Well, it is exciting to be here with you sweet ladies and we have had such a rich afternoon already. I love the fact that God uses such different personality types. Some of us are so sweet, like Jisela, absolutely the love of God just poured out of her. She gave the message and you saw the response from just the love of Jesus as the women responded to the altar call. And I pray that as Kim has shared how we walk in love and I'm going to talk about the lifestyle of love, I pray that that truly, once we start our relationship with Christ, I truly want that to become part of our lives as Christian women. You guys know just as much as I do how weak the church has been as of late. We've seemed to have lost the light and the commission that Jesus Christ himself has given to us as his church, as his bride to make a difference in this world. And we know that as God has called us to himself and he wants us to love because it's his love that's going to draw this dying world to himself, Lord, I pray that that becomes part of our hearts this day. I pray as we hear the same message over and over again this day, that that really sinks down and takes deep roots into our very souls. So will you bow your hearts with me as we ask for him to bless the message this morning or afternoon.  Father, and I do pray that you would allow our hearts to be so moved by you that we could really put ourselves aside and that we would do and be the women that you have called us to do and to be. I ask, Father, for your Holy Spirit to empower me, to fall afresh upon me, that you would breathe life into this message, that you are honored, that you are glorified and that we are motivated to draw closer to you. We love you, Father, and it's in Jesus' name that we pray. Amen. Amen. Well, our theme verse, Jeremiah 31.3, is a very powerful verse. We know that God has loved us with an everlasting love and he draws us to himself with loving kindness. Now I think that as we know about God's love, if we've been walking with the Lord for any amount of time, we have experienced his love. We know that we're forgiven. We know that we are going to spend eternity with him as believers in the Lord Jesus Christ. But sometimes I think we get caught up with thinking that maybe God isn't really as faithful as his word claims for him to be. The very same things that the children of Israel battled way back when, when Jeremiah was penning these words through the Holy Spirit, those insecurities that would creep up into their lives, thinking, you know, I know that God loved me before. I remember the works that we've heard when he led the children through the wilderness. We've heard of all of his power and the miracles and how he's protected and cared for his children. But right now I'm not really feeling very loved. So Jeremiah has given the words to say, you remind the people that I do love them and I love them with an everlasting love. Because of my loving kindness, I have drawn them to myself and he doesn't want us to forget those words or the work that he has done in our lives. We too suffer through those very same things that the children of Israel suffered. We've experienced hurts through broken relationships. We've had broken hearts. We've had broken dreams that have maybe caused us to put a guard over our heart. We may have even vowed that we would not trust anybody's love. And maybe we even included that to include our God. But ladies, if there is anything that we can trust on this earth is God's love for us. He promises that he will love us forever and ever and ever. And that is a very long time. And we can really count on him to be faithful to his word. God's love wasn't just a momentary whim. His love for us just didn't start a while ago. His love is from all eternity, from everlasting to everlasting. It's a love that will always last. He loved us before the world began, and his love will endure without change, and nothing can ever separate us from the love that he has for us. With his loving kindness, he says, he drew us to himself. And Colossians 1-3 says, for he rescued us from the dominion of darkness and brought us into the kingdom of the Son he loves, in whom we have redemption, the forgiveness of sins. As we acknowledge Christ as our Savior, and we begin our new lives in Christ, the knowledge of God's love for us through Jesus Christ motivates us to love other people even as we have been loved by him. His love motivates us to see others just like we were before we accepted him as our Lord and Savior. We were a people in desperate need of a loving Savior. In 1 John 4-7-12, Scripture says this, Dear friends, let us love one another. For love comes from God. Everyone who loves has been born of God and knows God. Whoever does not love does not know God, because God is love. This is how God showed his love among us. He sent his one and only Son into the world that we might live through him. This is love. Not that we love God, but that he loved us and sent his Son as an atoning sacrifice for our sins. Dear friends, since God so loved us, we also ought to love one another. No one has ever seen God, but if we love one another, God lives in us and his love is made complete in us. Love is a lifestyle, or it should be the lifestyle of a believer. It is a choice, like Kim said, and it is also sacrificial, and it is an action that we need to take. In his book, Dad, the Family Coach by Dave Simmons, he tells of an act of sacrificial love that occurred in a shopping mall. One day, he took his eight-year-old Helen and five-year-old Brandon to the Cloverdale Mall. He needed to go to Sears to buy some tools. When they pulled into the mall parking lot, there was a big sign that said, Petting Zoo. Immediately, the kids were jumping up and down and dancing all around. Daddy, daddy, can we go? Can we go? Please, daddy? Seeing that it would be no trouble at all and concluding that it might even make his trip easier, Dave said, sure, and he handed both of his kids a quarter. They bolted away as he headed for Sears. A few minutes later, he was making his way down the aisle when he spotted Helen, slowly walking up behind him. She looked up and said, well, daddy, it cost 50 cents. So I gave Brandon my quarter. Then she said the most beautiful thing of all, she repeated our family motto, love is action. What do you think he did? Not what you might think. Dave finished his shopping and then took Helen back to the Petting Zoo. They stood by the fence, watching Brandon go crazy, petting and feeding the animals. Helen stood with her hands in her chin, resting on the fence, just watching. Dave felt 50 cents in his pocket almost burning a hole, but he never offered it and she never asked for it. Helen was following through with the lesson, love is not just action. Love is sacrificial action. Love always pays a price. Love always cost us something. Love is expensive. When you love, benefits a crew to another's account. Love is for you, not for me. For love gives, it doesn't grab.  