After examining Cygnus X-1's blue supergiant companion up close, we now shift our view to the black hole and companion star system as a whole. We know these companions can either be high mass blue stars like HD 226868 or low mass stars like our sun. But how exactly does a black hole eat from a companion star. If the black hole is feeding from its companion star, then material from the star must be transferred to the black hole by some mechanism. The answer to this question is explained by the work of a French astronomer named Edouard Roche. He developed a model for the transfer of material between two massive objects such as a star and a black hole known as the Roche lobe. To understand the Roche lobe, let's consider two scenarios, a single star and a system of two stars. In either scenario the force of gravity will have some say on whether material will be drawn into the star or will not. In the case of a single star, if we were to draw lines or contours of constant gravitational potential, we would need to create a series of circles originating from the star. These drawings are similar to the topographical maps of mountains. How does this change when another star is nearby? When two stars are in a binary system, the gravity of the two will interact. These two stars would be in orbit around one another or rather around a common center of mass. However, we must also consider in addition to the gravitational force the force due to the relative motion of the stars, the centrifugal force. Think about a child's roundabout in a play park. Once you kickoff and begin spinning, you can feel a force pushing you outwards. This is called the centrifugal force. As a result of the rotation of the star system, we have gravity pulling inwards and centrifugal force pushing outwards. It is the combination of these two forces that are represented by the lines of constant potential in binary systems. If we now build up lines of equal potential around two stars, we will initially see circles around each of these stars. However, as these rings get larger and closer together, their shape begins to change. They are slowly stretched in the direction of the opposing star. The circles begin to morph into teardrops. This stretch or distortion increases until they connect forming a figure of eight around the two objects. Each of these teardrops or lobes is called a Roche lobe. It is the Roche lobe for the star it contains. Any material that is inside the lobe is gravitationally bound to that star. You can think about gravitational lobes like two lakes occupying adjacent valleys separated by mountains. The lakes' watersheds don't share any water unless they fill to mutual height which we typically call a watershed divide. Similarly, material within a Roche lobe is bound unless there is a point where the potential is equal between the two stars. The point where these teardrops meet is known as the Lagrange point. The first Lagrange point is commonly labeled L1. If you're an astronaut situated in L1, you would feel an equal gravitational pull towards each of these stars but there are other points where we could feel the equal pull between the two stars. If we continue to map the lines of equal potential, we find other Lagrange points. As you can see, there are four other points that surround a binary system. Although we've used an example of two stars, you can draw similar lines of constant potential around any other pair of massive bodies including the Sun and the Earth but more importantly between a black hole and its stellar lunch. Lagrange points are special regions in space where the gravitational potential is relatively flat making it easy for spacecraft to hover there. Considering the Earth-Sun system, the first Lagrange point lies on their connecting line. This region known as L1 is used extensively as a parking spot for solar telescopes since they can hover at L1 using small amounts of fuel. As such this location has been used as a prime spot for astrophysical observations which we will discuss further in future modules. Black hole binaries are a type of system which contain two massive bodies and so these systems also have Roche lobes and Lagrange points, but how does this help us understand the transfer of matter? The transfer of material from a star to a black hole is a gravitational effect. If material from the star moves outside its Roche lobe, gravity can pull it towards the black hole instead. But if material inside its Roche lobe is gravitationally bound to the star, then how can it move outside the boundary? The easiest way for this to happen is if the star begins to fill its Roche lobe. As stars like our sun get older, they will swell up to become red giants. At this time, the star can grow to fill its Roche lobe. At that point, material can spill over across the boundary at L1. The star stuff will end stop falling towards the black hole. Stars can also fill lab Roche lobe if the Roche lobe shrinks. How can this happen? The Roche lobes would get smaller if the bodies in the system move closer to each other or as astronomers call it the binary becomes more compact. There are many ways that binary systems can become compact. One method involving gravitational radiation will be discussed in module 10. This image shows 16 black hole binaries that all live in our galaxy. At the top of the image, you can see our sun and the distance between it and the planet mercury. Mercury orbits the sun at just over a third the distance between our sun and the earth yet most of these systems are much smaller or much more compact. So far we have looked at material being stripped away from the surface of the star as it overfills its Roche lobe. This material then crosses the first Lagrange point L1 and starts falling towards the black hole. This process is known as Roche Lobe overflow. It can provide a fairly stable way of feeding a black hole for quite some time. But is it the only way to transfer mass from the star to the black hole? No. There is another option. Many stars have winds. Our sun does but its winds are considered puny by stellar standards. Massive blue stars can blow away up to 100 million times more mass than our own sun does. Such strong winds can be captured by the gravity of the black hole and fall in towards the event horizon. This type of accretion is known as Wind Fed. While Wind Fed mass transfer is only really an option for high mass stars, Roche lobe overflow can occur with any type of star as long as the companion star fills its Roche lobe. Our good friend Cygnus X-1 is one such system with high mass companion that is both overflowing its Roche lobe and feeding the black hole with high velocity stellar winds.
