Why is it necessary to change the calculations if the repeated measurement is taken after a month or half a year?

			### Solution

			From the law of conservation of momentum $\vec{p}_1 = \vec{p}_2$:
			$$m_1 v_1 + m_2 u_1 = m_1 v_2 + m_2 u_2$$
			$$m_1/m_2 = (u_2 - u_1)/(v_1 - v_2)$$

			If measuring not after a year, it is necessary to introduce a correction for the change in the position and velocity of the Earth, since the radial velocities are relative to the velocity of the Earth.

			#### Answer
			$$\frac{m_1}{m_2} = \frac{u_2 - u_1}{v_1 - v_2}$$
			It is necessary to take into account the change in the Earth's velocity.

			#### Answer

			[Insert a concise answer or boxed result]