Huge example that the family was teaching their children in their family's life.  It's to give to others and it's sacrificial and at times it is going to cost us something. To walk in love cost us dying to self. Putting ourselves on the back burner and not looking to take care of ourselves. So does our lifestyle reflect a heart and an attitude that is filled with love on a daily basis? Not only at church, when we can put on our happy faces, but what about in our homes? Or in the marketplace or on the freeways or at work? Is our love sacrificial and compassionate to everybody, not just to the people that it's easy for us to love? It takes the work of the Holy Spirit to empower us and to teach us to love others the way the Lord has commanded us to. To learn of God's love and to apply his love to us is difficult. I think when we hear the messages that we've heard this morning already from Visala and from Kim, it sounds so easy to do. But for those of us that have been walking and learning what it means to really die to self and allow Christ to live in us and through us no matter what is going on in our lives. Isn't it so difficult? I mean, can we really say in the heat of the moment when that person that we're in contact with, just the one that we just don't like and you feel all that stuff just kind of like oozing up inside, can you just stop and say, oh, Father, I just love them? I mean, it's a battle, isn't it? I mean, it truly is. See, but because that battle is so strong, so many of us just want to throw our hands up and just forget it. She's not worth loving, but that's not true. See, we're giving into the enemy, into the enemy's lies and we're loving the way we think love should be operated with and through and in and God is saying no. He has empowered us to love the way he has called us to love others and we need to trust in his empowerment and not on our own strength. Love is not selfish. Love does what is right for the person instead of what we think would be okay for that person. Love means telling the truth to that person, even though it's going to hurt for a second. Love means putting somebody else over and above my own feelings or concerns. Love means I'm going to take that extra step to make sure that you're taken care of, even if that means I go without. A lifestyle of love reflects the love of Jesus Christ to all and that's something that we need to really own for ourselves. In Pastor Chuck's devotional book, Wisdom for Today, on September 27th, he says this, the heart of the Christian gospel is love. We are called to love God supremely and to love our neighbors as we love ourselves. This is the essence of the Christian message. God wants us to know and to offer the agape love that is described in 1 Corinthians chapter 13 verses 4 through 8. Now, many of us know this portion of Scripture. We hear it often at weddings. We might even recite it to our own spouse all the time when they're not being very patient or loving with us. This is a portion that we seem to know and we love it. We think it flows nice, very poetic, but do we live it? Are we applying it to our own lives? Love suffers long and is kind. Love does not envy. Love does not parade itself. It is not puffed up. It does not behave rudely. It does not seek its own. It is not provoked. Thinks no evil. Does not rejoice in iniquity, but rejoices in truth. Bears all things, believes all things, hopes all things, endures all things, love never fails. This verse basically sums up the characteristics of love. In these verses, we are able to see love in action. If we can live by this passage of Scripture, we will exemplify a lifestyle of love. So, as easy as this sounds, as poetic and beautiful as these verses are, why is it so difficult for us to put into practice? I think we all know the answer. We are so self-centered. We are very selfish people, especially in our human nature. Are we not? And I think that's such a huge red light for us. When we can feel ourselves becoming selfish and rude and impatient, that is not God's spirit working in us. That is our sinful nature that we are relying on and depending on in those moments. We battle with our own sinful, selfish desires every day. And many times, instead of persevering, we find ourselves just throwing in the red or the white flag surrendering, giving in to the enemy's whims. But Ephesians chapter four, verses 23 and 24 says, we are encouraged to put on the new nature in Christ Jesus so we can become the kind of person that he is producing in us. When Jesus says, love one another, he knew how difficult this would be for us, but he doesn't tell us something without teaching us and equipping us how to do it. As a new Christian, how do we learn that we weren't as loving as we thought we were? God has given us many tests over the years to show us what is really going on in our hearts. Now, when I was a brand new mama, and I can even back it up a little further than that, when I was a brand new wife, I loved Randy more than anything on this planet. He was like, he was it for me. But once we were married and we were living together and life took over, and I was not a believer at this moment, okay? But I found out that my knight in shining armor was a little rusty at times. And some of the things that he would do would just irritate the gulosh out of me. And it was like, how could that happen? Because before we were married and while we were dating, he could do nothing wrong. My mother would point out things that he would do wrong. Oh no, mom, that's not what he does. I mean, you could overlook everything. But once we had each other, I wasn't that willing to overlook those wrongs that I thought he was doing, those little irritants that would grow and fester. So when I would get so upset and he would come and ask for forgiveness for whatever reason, you know, no, not today, I won't forgive you yet.  You know, it's like, we're so selfish, I'm so selfish. So how did I learn to be a better wife? How did I learn to be a better mom? Because my, what's the word?  You know, so I didn't, maybe I didn't yell so much as my mom did or I didn't hit my kids on the head the way my mother always did with us. You know, but see, my standard had to be better than that. You know, how did I learn to be a good friend? Well, I could be a good friend when you're a good friend to me.  