Modeling short-range ballistic missile defense and Israel's Iron Dome system

This paper develops a model of short-range ballistic missile defense and uses it to study the performance of Israel’s Iron Dome system. The deterministic base model allows for inaccurate missiles, unsuccessful interceptions, and civil defense. Model enhancements consider the trade-offs in attacking the interception system, the difficulties faced by militants in assembling large salvos, and the effects of imperfect missile classification by the defender. A stochastic model is also developed. Analysis shows that system performance can be highly sensitive to the missile salvo size, and that systems with higher interception rates are more “fragile” when overloaded. The model is calibrated using publically available data about Iron Dome’s use during Operation Pillar of Defense in November 2012. If the systems performed as claimed, they saved Israel an estimated 1778 casualties and $80 million in property damage, and thereby made preemptive strikes on Gaza about 8 times less valuable to Israel. Gaza militants could have inflicted far more damage by grouping their rockets into large salvos, but this may have been difficult given Israel’s suppression efforts. Counter-battery fire by the militants is unlikely to be worthwhile unless they can obtain much more accurate missiles.

Charles Larned letter, April 3, 1813.

Charles Larned (1791-1834) was a lawyer and American military officer who served during the War of 1812. He was the son of Simon Larned (1753-1817), who served as a captain in the Revolutionary War and was a member of the United States Congress from Massachusetts from 1804-1805. Charles studied law in the office of Henry Clay in Kentucky, and was dining with a group of prominent citizens when word was received that General William Henry Harrison could soon be overpowered by General Henry Proctor. Colonel Owen, a member of the group, organized a regiment to reinforce Harrison’s troops. Larned became a member and subsequently survived the River Raisin Massacre and was later present at the Battle of the Thames. He was also part of a group of men who learned of General William Hull’s plan to surrender Detroit to the British and planned to overtake him should this occur. However, the plan failed, Hull did surrender and the men became prisoners of the British. After the war, Larned became a lawyer, and served as Attorney General of Michigan Territory during the Black Hawk War. During the cholera epidemic of 1834, he worked tirelessly to assist others, but was stricken with the disease and died. Letter Transcription: Pittsfield, April 8, 1813 I think that by this time my dear Charles you will allow I have some reason to give you a gentle reprimand for breach of duty—but I will not censure you upon suspicion maybe you have substantial reasons—at any rate one cannot very graciously reproach the other for negligence I for one am healthy as ham & that we have so seldom exchanged letters during your absence & on my honor promise to be a better girl in future—but the truth is my Dear Charles I am secretary for the Family—Mama you know never writes & James but seldom & they are all dispersed in different directions, consequently I have many calls upon my time—this to be sure is a pleasant duty & I urge it only as a slight palliation for my remissness if you should consider it as such—now I have finished my preface—I will try to be more interesting & doubtless I succeed. Our dear Father we hope & trust is now in Green Bush, where he will probably remain a month perhaps & from thence he expects to go to Sacket’s harbor—at which place you know our troops are fast collecting-- We shall hope to see him either here or there before he goes. Brother George I believe is [still] at Plattsburgh but expects soon to be removed to some other military part perhaps with Papa (I hope so at least). We have just got letters from Brothers Sylvester & Joseph at Middlebury—they are in good health. Mama has for some weeks been afflicted with an inflammation in her eyes but seems now to be convalescing. Sister Martha has been somewhat unwell for a few weeks but is now tolerably recovered. James & myself are both in our usual good health & at this time seated by the same stand, one reading, the other writing. Thus my Dear Charles have I given you an abstract history of our Family—but here indeed is a wonderful omission; not a word about Miss Harriet Hunt, who in truth ought to have been noted first but the last she’s not the least in my memory. She is much grown since you saw her, but does not speak as fluently as we could wish—a few word she can say. Probably before this you have been informed of the great loss your friend Sherrill has sustained in the death of his mother—also of the revolution that has taken place in Hackbridge as it respects the religion & morality of the place that more than one hundred on the plain have become religious converts & c—indeed I am at a loss what to say that will afford your pleasure—a narrative at this time must be gloomy indeed. The distressing situation of our country at this time would make almost any recital melancholy. The prevailing epidemic has swept off many of your acquaintance no doubt. Mrs. Dewey of Williamstown, the sister of Mrs. Danforth, has left a Husband, Children & many Friends sincerely to lament her loss—some few have died in our village, but we have escaped astonishingly –it has raged in every town about us--If we are unwilling to acknowledge a God in his mercies. I fear she shall be compelled to do it in the awfulness of his judgments.--------I am much [pleased] with our new neighbors the Parsons Wife & a Miss Woodward her cousin is a fine girl, I think, Mrs. Allen has not a handsome face but something in her manner that interests one her person I think the handsomest I ever saw & the Parson seems well pleased with his selection—Mrs. Ripley is with them this winter & will probably remain thro the summer—Her husband at [Sackett’s Harbor] little or no alteration is apparent since her marriage—she seems as gay & fond of company as ever.-------Mrs. [McKnight] it is expected will commence housekeeping in about three weeks in the house formerly occupied by Mr…. [Report] says that Mr. Goodman & Clarissa Weller are soon to be married & many other things that I must omit to mention for Mama wants a… PS reserved--now my Dear Charles remember you are considerably… & I am confident you have as much leisure as I have –… be ceremonious but write whenever I find time not & I beg… the same – I tell James I shall not send his love for he must write himself. I shall anxiously expect you to write & do not disappoint your affectionate, sister--H One word my Dear Charles from your affectionate Mother who longs to see Her Dear son Charles—but being deprived of that rich blessing at present—begs Him so to conduct that she may hope for it ere long—do you search the Scriptures and keep the Sabbath holy unto the Lord—and all the sacred Commandments of God—it is my ardent desire…He would protect, support and provide for your soul and body and believe me your affectionate friend and Mother. R Larned.

The salvo combat model with area fire

This paper analyzes versions of the salvo model of missile combat where area fire is used by one or both sides in a battle. While these models share some properties with the area fire Lanchester model and the aimed fire salvo model, they also display some interesting differences, especially over the course of several salvos. Whereas the relative size of each force is important with aimed fire, with area fire it is the absolute size that matters. Similarly, while aimed fire exhibits square law behavior, area fire shows approximately linear behavior. When one side uses area and the other uses aimed fire, the model displays a mix of square and linear law behavior.