That's my own way of loving. I had to learn God's way of loving and he taught me by putting things and people in my life to show me what was really going on in my own heart. You know, you say you love your husband, then I want you to show him what that means as you love him through me. That's hard and it's still hard sometimes. You know, you say you love your children and you'll sacrifice and do anything for them. Then you raise them and teach them the way I want you to. That's difficult. You say you love your friends, then it's gonna be a love that is my kind of love, not just so you get something out of that relationship like Kim said, it can be unconditional love. Can we learn to love the way Christ has called us to love? I was always blown away by the scriptures in Titus that said for the women to teach the women to love their children and their husbands. Why do we have to learn how to love our husbands and our children? We should know how to do that. But see, we all come to our relationship with Christ with our own idea of what love is. So we have to get those ideas out of our head and learn what it means to love the way Christ loves, not with our own standards of love, but with his standards. So, how did you learn how you were doing in your love test? Have you passed those tests now? Have you been walking with the Lord long enough to realize when you're loving in the flesh or if you're loving in the spirit? See, God is gonna continue to test us until we finally get it. I shared with my ladies the other night at Bible study that to pray for me because I'm gonna be sharing here with the women and my topic again is on love and it's like, I don't know if maybe I have love down or if I don't get it yet and God continues to have me to speak on this so it finally goes into my own heart. You know, we need to get and to understand what it means to truly be motivated to love the way God wants us to love and not to love the way we feel is the best way. Paul says in Ephesians chapter five verses one and two as Kim so eloquently shared, be imitators of God. As dearly loved children live a life of love. Just as Christ loved us and he gave himself up for us as a fragrant offering and a sacrifice to God. Imitate Christ. How unlovely we were yet Christ loved us anyway. How thankful that we have been as we accepted Christ's love despite how wretched we were. We know how unlovely we were. We know all the baggage that we brought to the cross and how wonderful it was, how freeing it was when we knew that Christ accepted us and loved us and forgived us and set our lives on a brand new path. His grace was wonderful. His forgiveness was wonderful. That new life that we were able to experience was wonderful. We love that so much for ourselves but yet to give that to someone else we find ourselves a little stingy and selfish with that. It's good for me but I'm not gonna share that with you. It's not what God wants for our lives. I know that as we have taken out this Saturday to come and be filled up through God's word that I'm speaking to women who really do wanna do it right. We want to please God with the way we love, the way we act, the way we live our lives. Preferably we're being devoted to the word of God. We're studying to show ourselves approved. We're involved in fellowship. We enjoy being with Christian sisters. And I know that as we desire to love God with everything that's within us, we want to know those hard things that he wants to teach us as well. Because that's when real growth and stretching is gonna happen in our own lives. We want relationships that are right and real and good in the Lord. But we also know the reality of strained relationships and the experience of both the joys and the pains that we've had in relationships. We found how easy it is to let bitterness and unforgiveness take residents up in our hearts. We've been hurt or we've been betrayed by someone so wanting to protect ourselves, we can just shut our hearts up and we will purpose in our hearts ourselves not to love anymore. And as a believer in Jesus Christ, that is not an option for us. We have to grow through those hard things no matter how we feel, no matter how justified we may be, no matter how great our excuse is, we show our love to God by doing what he commands us to do. A lifestyle of love is part of our new life in Christ Jesus. Second John 6 says, and this is love, that we walk in obedience to his commands as you have heard from the beginning. His command is that you walk in love. Love is very important to God. We know that God is love. We are to love God. We are to love one another. We are to love our husbands. We are to love our children. We are to love our neighbor. We are to love others better than ourselves. We are to even love our enemies. Love again is part of our new life in Christ Jesus. We can love now because his love has taught us how to love. Love is one way to know that God is alive and working in us as we love one another. Again, First John chapter four, verses 11 and 12. Dear friends, since God so loved us, we ought to love one another. No one has ever seen God, but if we love one another, God lives in us. If we love one another, we know that God is living in us and working through us.  This is the first commandment. And the second is love your neighbor as yourself. There is no other commandment greater than these. Our Christian walk depends on the quality of our love relationship with our God. We can love first because he first loved us. He gave himself for us. He died, Jesus Christ died on the cross for us. Nothing can show us more clearly how loved we are than that. If we are loving God with all that is within us, we can and we will be able to love God's people. Scripture says in John 13, 35, by this all will know that you are my disciples. If you have love one for another, love is the very tool that God uses to reach people, Christians and non-Christians alike. Among the Gentiles, the early Christians were known by the love that they had for one another. Tertullian, in his apology, even gives us their words when they say this. See how these people love one another. They are even ready to lay down their lives for each other. Could the world look at us and say the same thing about you and me? Look how they love one another. They're even willing and ready to lay down their lives for one another. Pure love makes an impact on others' lives. It's through love and unity of God's children that the world will be drawn to Christ. It's through his love that we are able to forgive and move past the wrongs that have been done to us.  Again, another scripture that it sounds so beautiful, but that is so hard. How do we move out of the way and die to self and choose to live for Christ wanting him to be glorified more than our hearts wanting to take revenge for the wrong that's been done to us?  Does that sound like us? When you know or you've heard, because somebody's gossiping to you, that this sister hates you, what's your first reaction?  What did I do? Can we heal this up? What's the first thing we wanna do?    See, God wants his love to motivate us to a point that we can, with all sincerity, do this scripture. That we can do this, just not in that, okay, I'm gonna do it like him was talking about it, okay, Father, because your love will compel me to move this way. See, I wanna please you so much that I don't matter. I want this woman reached or this man reached so desperately by the love of Jesus Christ that I can put myself aside. See, do we love like that? Can we love like that?  Not even being hated or spitefully used or persecuted. Think about your last argument with your spouse.  You know, that is very clear to that person who upset us that they made us mad. God says, I don't want that in you. I want you to move past that.  I read this somewhere and I wish I can give the proper credit to the one who wrote it, but I think it's wonderful. God does not determine whether you deserve his love before he loves you. Likewise, he does not want you to set up a test that others must pass before you love them. He wants you to love them and pray for them no matter how they treat you. He wants you to love them and pray for them no matter how they treat you. Now, we would never say that we would make up a test for somebody, but sometimes don't we do that in our heads? Especially if we have found that that person is always messing up, always hurting us. It's like, okay, next F you get on your test or out.  We need to be so careful. We have been polluted by the world's definition of love and we have to fight the enemy not to give in to that definition. At times it's hard to separate the emotions from agape love that God has empowered us with and he wants us to love with. Jesus himself said in Luke chapter six, verses 32 and 33, how hard is it for you to love those who love you or to do good to those who do good to you? Sinners can do that and isn't that true? You know, the test isn't there for us when we are loving somebody who loves us and when we love somebody so much with that pure love, agape love, we can overlook their faults, we can love them, we give them the benefit of the doubt, we don't think evil of them. We can really love them because they love us. We know that there's that unity and that safety in that relationship, but sinners do that. God wants us to take it a step further even if we're mistreated. The test is to love the unlovely in the love of the Lord. I believe we all have people in our lives that just rub us the wrong way. Maybe they haven't really done anything to us but they just bug us. Maybe they've spoken bad about us, maybe they've spoken bad about our husbands or our children, maybe they're a little off. You know, maybe they just talk too much, maybe they don't talk enough. You know, whatever it is, they just rub us the wrong way. There, many years ago, there used to be a book called, well, it could still be a book, called Irregular People. And I couldn't wait to read that book because I know a lot of irregular people. And I thought, okay, Father, what do you wanna show me through this book? Because this person is really an irregular person in my life. Can you please teach me how to deal with her? Well, you know what I learned? There are gonna be irregular people forever. There will be. But God wants to use those irregular people in your life to grow you up, to grow me up, to change me, to see how I am doing in the Agape Love Department. That's why those irregular people are in our lives. Well, maybe that's not the only reason but some of them are just in our families but you know, that's a whole other story. But as God allows these people on our paths, he does want us to search our own hearts, to see how we react, what are we going to do with the people that he allows to come into our lives? Are we going to treat them the way Jesus treated us? Will we love them unconditionally? Or do we have secret tests that they have to pass first? We are not above loving anyone. And maybe that very one that we shy away from is the very one who needs a touch of God's love from us more than anybody else. Usually the hardest person to love at the moment is the one who needs the love of God the most. I don't want you to mention any names but I know you're probably gonna have some flashing up in your brain right now. But who is it in your own personal life that just irritates the gulash out of you? Who is it that you constantly need to ask God to help you to love this person just to be around them?  How have you reacted to their interaction in your own life? What have you resolved to do in these circumstances? Have you really cried out to the Lord and asked for his strength to help you to love that unlovable person?  In Proverbs 27, verse 17, scripture says, as iron sharpens iron, so one man sharpens another. He uses people in our lives to develop and to mold our character. That's how we know what's even going on within our hearts is when we have people that rub us the wrong way. What are we gonna do with those feelings and those emotions?  We have decisions to make. First Corinthians 13.8 says, love never fails. Author Jerry Bridges says, to recognize that there is someone I do not love is to say to God, I do not love you enough to love that person. I'm gonna read that again. To recognize that there is someone that I do not love is to say to God, I do not love you enough to love that person. And when we are choosing not to love, that's exactly what we're doing. Our obedience to have a lifestyle of love is an outward expression of the love that we have for God. Growing in love is a continuing need for all of us. As we grow in that area, we will begin to see how much more we still need to grow in that area. If we refuse to love, then we are walking in disobedience with the love of the Lord. Don't ever forget that he loved us while we were yet sinners as Romans 5.8 says. In Max Cicado's book, A Love Worth Giving, Max Cicado says this, loving people is not easy. We deal with the vow breakers, the truth benders, the money grubbers and backstabbers that we meet, we may work with or we may even marry. How do you love someone who is hard to love? Our typical strategy is to try harder. Conventional wisdom says that a lack of love implies a lack of effort. So what we do is we dig in deeper and we're gonna strain more and we're going to love this person even if it kills us and it just might do that. Could it be that we have missed a step? Could it be that the first step of love is not toward them but toward him? After all, you cannot give what you don't have. The secret of loving, he says, is living love. You give love by first receiving his love. Once we receive the love of Christ, then it's gonna be easier to give his love out to others. The apostle Paul in Ephesians chapter four was always encouraging believers to live the life that was worthy of our calling, to live in unity, to walk worthy of the calling, to bear with one another in love, not to walk as a Gentiles walked.  Start putting on the new man that was created in Christ Jesus. Don't grieve the Holy Spirit by acting in our flesh. Instead, be kind and tenderhearted for giving one another just as Christ has forgiven us. And then in Ephesians five, the first verse, he says, imitate God as dearly loved kids because that's the only way we're gonna be able to do Ephesians chapter four. We need to imitate the Lord, mimic his character as we walk with him. As we recognize God's love, we live in his love and see ourselves daily as God's child, it does make it easier to imitate him. The Lord's way of life was love and in that love he served others. But we have to be willing to let his love move through us and in us. I love Romans five, five, and Kim and Gisela also shared the scripture where it says that the love of God has been poured out within our hearts through the Holy Spirit who was given to us. And I love that because we see the love of God, I mean, and his love is never ending. So his love has been poured into our hearts. So when we say we can't love, we're lying if we're a believer because God has given us his love, right? The scripture just says it's been poured into our hearts. His love has been poured here and we've got to rely and trust in his love and not on our own. So if we are willing to allow his love to live in us and through us and to move us to love the way he wants us to love, we're gonna experience a victory that he wants us to have. We have to follow the pattern that has been given to us and that pattern is Jesus Christ himself. That's God's love, not the world's love. He loved us and he gave himself up for us. Giving up ourselves for one another is the epitome of agape love. Biblical love isn't just being in a good mood or feeling good about someone. It is actually dying to your own flesh and giving of yourself for somebody else's welfare. In 1st John 316, scripture says this is how we know what love is, that Jesus Christ laid down his life for us and we ought to lay down our lives for our brothers. Are we willing to go that far? Are we willing to lay ourselves aside for the sake of our brothers? Walking in love is an act of our will. It's a choice. We are equipped because the Holy Spirit resides within us but it's still our choice to obey or not to obey. Love has to be more than words. It has to be moved to action. 1st John 318 says not to love with words or tongue but with actions and in truth. Edith Schaefer asked this question, to whom can you demonstrate love in the day to day life, in the mundane circumstances of your life? Every person provides you with an opportunity to put on the work clothes of love and to serve. Every day, day to day life, who is it that we can demonstrate God's love to? Again, Max Lucado says, are we living in the overflow of God's love? Does the way that I treat other people reflect the way that God has treated me? Those are huge questions that we need to be asking ourselves. In closing, I wanna end with a portion of Max Lucado's book, A Love Worth Giving and he says this. I want us to rewrite 1st Corinthians chapter 13 verses four through eight. Not with your name or with Jesus' name but with both. Like you to read it aloud with your name in the blank and see what you think. Christ in Jeanette is patient. Christ in Jeanette is kind. Christ in Jeanette does not envy. Christ in Jeanette does not boast. Christ in Jeanette is not proud. Christ in Jeanette is not rude. Christ in Jeanette is not self-seeking. Christ in Jeanette is not easily angered. Christ in Jeanette keeps no records of wrong. Christ in Jeanette does not delight in evil, but rejoices with truth. Christ in Jeanette always protects, always trusts, always hopes, always persevere. Christ in Jeanette never fails. Oh, how I wish that would ring true in my life every day of my life. In these areas that I fall short, that's when I know that Christ is not working in me. I'm not allowing him to work in me. Because if he was, I would be all those things. Will we ever love like that? Will we ever love perfectly? No. But we can sure love better today than we did yesterday, and we can love better tomorrow than we do today as we allow God to do the work that he wants to do in our lives. Having a lifestyle of love is as simple as loving God with everything that's within you and then doing anything that you want. Because you see, if I love God with everything that's within me, everything that I wanna do is gonna be wanting to bring glory and honor to him. God help us to walk the walk of love that God can really use our lives to shine his light on this dying world. Time is short, and we need to be busy about our father's business and quit wasting time thinking of how we can be satisfied and filled and happy and all this stuff. How can we learn to love God more, to be used more for his kingdom? How can we show the true agape love of God to this dying world? He wants to use us and he will if we're willing. Amen? Let's pray. Father, I pray that you will help us to truly reflect a lifestyle of love. Help us to give to others what you have so freely given to us. Please make us more like you so others may come to know you. And we ask these things in the precious love of our savior, Jesus Christ. Amen.


Welcome to our segment on distant stars. By distant stars, I mean those stars that are so far away that parallax doesn't work anymore. Stars like this one, V838 monocerotis. In order to know how far away V838 monocerotis is, we'll need a new way to determine distance. To do that, we're going to build out a diagram that maps the star's color to its luminosity. Then we're going to study the nature of light that comes from the stars. And with that knowledge, we'll be able to use that diagram to determine the intrinsic luminosity of stars like this one. And with that, we'll know how far away they are. So let's begin with our study of light. We know a few things about light. We know that light is electromagnetic radiation created by moving electrons. With help from Jupiter's moon Io, as we discussed in our segment on the solar system, light travels at 300,000 km per second, or 186,000 miles per second. And as we have seen, stars vary in apparent brightness, and knowing the star's distance, we can use the inverse square law to find its intrinsic luminosity. The other thing we know is that stars have different colors. And very interestingly, color tells us a lot more than you might think. But to understand color, we need to know more about light. Light, as we currently understand it, has a dual nature. We can view it as a particle, like a photon, or we can view it as a wave that can interfere with itself. Although we haven't been able to reconcile these two incompatible views with one underlying understanding, what we do know has turned out to be surprisingly sufficient for the distance ladder. For color, we view light as a wave. Here's a simple wave. It has a repeating cycle, a wavelength, and a frequency in cycles per second. Newton showed that the sun's light can be dispersed into the colors of the rainbow with a crystal. This effect comes from the wave nature of light. Different colors represent different light frequencies. The higher the frequency, or inversely, the shorter the wavelength, the more it's bent by the crystal. This produces a spectrum of light with blue and violet at the high frequency end and red at the low frequency end. An important relationship between energy and light is that a light's energy is directly proportional to the frequency. So when physicists see color, they think energy. Red is low energy light and blue is high energy light. Here we see the full electromagnetic spectrum with visible light in the middle. Radiation with longer wavelengths and smaller frequencies than red light is called infrared and still longer wavelengths are called radio waves. Moving up the energy scale, radiation with shorter wavelengths than violet light is called ultraviolet. Still shorter wavelengths are called x-rays and the maximum energy radiation is called gamma rays. Celestial objects shine in radio to infrared visible light and ultraviolet to x-rays. One of the very important relationships between light and matter is called black body radiation. Turns out that the color of most matter at high temperatures depends completely and totally on the temperature. Nothing else really matters. Take a look at this iron rod as it heats up. See how it goes from red hot at the outer edges through yellow to white hot at the center? If we could get it hot enough, you'd see it turning blue. Here's why. As temperature increases and the electrons start moving more rapidly, two things happen. One, the object emits more radiation at all wavelengths. And two, the peak emission frequency shifts towards the shorter, higher energy blue wavelengths. As the heating starts, the radiation is all in the infrared range so we can't see it. As the temperature approaches 2,000 degrees Celsius, we begin to see red. We've seen the red star Aldebaran. It's a good example of this. By 3,000 degrees, the red has morphed to orange. Arcturus is an example of this. By 4,000 degrees, it is quite yellow. Capella and our own sun are yellow. Around 6,000 degrees, it is turning white. The star Sirius A is an example of this. And by 10,000 degrees, it has a distinct blueish color. Spica is a good example of a blue star. So the bottom line is, through the color of a star, by the frequency of the light it emits, then you've determined its temperature. It's that simple. Now that we know star temperatures via their color and luminosity via their parallax distance, we can build a diagram I mentioned in the introduction. In 1913, Engnarr, Hertzsprung, and Henry Russell began mapping these star temperatures Note that the horizontal axis is mapping temperatures in the decreasing direction. If we begin with the stars we use to illustrate black body radiation, Aldebaran, Arcturus, Capella, our own sun, Sirius A, and Spica, and throw in a few others like Sirius B, Wolf 359, Polaris and Vega, we get a graph that looks like this. With this small sample, it looks like any combination of temperature and luminosity is possible. But Hertzsprung and Russell meticulously plotted all the stars with known distance and luminosities. And they got this. Here we see that most stars fall on the diagonal line from the upper left hot blue luminous stars down to the lower right, cooler, dimmer red stars. But there is also a grouping of stars well below the main line and two groupings of stars well above the main line. This is the Hertzsprung-Russell diagram or HR diagram for short. It is one of the most important tools in understanding stars ever devised. It tells us a great deal about the life, death, and age of stars. And, more importantly, for our purposes, it can tell us how far away stars are. But in order for the HR diagram to do this, we need to know more about what makes stars shine and we need to know more about the full spectrum of light we receive from these stars. In our segment on star birth nebula, we'll cover how stars form from giant hydrogen clouds collapsing under the force of gravity. They start shining once the pressure and temperature at the core reaches the level needed for the hydrogen fusion. The fusion of hydrogen and helium converts some of the mass into energy. And because E equals MC squared and C is a very big number, the process generates a great deal of energy. The more hydrogen there is in the collapsing cloud, the more mass of the star. The more mass of the star, the more intense the pressure in its core. The more intense the pressure, the higher the temperature. The higher the temperature, the greater the star's luminosity. Thus, the diagonal line on the HR diagram represents the main sequence for stars burning hydrogen. The upper left blue and white hot stars are high mass stars, many times more massive than the sun. The middle region, yellow and orange stars of the sun. The lower right stars are cool, low mass stars that are a fraction of the mass of the sun. When a star runs out of hydrogen fuel, the core contracts and gets hotter. This heat expands the outer layers, reducing the density and turning the star red. The star moves off the main sequence and enters the realm of giants or super giants of low mass. When a red giant with a mass less than 5 times the mass of the sun runs out of fuel, it explodes and leaves behind a dim, hot, tiny star called a white dwarf. We'll discuss these processes more in our segment on Planetary Nebula. We'll cover the endgame for more massive stars in our segment on supernova. Now that we understand the meaning of the HR diagram, let's see how we can use it to find out how far away a star is. For that, we need to view light as a particle and examine its spectrum. Early in the 19th century, German chemist Joseph Fraunhofer invented the spectroscope, an instrument to automatically separate light and mark the wavelengths. In so doing, he discovered that when he spread sunlight into a spectrum, the spectrum was crossed by a great number of fine dark lines. He had no idea what these dark lines were, but today we know that they were the key to learning what stars are made of. Remember the red and green light of the aurora borealis and the structure of molecules we discussed in our segment on the heliosphere? The aurora is a good example of light being emitted as electrons change energy levels. But for our purposes here, we want to examine what happens when light from the center of a star passes through the gases in the outer layers on its way to us. Here's how it works. When a photon with exactly the right energy level hits an electron orbiting a nucleus, its entire energy is transformed to the electron which jumps to a higher energy level with a larger quantum number. The photon is eliminated. This creates absorption lines in a star's spectrum as light from the star travels through the star's atmosphere. Every atom and molecule has its own spectral line signature. So by observing the absorption lines in a star's spectrum, we can tell what elements are present. When scientists discovered connections between groups of spectral lines and star temperatures, they developed a set of spectral classifications to highlight this connection. Every star we have seen so far fits into one of these classifications. Our sun is spectral class G and has around 67 elements in its photosphere. Here are a few identified by their spectral signature. Turns out that hydrogen is 50-80% of most stars and combined with helium they make up 96-99% of all stars. Star spectra has one more characteristic called luminosity class that enable us to determine whether a star is on the main sequence or not. This is the key to using the HR diagram to determine the star's distance. If you recall the evolution of a star off the main sequence involves the expansion of the outer layer to gargantuan proportions. This makes the density of the gas in the outer layer of a giant much less than the density in the outer layer of a star on the main sequence. It turns out that the photon absorption characteristics of closely packed atoms makes the spectral lines fuzzier. For a given spectral classification the fuzzier the spectral line the smaller the star. Roman numerals are used to identify luminosity classes. Our sun is class 5 a main sequence star. We'll use Betelgeuse to illustrate how star spectra works with the HR diagram to determine a star's distance. First we use the star's color, temperature and spectra to find its point on the horizontal axis. Looking up the vertical luminosity axis we see Betelgeuse could either be a main sequence star or a giant. Examining the luminosity class we see that it is very sharp implying that Betelgeuse is a super giant. Now drawing the line to the vertical axis the star's intrinsic luminosity is 120,000 times greater than our sun's luminosity. Measuring the apparent luminosity and using the inverse square law we get the distance. If stars everywhere behave like the stars in our neighborhood then the HR diagram can show us how far away they are. Astronomers call this technique spectroscopic parallax but we'll just stick with HR diagram. Now let's take a look at some distant stars. This star, known as V1331CYG is a young star that is starting to contract to become a main sequence star similar to the sun. It lies inside a dust cloud. We're looking down on the star at one of its poles which allows us to see the dust cloud enveloped around it. Usually for young stars like this all we get to see is the dust cloud. The star at the center of this picture is a red super giant called UY SCUTTY. It is very dim but appearances can be deceptive in astronomy. This star is actually about 340,000 times more luminous than the sun. In fact this is a candidate for being the largest star in the entire Milky Way galaxy. Astronomers believe the actual size of UY SCUTTY is big enough to hold 5 billion suns. Here we see the super hot star WR124 and the hot clumps of gas it is ejecting into the space around it. Ejection gases are traveling at over 150,000 kilometers per hour. That's 93,000 miles per hour. The cloud known as Nebula M167 is estimated to be no more than 10,000 years old. That's very young in astronomical terms. Here's a Hubble image of the luminous blue variable star AG Karina. It has evolved from the main sequence with 20 times the mass of the sun. AG Karina is losing its mass at a phenomenal rate. Its mass-to-energy conversion is creating powerful stellar winds with speeds up to 7 million kilometers per hour or 4.3 million miles per hour. These powerful winds are responsible for the shroud of material visible in this image. You'll recall how we used the inverse square law when we covered star luminosity in the nearby star segment. We measured the brightness distance from parallax measurements to get the luminosity. But if we had a way to know what the intrinsic luminosity of a star was, we could use that along with the apparent brightness to get the distance. For example, if we measured the brightness of a 10 watt candle from some distance away to be a tenth of a watt per square meter, we could calculate that the candle is just under 3 meters away. A celestial object with a known luminosity is called a standard candle. But until 1912, there were no known standard candle stars. That changed when Henrietta Leavitt published her findings on Cepheid Variable Stars. Like Polaris, Delta Cepheid is a binary star system and a Cepheid Variable Star. Cepheid stars undergo periodic changes in luminosity. Delta Cepheid is among the closest stars of this type of variable with only Polaris being closer. Most stars have some variability in their luminosity. Even our sun varies on an 11-year cycle of sunspots. But Delta Cepheid's variability is caused by regular pulsation in the outer layer of the star. Here's its light curve showing luminosity changes over time. The pattern is quite regular. Early in the 1900s, Henrietta Leavitt thought to plot Cepheid luminosity cycle periods against luminosity. She found that the period of these stars varied in proportion to their absolute brightness. This was very interesting because as we have discussed once we know the intrinsic luminosity of a star, we can easily calculate its distance. Leavitt's discovery made Cepheid stars true standard candles and changed the history of astronomy. Rr Lyra is a variable star like Delta Cepheid. As the brightest star in its class it became the namesake of the Rr Lyra Variable Class of Stars. The relationship between pulsation period and absolute magnitude of the Rr Lyras make them good standard candles. They are not as bright as Cepheid variables but there are a lot more of them. They are extensively used in globular cluster studies including the studies that helped us understand the form and size of our Milky Way Galaxy. Zetagenenorm is an intermediate luminous supergiant. It is also a classical Cepheid variable. Chi P13044 came from outside our galaxy. It was part of a former dwarf galaxy that merged with the Milky Way between 6 and 9 billion years ago. T Lyra is a carbon star. It is used up most of its hydrogen fuel and is now fusing helium into carbon. Here we see the summer triangle a giant triangle in the sky composed of three bright stars Vega Altair and Deneb. Deneb is a blue-white supergiant. It is one of the biggest white stars known at 203 times the size of the sun and around 19 times more massive. Deneb's solar wind is blowing away material at a rate that results in its losing mass 100,000 times faster than our sun. The star is very faint.  We estimate that the star is probably more than 13 billion years old. That would make it one of the oldest stars in the universe. Mu Cepheid is a red supergiant star. In fact, it is one of the largest and most luminous stars known in the Milky Way. This star could fit a billion suns into its volume. Ethicarina 10,000 light years away is estimated to be 100 times more massive than our sun. It is one of the most massive stars in the galaxy. It radiates about 5 million times more power than our sun and its mass as you can see also makes it very unstable. Hubble's latest image of the star V838 Montserratus reveals dramatic changes in the illumination of surrounding dusty cloud structures. The effect, called a light echo has been unveiling never-before-seen dust patterns ever since the star suddenly brightened for several weeks in early 2002. During the outburst the normal faint star suddenly brightened becoming 600,000 times more luminous than our sun. Hubble has captured this image of a hypervelocity star over a quarter of a million light years away. It's 200,000 light years above the galactic plane and traveling at 722 kilometers per second. That's 450 miles per second. That's fast enough to escape the galaxy's gravitational grip. Astronomers think it was a member of a multi-star system and was jettisoned by the black hole in our galaxy's central bulge. We'll cover black holes when we get to the segment on the Milky Way. The black hole's tremendous gravitational pull stripped one member while violently ejecting the other member deep into space at these high velocities conserving the system's momentum. The first example of hypervelocity star was discovered in 1995. This one, US-708, is a second such star to be discovered. It is an extremely rich helium hot white dwarf moving at 1,200 kilometers per second. That's 746 miles per second. That makes it the fastest star ever discovered. It crossed the galactic plane around 14 million years ago and is thought to be the companion of a star that sent it out into intergalactic space. In our segment on nearby stars we reached as far as parallax can take us by using the space-based satellites Parcus and Gaia. But if that's all we had we'd know very little about our galaxy and almost nothing about the universe beyond. But in this segment on distant stars we introduced two new rungs for our cosmic distance ladder. One is the Hertzbrunn-Russell diagram for estimating a star's luminosity and therefore its distance. The other is variable stars that work as standard candles, stars that tell us their intrinsic luminosity by the period of their variable luminosity cycles. In particular, we covered Cepheid and RR Lyra variables. These rungs in our distance ladder have taken us across the entire Milky Way. In subsequent sections we'll use these distance ladder techniques on star clusters, star birth nebula, star death nebula also known as planetary nebula. Our next segment.


Hello and welcome to the session. In this session we are going to discuss the following question and the question says that find a fourth degree polynomial function with real coefficients that has minus 3, 3 and iota as zeros such that f of 2 is equal to 150. We know that linear factor theorem states if f of x has degree n greater than or equal to 1 with nonzero leading coefficient a n then f of x has exactly n linear factors and can be written as f of x is equal to a n into x minus c 1 the whole into x minus c 2 the whole into and so on up to x minus c n the whole where c 1 c 2 c 3 and so on up to c n are real or complex zeros with this key idea we shall move on to the solution. In this question we have to find a fourth degree polynomial function with real coefficients and we are given that minus 3, 3 and iota are its zeros we know that complex zeros always occur in conjugate pairs so if iota is the root then minus iota will also be the root hence the function has four roots or zeros and these are minus 3, 3 iota and minus iota. Now we use linear factor theorem given in the key idea we write the polynomial function as f of x is equal to a n into x minus 3 the whole into x minus of minus 3 the whole into x minus iota the whole into x minus of minus iota the whole this implies that f of x is equal to a n into x minus 3 the whole into x minus of minus 3 that is x plus 3 the whole into x minus iota the whole into x minus of minus iota that is x plus iota the whole which implies f of x is equal to a n into now x minus 3 the whole into x plus 3 the whole can be written as x square minus 3 square the whole by using the formula a minus b the whole into a plus b the whole is equal to a square minus b square into now using the same formula here we get x square minus iota square the whole since iota square is equal to minus 1 so we get f of x is equal to a n into x square minus 9 the whole into x square minus of minus 1 the whole which implies f of x is equal to a n into x square minus 9 the whole into x square plus 1 the whole now solving it further we get f of x is equal to a n into now x square into x square will be equal to x raised to the power 4 now x square into 1 is plus x square now minus 9 into x square is minus 9 x square minus 9 into 1 will be minus 9 the whole this gives f of x is equal to a n into x raised to the power 4 now plus x square minus 9 x square can be written as minus 8 x square minus 9 the whole now to determine this function we have to find the value of the leading coefficient a n now in the question it is also given that f of 2 is equal to 150 so we put x is equal to 2 in the function f of x and equate it to 150 so we get a n into 2 raised to the power 4 minus 8 into 2 square minus 9 is equal to 150 which implies a n into now 2 raised to the power 4 is 16 minus 8 into 2 square that is 4 minus 9 the whole is equal to 150 which implies that a n into 16 minus 32 minus 9 the whole is equal to 150 which implies a n into minus 25 is equal to 150 which further implies a n is equal to 150 upon minus 25 this implies a n is equal to minus 6 now substituting a n is equal to minus 6 in the function f of x is equal to a n into x raised to the power 4 minus 8 x square minus 9 the whole we get f of x is equal to minus 6 into x raised to the power 4 minus 8 x square minus 9 the whole which implies f of x is equal to minus 6 x raised to the power 4 plus 48 x square plus 54 thus the required polynomial function is f of x is equal to minus 6 x raised to the power 4 plus 48 x square plus 54 which is the required answer this completes our session hope you enjoyed this session